Editors in Chief Editorial Board

Editors in Chief

Adelina Georgescu, Professor, Ilie Burdujan, Professor, Member of Academy of University of Agricultural Sciences and Romanian Scientists, Veterinary Medicine “Ion Ionescu de la 54, Splaiul Independentei, 050094, Brad” Iaşi Bucharest, ROMANIA 3, Mihail Sadoveanu, 700490 Iaşi, ROMANIA

Editorial Board Nuri Aksel, Professor, Dumitru Botnaru, Professor, Faculty of Applied Sciences, Chair of Superior Mathematics, Department of Applied Mechanics Technical University of Moldova, Bv. and Fluid Dynamics, University of Stefan cel Mare, 168, MD 2004, Bayreuth, D-95440 Chişinău, REPUBLIC OF MOLDOVA Bayreuth, GERMANY

Tassos Bountis, Professor , Mitrofan Choban, Professor, Centre for Research and Applications Member of Academy of Sciences of of Nonlinear Systems and Department of Moldova, Mathematics, University of Patras, 26500, State University of Tiraspol, Iablocikin, 5, Patras, GREECE MD 2069, Chişinău, REPUBLIC OF MOLDOVA

Sanda Cleja-Ţigoiu, Professor, Constantin Fetecău, Professor, Faculty of Mathematics and Computer Faculty of Mechanical Engineering, The Science, University of Bucharest, Gh. Asachi Technical University of Iaşi, Academiei, 14, 010014, Bv. Dimitrie Mangeron, 61-63, 700050, Bucharest, ROMANIA Iaşi, ROMANIA

Peter Knabner, Professor, Boris V. Loginov, Professor, Chair for Applied Mathematics, Faculty Faculty of Natural Sciences, Ulyanovsk of Science, Friedrich Alexander State Technical University, Severny University Erlangen-Nuremberg, Venetz, 32, 432027, Martensstr. 3, 91058, Ulyanovsk, RUSSIA Erlangen, GERMANY

Nenad Mladenovici, Professor, Toader Morozan, Senior Researcher, Mathematical Sciences, John Crank 210, Institute of Mathematics Simion Stoilow Brunel University, Uxbridge, UB8 3PH, of The Romanian Academy, Calea Uxbridge, UNITED KINGDOM Griviţei, 21, 010702, Bucharest, ROMANIA

Emilia Petrişor, Professor, Mihail Popa, Senior Researcher, Department of Mathematics, University Institute of Mathematics and Computer Politehnica of Timişoara, Victoriei Science of The Academy of Sciences Square, 2, 300006, of Moldova, Academiei, 5, MD 2028, Timişoara, ROMANIA Chişinău, REPUBLIC OF MOLDOVA

Kumbakonam Rajagopal, Professor, Mefodie Raţă, Professor, Senior Department of Mathematics, Texas Researcher, A&M University, Corr. Member of Academy of Sciences Mailstop 3368, College Station, TX of Moldova, Institute of Mathematics 77843-3368, and Computer Science of The UNITED STATES OF AMERICA Academy of Sciences of Moldova, Academiei, 5, MD 2028, Chişinău, REPUBLIC OF MOLDOVA

Panayotis Siafarikas, Professor, Mirela Ştefănescu, Professor, Department of Mathematics, Faculty of Mathematics and Computer University of Patras, 26500, Science, Ovidius University, Patras, GREECE Bv. Mamaia, 124, 900527, Constanţa, ROMANIA

Nicolae Suciu, Senior Researcher, Kiyoyuki Tchizawa, Professor, Tiberiu Popoviciu Institute of Numerical Kanrikogaku Kenkyusho, Ltd., Analysis, 2-2-2 Sotokanda, Chiyoda-ku, 101-0021, P.O. Box 68-1, 400110, Tokyo, JAPAN Cluj-Napoca, ROMANIA

Vladilen A. Trenogin, Professor, Constantin Vârsan, Senior Researcher, Moscow Institute of Steel and Alloys, Institute of Mathematics Simion Stoilow B-49, Leninsky Prospect, 4, 119049, of The Romanian Academy, Calea Moscow, RUSSIA Griviţei, 21, 010702, Bucharest, ROMANIA

IN MEMORIAM

ADELINA GEORGESCU (1942-2010)

The founder and first President of the Romanian Society of Applied and Industrial Mathematics – ROMAI passed away on May 1st, 2010, at the age of 68

- i - - ii - Adelina Georgescu was born in Drobeta Turnu-Severin on April 25, 1942 in a family of intellectuals (her mother, Maria Berindei, was a teacher of history and geography and her father, Constantin Georgescu, was a lawyer). She finished the high school with excellent results and in 1960 she began the universitary studies, at the Faculty of Mathematics and Mechanics of the University of Bucharest. In 1965 she successfully defended her graduation thesis of the bachelor degree. After graduation she started to work at the Institute of Fluid Mechanics and Aerospace Constructions of the Romanian Academy. The theory of hydrodynamic stability was her first domain of research. This theory is mainly known to engineers, meteorologists, hydrologists, physicists, due to its relevance for practical applications. In 1970, Adelina Georgescu defended at the Institute of Mathematics of the Romanian Academy, her PhD thesis on linear hydrodynamic stability under the supervision of the Academician Caius Iacob, one of the leaders of the Romanian School of Mechanics. In 1976 she published the first Romanian monograph on hydrodynamic stability. In 1985 Kluwer issued its enlarged and improved English version, being a bridge between the classical theory, addressed mainly to engineers, and the pure mathematical one, belonging to applied functional analysis. This monograph was cited by many authors, since it was widely used as an universitary textbook and by researchers, being a pioneer work in this field. After 1985, due to her interest in the still unsolved problem of the origin of turbulence – a topic related to the loss of stability – the spectrum of scientific work of Adelina Georgescu became very broad, including closely interrelated areas as: the stability and the multiparametric bifurcation in fluid dynamics, spectral problems for ODE in the hydrodynamic stability theory, dynamical systems associated with fluid flows, the transition to turbulence as a deterministic chaos, fractals, models of asymptotic approximation in fluid dynamics, synergetics, nonlinear dynamics. Besides her research work, she devoted a lot of time to teaching. Since 1974 at the Faculty of Mathematics of Bucharest University she led courses devoted to the theory of Navier-Stokes equations, dynamics of rivers, boundary layer theory, turbulence in fluids, bifurcation theory (the first such course in Romania), analytical mechanics, mechanics of continua, nonlinear dynamics, sinergetics. During the communist regime, in spite of her non-proletarian origin rising many obstacles to her career, she did not join the communist party, giving up the opportunities of quick advancement her adhesion would have provided.

- iii - In 1990, as soon as the conditions following the fall of the communism allowed the affirmation of personal initiative in science, Adelina Georgescu became the initiator of the project of a new research institute – the Institute of Applied Mathematics of the Romanian Academy. The Institute was created in 1991 and Adelina Georgescu became its first director. Simultaneously, she initiated the foundation of the Romanian Society of Applied and Industrial Mathematics – ROMAI. Adelina Georgescu was President of ROMAI from the moment of the birth of this Society up to 2009. In 1997 she enforced her the didactic activity as a professor of the University of Piteşti, whose Department of Applied Mathematics she led between 1999 and 2005. At the Faculty of Mathematics of Piteşti, Adelina Georgescu led the courses at the 3rd, 4th year and the master level in Analytical Mechanics, Mechanics of continua, Fluid dynamics, Dynamical systems, Bifurcation theory. She also founded and led the ”Victor Vâlcovici” scientific seminar for master students and PhD students. Adelina Georgescu was involved in the supervision of many master theses and of 19 PhD theses. Throughout the time spent at the University of Piteşti, she enthusiastically encouraged the work of young researchers in her fields of interest and especially in dynamical systems and bifurcation theory. New fields as the application of bifurcation theory in the economy and the biology were explored by some of her PhD students with remarkable, internationally recognized results. The results of Professor’s Adelina Georgescu scientific activity were published in 18 books (some of them by well-known publishing houses as Kluwer, Chapman and Hall or World Scientific) and more than200 scientific papers in prestigious journals of mathematics.) A great energy and a lot of work were invested by Professor Adelina Georgescu in gathering the material and writing, with a few collaborators, two editions (2004 and 2006) of a Dictionary of Romanian Mathematicians, a precious tool in making a genuine image of the Romanian mathematical school. This was one of Professor’s Adelina Georgescu many acts of patriotism. As a recognition of her scientific value, she was invited as visiting professor at foreign universities or research institutes, delivering a large number of conferences, leading doctoral courses and research seminars at the Politecnico di Bari (2000); Istituto per le Applicazioni del Calculo, Roma (1997, 1996); University of Bari, (1991, 1992, 1993, 1994, 1995, 1996, 2004, 2005, 2006, 2007); Instituto Pluridisciplinar, University Complutense, Madrid (1996); Institut für Chemie und Dynamik der Geosphäre (ICG-4), KFA, Julich (1999, 1998, 1996, 1995); University of

- iv - Le Havre (1994); University of Paris VI (1994, 1995), Istituto per le Ricerche di Matematica Applicata (IRMA), Bari (1992,1994); University of Metz (1991); Polytechnical Institute of Poznan (1991); Institute of Mathematics and its Applications, Minneapolis (1990); Institut für Strömungslehre und Strömungsmechanik, Karlsruhe (1990); University of Catania (2002); University of Lecce (2001, 2002); University of Messina (2001, 2002, 2003, 2004, 2005, 2006, 2007); Institute of Mathematics, Belgrad (2003); CREATIS-INSA, Lyon (2003, 2004), University of Patras (2001, 2007). In addition to her own scientific and didactic activity, Adelina Georgescu has brought a significant contribution to the developing of Romanian applied and industrial mathematics as a talented manager of sciences. She was one of the main organizers of the first (1990) and second (1992) Preparatory Conference for the International Congress of Romanian Mathematicians. Since 1993 annually, ROMAI and some universities (of Piteşti, of Oradea, Tiraspol State University) organized the already well-known International Conferences on Applied and Industrial Mathematics – CAIM, important forums of genuine debates and exchanges of ideas in mathematics and its applications. These conferences widened the traditional lines of Romanian mathematical research to recent topics developped by a large number of professionals, mainly from Romania and the Republic of Moldova, ranging from pure mathematics to engineering and high-school mathematics. As President of ROMAI, Adelina Georgescu led a policy of constant support to the mathematical community of the Republic of Moldova. This represented another act of patriotism, which led to the creation of strong collaboration and friendship relations between the mathematical schools from the two sides of the Prut. Besides the participants from Romania and Republic of Moldova, mathematicians from countries like Canada, France, Greece, Italy, Russia, Slovakia, Ukraine, Uzbekistan, took part at the successive editions of CAIM. From 1997 until 2004, the CAIM Proceedings were published mainly in the journal Buletinul Ştiinţific al Universităţii din Piteşti, (Seria Matematica şi Informatica) this bringing to this journal a good place in the CNCSIS classification of scientific journals (namely a category B journal). Adelina Georgescu was practically the Main Editor of this journal in the mentioned period (with the exception of vol. 5). Since 2005, ROMAI began to issue ROMAI Journal that publishes mainly papers presented at CAIM, but also other valuable mathematical papers and Educational ROMAI Journal, which is devoted to preuniversitary mathematics. Adelina

- v - Georgescu was Editor in Chief of ROMAI Journal. She was also a member of the editorial board of some scientific journals like Int. J. Chaos Theory and Applications, Applied and Numerical Mathematics, reviewer for Math. Reviews, Zentralblatt für Mathematik., and peer-reviewer for Rev. Roum. Math. Pures Appl., Rev. Roum. Phys., Rev. Roum. Sci. Tech.-Mec. Appl.; St. Cerc. Mat.; St.Cerc. Mec. Appl. As a Professor at the University of Piteşti, Adelina Georgescu initiated the Series of Applied and Industrial Mathematics of the University of Piteşti, a collection of books containing 29 titles up to now. Five among her former PhD students, are now teaching at this University. Her activity was devoted to the formation of a school of Dynamical Systems Theory at the University of Piteşti, when no similar school existed in our country, as well as the permanent attempt of inducing high universitary standards at this University. After her first visit to Chişinău in 1990, Adelina Georgescu played an important role in the mathematical life of the Republic of Moldova: she became an active participant to many mathematical forums organized in Moldova; she took part in the expert commissions in PhD dissertations; she supervised several moldavian PhD and masters theses. Her valuable and multilateral contribution to the progress of mathematics received a high recognition from the highest scientific entitities of Moldova, by receiving the Diplom of Honour of the Academy and the title of Doctor Honoris Causa of the University of Tiraspol. Other international signs of recognition of Professor Adelina Georgescu activity were her election as member of the Russian Academy of Nonlinear Sciences, of the Accademia Peloritana dei Pericolanti (Messina) and of the AOSR (Academia Oamenilor de Ştiinţă din România). Her tremendous energy and will power, unyielding integrity and idealistic dedication to the aim of opening new research lines in the field of Romanian applied mathematics will remain for ever engraved in the memories and hearts of her many friends, collaborators and students.

Directory Committee of the Romanian Society of Applied and Industrial Mathematics – ROMAI .

- vi - .

- vii -

- viii - ROMAI J., 6, 1(2010), 1–4

A NOTE ON DIFFERENTIAL SUBORDINATIONS USING SAL˘ AGEAN˘ AND RUSCHEWEYH OPERATORS Alina Alb Lupa¸s Department of Mathematics and Computer Science, University of Oradea, Romania [email protected] Abstract In the present paper we deﬁne a new operator using the Sal˘ agean˘ and Ruscheweyh operators. Then we study some diﬀerential subordinations regarding the new operator.

Keywords: diﬀerential subordination, convex function, best dominant, diﬀerential operator, convolution product. 2000 MSC: 30C45, 30A20, 34A40.

1. INTRODUCTION Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1}, and by H(U) the space of holomorphic functions in U. Let n+1 An = { f ∈ H(U), f (z) = z + an+1z + ..., z ∈ U}, for n ∈ N = {1, 2, ...}. Denote by ( ) z f 00(z) K = f ∈ A , Re + 1 > 0, z ∈ U n f 0(z) the class of normalized convex functions in U. If f and g are analytic functions in U, we say that f is subordinate to g, written f ≺ g, if there is a function w analytic in U, with w(0) = 0, |w(z)| < 1, for all z ∈ U, such that f (z) = g(w(z)) for all z ∈ U. If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (U) ⊆ g(U). Let ψ : C3 × U → C and h be an univalent function in U. If p is analytic in U and satisfies the (second-order) differential subordination ψ(p(z), zp0(z), z2 p00(z); z) ≺ h(z), for z ∈ U, (1) then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (1).

1 2 Alina Alb Lupa¸s

A dominant eq that satisﬁes eq ≺ q for all dominants q of (1) is said to be the best dominant of (1). The best dominant is unique up to a rotation of U.

m Deﬁnition 1. (S˘al˘agean [4]) For f ∈ An, n ∈ N, m ∈ N ∪ {0}, the operator S is m reccurently deﬁned by S : An → An,

S 0 f (z) = f (z) S 1 f (z) = z f 0(z) ... S m+1 f (z) = z S m f (z) 0 , for z ∈ U.

P∞ j m P∞ m j Remark 1.1. If f ∈ An, i.e. f (z) = z + j=n+1 a jz , then S f (z) = z + j=n+1 j a jz , for z ∈ U.

m Deﬁnition 2. (Ruscheweyh [3]) For f ∈ An, n ∈ N, m ∈ N ∪ {0}, the operator R is m deﬁned by R : An → An,

R0 f (z) = f (z) R1 f (z) = z f 0 (z) ... (m + 1) Rm+1 f (z) = z Rm f (z) 0 + mRm f (z) , for z ∈ U.

P∞ j m P∞ m j Remark 1.2. If f ∈ An, i.e. f (z) = z+ j=n+1 a jz , then R f (z) = z+ j=n+1 Cm+ j−1a jz , for z ∈ U.

Lemma 1. (Miller and Mocanu [1]) Let g be a convex function in U and let

h(z) = g(z) + nαzg0(z), for z ∈ U, where α > 0 and n is a positive integer. If n n+1 p(z) = g(0) + pnz + pn+1z + ..., for z ∈ U is holomorphic in U and

p(z) + αzp0(z) ≺ h(z), for z ∈ U, then p(z) ≺ g(z), for z ∈ U, and this result is sharp. A note on diﬀerential subordinations using S˘al˘agean and Ruscheweyh operators 3 2. MAIN RESULTS Deﬁnition 3. ([1]) Let m ∈ N ∪ {0}. Denote by SRm the operator given by the Hadamard product (the convolution product) of the S˘al˘agean operator S m and the m m Ruscheweyh operator R , SR : An → An, SRm f (z) = S m ∗ Rm f (z) .

P∞ j Remark 2.1. If f ∈ An, i.e. f (z) = z + j=n+1 a jz , then

∞ m P m m 2 j SR f (z) = z + Cm+ j−1 j a j z , j=n+1 for z ∈ U. Theorem 2.1. Let g be a convex function such that g (0) = 1 and let h be the function 0 h (z) = g (z) + zg (z), for z ∈ U. If m ∈ N ∪ {0}, f ∈ An and the differential subordination 1 m SRm+1 f (z) + z SRm f (z) 00 ≺ h (z) , for z ∈ U, (2) z m + 1 holds, then SRm f (z) 0 ≺ g (z) , for z ∈ U, and this result is sharp. m 0 P∞ m m+1 2 j−1 Proof. With notation p (z) = (SR f (z)) = 1 + j=n+1 Cm+ j−1 j a j z and p (0) = P∞ j 1, we obtain for f (z) = z + j=n+1 a jz , 0 1 m+1 m m 00 p (z) + zp (z) = z SR f (z) + z m+1 (SR f (z)) . We have p (z) + zp0 (z) ≺ h (z) = g (z) + zg0 (z), for z ∈ U. By using Lemma 1, we obtain p (z) ≺ g (z), for z ∈ U, i.e. (SRm f (z))0 ≺ g (z), for z ∈ U, and this result is sharp. Theorem 2.2. Let g be a convex function, g (0) = 1, and let h be the function 0 h (z) = g (z) + zg (z), for z ∈ U. If m ∈ N ∪ {0}, f ∈ An and verifies the differential subordination SRm f (z) 0 ≺ h (z) , for z ∈ U, (3) then SRm f (z) ≺ g (z) , for z ∈ U, z and this result is sharp. P∞ j m P∞ m m 2 j Proof. For f ∈ An, f (z) = z+ j=n+1 a jz , we have SR f (z) = z+ j=n+1 Cm+ j−1 j a j z , for z ∈ U. P∞ m z+ Cm jma2z j SR f (z) j=n+1 m+ j−1 j P∞ m m 2 j−1 Consider p (z) = z = z = 1 + j=n+1 Cm+ j−1 j a j z . 4 Alina Alb Lupa¸s

We have p (z) + zp0 (z) = (SRm f (z))0, for z ∈ U. Then (SRm f (z))0 ≺ h (z), for z ∈ U, becomes p (z)+zp0 (z) ≺ h (z) = g (z)+zg0 (z), SRm f (z) for z ∈ U. By using Lemma 1, we obtain p (z) ≺ g (z), for z ∈ U, i.e. z ≺ g (z), for z ∈ U. Theorem 2.3. Let g be a convex function such that g (0) = 1 and let h be the function 0 h (z) = g (z) + zg (z), for z ∈ U. If m ∈ N ∪ {0}, f ∈ An and veriﬁes the diﬀerential subordination ! zS Rm+1 f (z) 0 ≺ h (z) , for z ∈ U, (4) SRm f (z) then SRm+1 f (z) ≺ g (z) , for z ∈ U, SRm f (z) and this result is sharp.

P∞ j m P∞ m m 2 j Proof. For f ∈ An, f (z) = z+ j=n+1 a jz , we have SR f (z) = z+ j=n+1 Cm+ j−1 j a j z , for z ∈ U. P∞ P∞ m+1 m+1 m+1 2 j m+ m+1 2 j−1 SR f (z) z+ j=n+1 Cm+ j j a j z 1+ j=n+1 Cm+ j j a j z ( ) P∞ P∞ Consider p z = SRm f (z) = m m 2 j = m m 2 j−1 . z+ j=n+1 Cm+ j−1 j a j z 1+ j=n+1 Cm+ j−1 j a j z m+1 0 0 0 (SR f (z)) (SRm f (z)) We have p (z) = SRm f (z) − p (z) · SRm f (z) . 0 0 zS Rm+1 f (z) Then p (z) + zp (z) = SRm f (z) . Relation (4) becomes p (z) + zp0 (z) ≺ h (z) = g (z) + zg0 (z), for z ∈ U, and by using SRm+1 f (z) Lemma 1, we obtain p (z) ≺ g (z), for z ∈ U, i.e. SRm f (z) ≺ g (z), for z ∈ U. References

[1] A. Alb Lupas¸, Certain diﬀerential subordinations using S˘al˘agean and Ruscheweyh operators, Mathematica (Cluj), Proceedings of International Conference on Complex Analysis and Related Topics, The 11th Romanian-Finnish Seminar, Alba Iulia, 2008 (to appear). [2] S.S. Miller, P.T. Mocanu, Diﬀerential Subordinations. Theory and Applications, Marcel Dekker Inc., New York, Basel, 2000. [3] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109- 115. [4] G. St. Sal˘ agean,˘ Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. ROMAI J., 6, 1(2010), 5–14

ON TOPOLOGICAL AG-GROUPOIDS AND PARAMEDIAL QUASIGROUPS WITH MULTIPLE IDENTITIES Natalia Bobeic˘a,Liubomir Chiriac Tiraspol State University, Chi¸sin˘au,Republic of Moldova [email protected], [email protected] Abstract This paper studies some properties of (n, m)-homogeneous isotopies of topological AG- groupoids and paramedial quasigroups with (n, m)-identities. We study properties of paramedial groupoids with multiple identities. We extend some aﬃrmations of the theory of topological groups on the class of topological (n, m)-homogeneous quasigroups.

Keywords: AG-groupoid, quasigroups, (n, m)-identity, (n, m)-homogeneous isotope. 2000 MSC: 97U99.

1. INTRODUCTION In this work we study the (n, m)-homogeneous isotopies of AG-topological groupoid with multiple identities and some topological properties of topological paramedial quasigroups.The results established in this paper are related to the results of M. Choban and L. Kiriyak in [2] and to the research papers [6-14]. In section 3 we expand on the notions of multiple identities and (n, m)-homogeneous isotopies introduced in [5]. This concept facilitates the study of topological groupoids with (n, m)-identities. In this section we prove that if (G, +) is an AG-topological groupoid and e is an (1, p)-zero, then every (n, 1)-homogeneous isotope (G, ·) of (G, +) is an AG-groupoid, with (1, np)-identity e in (G, ·). In section 4 we study the paramedial groupoids with multiple identities. We present some interesting properties of a class of paramedial groupoids with (2, 1)-identities. We prove that if P is an open compact set of a right identity of a topological right paramedial loop G, then P contains an open compact topological right paramedial subloop Q. This result was obtained by L.S. Pontrjagin for topological groups in (Theorem 16, [13]) and by L.L. Chiriac for topological left medial quasigroups in [14]. We shall use the notations and terminology from [1-3, 13].

2. BASIC NOTIONS A non-empty set G is said to be a groupoid relatively to a binary operation denoted by “ · ”, if for every ordered pair (a, b) of elements of G there is a unique element ab = a · b ∈ G.

5 6 Natalia Bobeic˘a,Liubomir Chiriac

If the groupoid G is a topological space and the binary operation (a, b) → a · b is continuous, then G is called a topological groupoid. An element e ∈ G is called an identity if ex = xe = x for every x ∈ X. A quasigroup with an identity is called a loop. A groupoid G is called medial if it satisfies the law xy · zt = xz · yt for all x, y, z, t ∈ G. A groupoid G is called paramedial if it satisfies the law xy · zt = ty · zx for all x, y, z, t ∈ G. If a paramedial guasigroup G contains an element e such that e · x = x(x · e = x) for all x in G, then e is called a left (right) identity element of G and G is called a left (right) paramedial loop. A groupoid (G, ·) is called a groupoid Abel-Grassmann or AG-groupoid if it satisfies the left invertive law (a · b) · c = (c · b) · a for all a, b, c ∈ G. A groupoid G is said to be hexagonal if it is idempotent, medial and semisymetric, i.e. the equalities x · x = x, xy · zt = xz · yt, x · zx = xz · x = z hold for all of its elements. The last identities (semisymetric law) can be represented in the form of an equivalence x · z = t ⇔ x = z · t. A groupoid G is called bicommutative if it satisfies the law xy · zt = tz · yx for all x, y, z, t ∈ G.

3. GROUPOIDS WITH MULTIPLE IDENTITIES Consider a groupoid (G, +). For every two elements a, b from (G, +) we denote: 1(a, b, +) = (a, b, +)1 = a + b, and n(a, b, +) = a + (n − 1)(a, b, +), (a, b, +)n = (a, b, +)(n − 1) + b for all n ≥ 2. If a binary operation (+) is given on a set G, then we shall use the symbols n(a, b) and (a, b)n instead of n(a, b, +) and (a, b, +)n. Deﬁnition 3.1. Let (G, +) be a groupoid and let n, m ≥ 1. The element e of the groupoid (G, +) is called: - an (n, m)-zero of G if e + e = e and n(e, x) = (x, e)m = x for every x ∈ G; - an (n, ∝)-zero if e + e = e and n(e, x) = x for every x ∈ G; - an (∝, m)-zero if e + e = e and (x, e)m = x for every x ∈ G. Clearly, if e ∈ G is both an (n, ∝)-zero and an (∝, m)-zero, then it is also an (n, m)- zero. If (G, ·) is a multiplicative groupoid, then the element e is called an (n, m)- identity. The notion of (n, m)-identity was introduced in [5]. Example 3.1. Let (G, ·) be a paramedial groupoid, e ∈ G and ex = x for every x ∈ G. Then (G, ·) is a paramedial groupoid with (1, 2)-identity e in G. Indeed, if x ∈ G, then xe · e = xe · ee = ee · ex = e · ex = e · x = x. Example 3.2. Let G = {1, 2, 3, 4, 5}. We deﬁne the binary operation ” · ”: On topological AG-groupoids and paramedial quasigroups with multiple identities 7

· 1 2 3 4 5 1 1 5 4 3 2 2 3 2 1 5 4 3 5 4 3 2 1 4 2 1 5 4 3 5 4 3 2 1 5

Then (G, ·) is a non-commutative, hexagonal and AG-quasigroup and each element from (G, ·) is a (2, 4)-identity in G. Definition 3.2. Let (G, +) be a topological groupoid. A groupoid (G, ·) is called a homogeneous isotope of the topological groupoid (G, +) if there exist two topological automorphisms ϕ, ψ :(G, +) → (G, +) such that x · y = ϕ(x) + ψ(y), for all x, y ∈ G. For every mapping f : X → X we denote f 1(x) = f (x) and f n+1(x) = f ( f n(x)) for any n ≥ 1. Definition 3.3. Let n, m ≤ ∞. A groupoid (G, ·) is called an (n, m)- homogeneous isotope of a topological groupoid (G, +) if there exist two topological automorphisms ϕ, ψ :(G, +) → (G, +) such that: 1. x · y = ϕ(x) + ψ(y) for all x, y ∈ G; 2. ϕψ = ψϕ; 3. If n < ∞, then ϕn(x) = x for all x ∈ G; 4. If m < ∞, then ψm(x) = x for all x ∈ G. Definition 3.4. A groupoid (G, ·) is called an isotope of a topological groupoid (G, +), if there exist two homeomorphisms ϕ, ψ :(G, +) → (G, +) such that x · y = ϕ(x) + ψ(y) for all x, y ∈ G. Under the conditions of Definition 3.4 we shall say that the isotope (G, ·) is generated by the homeomorphisms ϕ, ψ of the topological groupoids (G, +) and write (G, ·) = g(G, +, ϕ, ψ). Example 3.3. Let (R, +) be the topological Abelian group of real numbers. 1. If ϕ(x) = 5x, ψ(x) = x and x · y = 5x + y, then (R, ·) = g(R, +, ϕ, ψ) is a locally compact medial quasigroup. By virtue of Theorem 7 from [2], there exists a left invariant Haar measure on (R, ·). 2. If ϕ(x) = 5x, ψ(x) = 7x and x · y = 5x + 7y, then (R, ·) = g(R, +, ϕ, ψ) is a locally compact medial quasigroup and on (R, ·) as above, by virtue of Theorem 7 from [2], does not exist any left or right invariant Haar measure.

Example 3.4. Denote by Zp = Z/pZ = {0, 1, ..., p − 1} the cyclic Abelian group of order p. Consider the commutative group (G, +) = (Z7, +), ϕ(x) = 3x, ψ(x) = 4x and 8 Natalia Bobeic˘a,Liubomir Chiriac

x·y = 3x+4y. Then (G, ·) = g(G, +, ϕ, ψ) is a medial, paramedial and bicommutative quasigroup and the neutral element of (G, +) is a (3, 6)-identity in (G, ·).

Example 3.5. Consider the commutative group (G, +) = (Z5, +), ϕ(x) = 3x, ψ(x) = 2x and x · y = 3x + 2y. Then (G, ·) = g(G, +, ϕ, ψ) is a medial, paramedial and bicommutative quasigroup and the neutral element of (G, +) is a (4, 4)-identity in (G, ·).

Example 3.6. Consider the commutative group (G, +) = (Z7, +), ϕ(x) = 2x, ψ(x) = 5x and x · y = 2x + 5y. Then (G, ·) = g(G, +, ϕ, ψ) is a medial, paramedial and bicommutative quasigroup and the neutral element of (G, +) is a (6, 3)-identity in (G, ·).

Example 3.7. Consider the commutative group (G, +) = (Z7, +), ϕ(x) = 3x, ψ(x) = 5x and x · y = 3x + 5y. Then (G, ·) = g(G, +, ϕ, ψ) is a medial and hexagonal quasigroup and each element of (G, ·) is a (6, 6)-identity.

Theorem 3.1. If (G, +) is an AG-groupoid and e ∈ G is an (1, p)-zero, then every (n, 1)-homogeneous isotope (G, ·) of the topological groupoid (G, +) is AG-groupoid with (1, np)-identity e in (G, ·) and a + bc = b · (a + c) for all a, b, c ∈ G and n, p ∈ N.

Proof. We will prove that e is an (1, np)-identity in (G, ·) by the method described in [2]. Let (G, ·) be an (n, 1)-homogeneous isotope of the groupoid (G, +) and e be an (1, p)-zero in (G, +). We mention that ϕq(e) = ψq(e) = e for every q ∈ N. Then in (G, +) we have q · 1(e, x, +) = x for each x ∈ G and for every q ∈ N. Since we have (n, 1)-homogeneous isotope (G, ·) then m = 1 and ψ(x) = x for all x ∈ G. In this case

1(e, x, ·) = 1(e, ψ(x), +) and

q(e, x, ·) = q(e, ψq(x), +) for every q ≥ 1. Therefore

1(e, x, ·) = 1(e, ψ(x), +) = 1(e, x, +) = x.

Analogously we obtain that

(e, x, ·)np = (e, ϕnp(x), +)np = (e, x, +)np = x.

Hence e is an (1, np)-identity in (G, ·). The mediality of the AG-topological groupoid (G, +) follows from [4]. Really,

(x + y) + (z + t) = ((z + t) + y) + x = ((y + t) + z) + x = (x + z) + (y + t). On topological AG-groupoids and paramedial quasigroups with multiple identities 9

Since e is an (1, p)-zero in (G, +), hence e is left zero and e + x = x for every x ∈ (G, +). In this case for each AG-groupoid (G, +) with (1, p)-zero we get

x + (z + t) = (e + x) + (z + t) = (e + z) + (x + t) = z + (x + t).

We will prove that (n, 1)-homogeneous isotope (G, ·) of the topological groupoid (G, +) is AG-groupoid and x·(zt) = z·(xt). Since (G, ·) is (n, 1)-homogeneous isotope of (G, +), then ψ(x) = x. Hence

x · zt = ϕ(x) + ψ(zt) = = ϕ(x) + zt = ϕ(x) + (ϕ(z) + ψ(t)) = ϕ(z) + (ϕ(x) + ψ(t)) = = ϕ(z) + xt = ϕ(z) + ψ(xt) = z · xt.

Therefore the identity x · zt = z · xt holds in groupoid (G, ·). We will show that a + bc = b · (a + c). Really,

a + bc = (ea) + (bc) = (ϕ(e) + ψ(a)) + (ϕ(b) + ψ(c)) = = (ϕ(e) + ϕ(b)) + (ψ(a) + ψ(c)) = ϕ(e + b) + ψ(a + c) = = ϕ(b) + ψ(a + c) = b · (a + c).

The proof is complete. Corollary 3.1. If (G, +) is an AG-groupoid and e is a left zero, then every (1, 1)- homogeneous isotope (G, ·) of the topological groupoid (G, +) is AG-groupoid with left identity e in (G, ·) and a + bc = b · (a + c), for all a, b, c ∈ G.

4. SOME PROPERTIES OF PARAMEDIAL QUASIGROUPS We provide an example of a paramedial quasigroup which is not medial. Example 4.1. Let G = {1, 2, 3, 4}. We deﬁne the binary operation ” · ”:

· 1 2 3 4 1 2 1 3 4 2 3 4 2 1 3 4 3 1 2 4 1 2 4 3

Then (G, ·) is a paramedial quasigroup but it is not medial. Indeed, for example

(2 · 3) · (1 · 4) , (2 · 1) · (3 · 4). 10 Natalia Bobeic˘a,Liubomir Chiriac

Theorem 4.1. If (G, ·) is a multiplicative groupoid, e ∈ G and the following conditions hold: 1. xe = x for every x ∈ G, 2. x2 = x · x = e for every x ∈ G, 3. xy · z = xz · y for all x, y, z ∈ G, 4. for every a, b ∈ G there exists an unique point y ∈ G such that ya = b, then e is a (2, 1)-identity in G. Proof. Fix x ∈ G. Pick y ∈ G such that y · ex = x. By conditions 2 of Theorem 4.1 we have (y · ex) · x = x · x = e. (1) By condition 3 of this theorem (y · ex) · x = yx · ex. (2) From (1) and (2) we obtain yx · ex = e. (3) It is clear that ex · ex = e. (4) Thus, from (3) and (4) yx · ex = ex · ex. Hence yx = ex and y = e. Therefore y · (e · x) = e(ex) = x and e is a (2, 1)− identity in G. The proof is complete. Theorem 4.2. If (G, ·) is a multiplicative groupoid, e ∈ G and the following conditions hold: 1. xe = x for every x ∈ G, 2. x2 = x · x = e for every x ∈ G, 3. x · zt = t · zx for all x, y, z ∈ G, 4. for every a, b ∈ G there exists a unique point y ∈ G such that ya = b, then e is a (2, 1)-identity in G. Proof. Similar to Theorem 4.1. Remark 4.1. The proofs of Theorems 4.1. and 4.2. are based on a rather general method. Professor I. Burdujan noticed that there is an easier way to prove Theorem 4.2. He observed that e · ex = x · ee = xe = x and e is a (2, 1)-identity in G. Theorem 4.3. If (G, ·) is a multiplicative groupoid, e ∈ G and the following conditions hold: On topological AG-groupoids and paramedial quasigroups with multiple identities 11

1. xe = x for every x ∈ G, 2. x2 = x · x = e for every x ∈ G, 3. xy · uv = vy · ux for all x, y, u, v ∈ G, 4. if xa = ya then x = y for all x, y, a ∈ G, then (G, ·) is a paramedial quasigroup with a (2, 1)-identity e. Proof. If x ∈ G, then

e · ex = ee · ex = xe · ee = xe · e = x · e = x.

Thus e is (2, 1)− identity. Consider the equation xa = b. Then xa · e = b · e, xa · ee = be, ea · ex = be, (ea · ex) · be = be · be. Thus, (ea · ex) · (be) = e. By condition (3) of theorem 4.3 we have

(e · ex) · (b · ea) = e. (5)

From (5) we obtain

x · (b · ea) = e. (6)

By condition (2) of theorem 4.3

(b · ea) · (b · ea) = e. (7)

From (6) and (7) have x = b · ea. Since xa = b we can verify that (b · ea) · a = (b · ea) · (ae) = (e · ea) · (ab) = a · ab = ae · ab = be · aa = be · e = b. In this case the element x = b · ea is a unique solution of the equation xa = b. Now we consider the equation ay = b. We have e · b = e · ay = ee · ay = ye · ae = y · a. Thus ya = eb and, by considering the solution of equation xa = b we have y = eb · ea = ab · ee = ab · e = ab. It follows that y = ab is a unique solution of the equation ay = b. The proof is complete. Corollary 4.1. If (G, ·) is a paramedial quasigroup with a (2, 1)-identity e, then solutions of the equations xa = b and ay = b are respectively x = b · ea and y = ab for every a, b ∈ G. Lemma 4.1. Let P be a subset of topological right paramedial loop (G, ·) and e ∈ P. If P1 = P ∩ eP, then 12 Natalia Bobeic˘a,Liubomir Chiriac

1. eP1 = P1. 2. If P is open, then P1 is open. 3. If P is closed, then P1 is closed. 4. If P is compact, then P1 is compact.

Proof. The mapping f : G → G, where f (x) = ex, is a homeomorphism and P1 = P ∩ eP. That proved the assertions 2, 3 and 4. For every x ∈ G we have e · ex = x. Therefore, eP1 = eP ∩ (e · eP) = eP ∩ P = P1. The proof is complete.

Proposition 4.1. Let (G, ·) be a right paramedial loop. Then the mapping

f : G → G, where f (x) = ex, is an involutive mapping, i.e. f = f −1.

Proof. It is obvious that f is an one-to-one mapping. The solution of a equation ay = b is y = ab. Hence a · ab = b for every y ∈ G. In particular e · ex = x and f ( f (x)) = x. Hence f = f −1.The proof is complete.

Theorem 4.4. Let (G, ·) be a topological right paramedial loop with the identity x2 = e. If P is an open compact subset such that e ∈ P, then P contains an open compact right paramedial subloop (Q, ·) of (G, ·).

Proof. In virtue of Lemma 4.1 we consider that eP = P. Note Q = {q ∈ G : qP∪Pq ⊂ P}. We prove that Q is an open compact right paramedial subloop. Now we show that Q is the open set. Let q be a ﬁxed point of Q and x be an arbitrary point of P. Since xq ∈ P and P is an open set, then there exists such neighborhoods Ux 3 x and Vx 3 q, S∞ such that UxVx ⊂ P. In this case we have P = i=1 Uxi . Because P is a compact Sk set we can extract an open ﬁnitely subcovering Ux1 , ..., Uxk such that P = i=1 Uxi . Tk Note V = i=1 Vxi . Then PV ⊂ P. Let us consider qx ∈ P. By analogy we prove that there exists such neighborhood W 3 q so that WP ⊂ P. Note V ∩ W = U. Then we have that UP ⊂ P and PU ⊂ P. Hence for the open set U 3 q we have that U ⊂ Q. Therefore Q is the open set. Let us show that Q is a closed set. Suppose that p < Q. Then for some q ∈ P we have that pq < P or qp < P. Let us assume that pq < P. Then there exists an open set U such that p ∈ U and Uq ⊂ G \ P. Therefore U ∩ Q = ∅ and q is not a limit point of a set Q. It follows that the set Q is closed. We show that Q ⊂ P. If q ∈ Q, then q ∈ qP ∩ Pq. Since if qP ∪ Pq ⊂ P, then q ∈ P. Therefore Q ⊂ P. By condition of theorem eP ∪ Pe = P ∪ P = P ⊂ P. Hence e ∈ Q. We will prove that Q is a right paramedial loop. Fix a, b ∈ Q. Then

P · ab = Pe · ab = be · aP = b · aP = b · P ⊂ P On topological AG-groupoids and paramedial quasigroups with multiple identities 13 and

ab · P = ab · eP = Pb · ea = P · ea = Pe · ea = ae · eP = a · P ⊂ P.

Therefore ab ∈ Q and Q is a subgroupoid of G. If a, b ∈ Q then for equation xa = b his solution x = b · (ea) is in Q. Really, since e, a ∈ Q we have ea ∈ Q, and b · ea ∈ Q. For equation ay = b his solution y = ab is also in Q. Hence Q is a right paramedial subloop of G. The proof is complete.

Theorem 4.5. Let (G, ·) be a topological paramedial quasigroup with a (2, 1)-identity e and x2 = e for every x ∈ G. If P is an open compact subset from G such that e ∈ P, then P contains an open compact paramedial subquasigroup (Q, ·) with a (2, 1)-identity e. Proof. It follows from Theorem 4.3, Lemma 4.1 and Theorem 4.4. The proof is complete.

Remark 4.2. For topological groups a series of properties that are based on the notion of open compact are proved (see[3, 13]).

5. SOME REMARKS OF MEDIAL QUASIGROUPS The ideas used in the proofs of Theorem 4.3, Lemma 4.1, Theorem 4.4 can be adopted for topological right medial loops. We mention the following properties. Remark 5.1. If (G, ·) is a right medial loop, then the mapping f : G → G, where f (x) = ex, is an involutive automorphism, i.e. f = f −1 and f (x · y) = f (x) · f (y) for every x, y ∈ G. Remark 5.2. If (G, ·) is a right medial loop, e ∈ G and x2 = e for every x ∈ G, then e is a (2, 1)-identity. Remark 5.3. Let (G, ·) be a right medial loop and x2 = e for every x ∈ G. The related operation x ◦ y = ex · ey in G satisﬁes the following properties: 1. (G, ◦) is a medial quasigroup. 2. e ◦ x = x for every x ∈ G. 3. H = {x ∈ G : ex = x} is a commutative group and is a subloop of the loops (G, ·) and (G, ◦). Example 5.1. Let (G, +) be a commutative group. We deﬁne in G the operation ” · ”: x · y = x − y for every x, y ∈ G. Then (G, ·) is a right medial loop and the identity element e of (G, +) is a (2, 1)-identity for it. Acknowledgements. The authors gratefully acknowledge helpful suggestions of the referees. 14 Natalia Bobeic˘a,Liubomir Chiriac References

[1] V. D. Belousov, Foundation of the theory of quasigroups and loops, Moscow, Nauka, 1967. [2] M. M. Choban, L. L. Kiriyak, The topological quasigroups with multiple identities, Quasigroups and Related Systems, 9(2002), 19-31. [3] E. Hewitt, K. A. Ross, Abstract harmonic analysis, Vol. 1: Structure of topological groups. Inte- gration theory. Group representation, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1963. [4] M. A. Kazim, M. Naseeruddin, On almost-semigroups, Aligarh Bulletin of Mathematics, Aligarh Muslim Univ., Aligarh, India, 8(1978), 67-70. [5] M. M. Choban, L. L. Kiriyak, The Medial Topological Quasigroup with Multiple identities, The 4 th Conference on Applied and Industrial Mathematics, Oradea-CAIM, 1995, p.11. [6] L. L. Chiriac, On Topological Quasigroups and Homogeneous Isotopies, Analele Universitat¸ii˘ din Pites¸ti, Buletin S¸tiint¸iﬁc, Seria Matematica˘ s¸i Informatica,˘ 9(2003), 191-196. [7] L. L. Chiriac, N. Bobeica, Some properties of the homogeneous isotopies, Acta et Commenta- tiones, Universitatea de Stat Tiraspol, Chis¸inau,˘ 3(2006), 107-112. [8] L. L. Chiriac, N. Bobeica, Paramedial topological groupoids, 6th Congress of Romanian Mathe- maticians, Bucharest, Romania, June 28-July 4, 2007, 25-26. [9] L. L. Chiriac, L. Chiriac Jr, N. Bobeica, On topological groupoids and multiple identities, Bulet- inul Academiei de S¸tiint¸e a RM, Matematica, 1(59)(2009), 67-78. [10] L. L. Chiriac, Topological paramedial groupoids with multiple identities, The 17-th Conference on Applied and Industrial Mathem., Constantza-CAIM, Romania, 2009, p. 28. [11] N. Bobeica, On Paramedial Loops, The 17-th Conference on Applied and Industrial Mathem., Constantza-CAIM, Romania, 2009, p. 20. [12] N. Bobeica, Topological Hexagonal Groupoids with Multiple Identities, MITRE-2009, State Uni- versity of Moldova, October, 2009, 5-6. [13] L. S. Pontrjagin, Neprerivnie gruppi, Nauka, Moskow, 1984. [14] L. L. Chiriac, Topological Algebraic System, S¸tiint¸a, Chis¸inau,˘ 2009. ROMAI J., 6, 1(2010), 15–28

AUTOMORPHISMS AND DERIVATIONS OF HOMOGENEOUS QUADRATIC DIFFERENTIAL SYSTEMS Ilie Burdujan Department of Mathematics, University of Agricultural and Veterinary Medicine ”Ion Ionescu de la Brad” Ia¸si,Romania burdujan [email protected] Abstract A binary algebra is associated with any homogeneous quadratic differential system. Derivations and automorphisms of a homogeneous quadratic differential system are just derivations and automorphisms of the algebra associated to it. The real 3-dimensional commutative algebras without nilpotent elements of order two, which have a derivation, are classified up to an isomorphism. Consequently, the corresponding homogeneous quadratic differential systems are classified up to a center-affine equivalence.

Keywords: homogeneous quadratic diﬀerential system. 2000 MSC: 37C99.

1. INTRODUCTION The theory of quadratic differential system (shortly, QDS) is the first step toward a systematic study of nonlinear systems of differential equations. There exist almost exhaustive studies on QDSs with two unknown functions [8], [5], [13], [14]. More exactly, the classification up to an affine equivalence of these systems was achieved. In fact, several classifications were made according with various classification criteria. Among them we quote the classifications which were achieved by Markus [8] (with a completion due to Date [5]) and those due to Vulpe-Sibirschii [13] for quadratic differential systems in plane. Unfortunately, the similar studies for similar systems in R3 creates important technical difficulties. A direction which had opened large perspectives for the study of systems of quadratic differential equations (and, more general, of systems of polynomial differential equations) was the one revealed by L. Markus [8] in 1960. Markus realized that a commutative algebra, which may be a non-associative one (in general), can be naturally associated to each homogeneous quadratic differential system (shortly, HQDS). Along this way, the homogeneous quadratic differential systems can be studied on the multidimensional spaces and on Banach spaces as well. The derivations and the automorphisms of a HQDS (i.e. of the quadratic vector form which defines it) are defined. The existence of such automorphisms entails the existence of certain symmetries that - in their turn - imply the existence of some

15 16 Ilie Burdujan suitable coordinates in which the system would take the simplest form (i.e., a large number of system’s coefficients would be either 0 or ”small” integers). Notice that any derivation (resp. automorphism) of a HQDS is a derivation (resp. automorphism) of its associated algebra. In the first part of this paper we present several results concerning the HQDSs on Banach spaces. They generalize the similar results that were already proved for HQDSs on Rn (see, for example, [6], [7], [9], [14]). The last part of our paper deals with the classification of some HQDSs on R3. To this end, we firstly remark that the set of important general results, concerning the derivation algebra of a real commutative algebra, is not too large. However, there exist notable progresses on this subject for the so-called NN-algebras, i.e. algebras without nilpotent elements of order 2 (see [1], [2]). Some results concerning the derivation algebra of an NN-algebra are presented and they are used for classifying the real 3-dimensional commutative NN-algebras having at least a derivation. It was proved that there exist nine multiparametric classes of nonisomorphic such NN-algebras. Accordingly, there exist nine multiparametric classes of HQDSs on R3, whose associated algebras are NN-algebras with a derivation, that are mutually nonequivalent up to an affinity.

2. PRELIMINARIES

Let E(k · k1) be a real Banach space. Definition 2.1. Any equation of the form dX = F(X) (1) dt where F : E → E is a continuous quadratic vector form on E is called a homogeneous quadratic differential equation (shortly, HQDE) on E. Recall that a vector form F is quadratic if F(sX) = s2F(X) for all s ∈ R and X ∈ E. With any quadratic vector form F is associated its polar form which is a symmetric bilinear vector form G : E × E → E defined by 1 G(X, Y) = [F(X + Y) − F(X) − F(Y)] , (2) 2 for all X, Y ∈ E. The definition of tensor product assures us that there exists a unique linear mapping µF : E ⊗ E → E such that µ ◦ ιE = G (here ιE(x, y) = x ⊗ y). In its turn, µF is naturally identified with a (1,2)-tensor. Since G is symmetric, it results that µF is a covariant symmetric (1,2)-tensor. On the other hand, G allows to define a commutative binary operation ”·” on E by x · y = G(x, y) for all x, y ∈ E. (3) The obtained commutative algebra is denoted by E(·). In general, E(·) is a non- associative algebra. Taking into account that x · x = x2 = F(x), HQDE (1) can be Automorphisms and derivations of homogeneous quadratic differential systems 17 written in the form dX = X2, (4) dt for pointing out its connection with algebra E(·). Now, let us consider another HQDE on the real Banach space E0, namely dY = F (Y), (5) dt 1 0 0 where F1 : E → E is a continuous quadratic vector form. The polar form G1 of F1 can be identified with a covariant symmetric (1,2)-tensor on E0. The binary operation ” ∗ ”, defined on E0 by

x ∗ y = G1(x, y) for all x, y ∈ E, (6) organizes E0 as a commutative algebra denoted by E0(∗). Deﬁnition 2.2. It is said that HQDE (1) is center-aﬃnely equivalent (or, CA-equivalent) with the HQDE (5) if and only if there exists a continuously invertible linear mapping h : E0 → E such that X = h(Y) is a solution for (1) whenever Y is a solution for (5); in this case it is said that h is an equivalence of equation (1) with equation (5). An equivalence of equation (1) with itself is called an automorphism of equation (1) or an automorphism of F. Proposition 2.1. [3] HQDE (1) is CA-equivalent with the HQDE (5) if and only if there exists a continuously invertible linear mapping h : E0 → E such that

h ◦ F1 = F ◦ h. (7)

−1 −1 −1 Since h is continuous and the equality h ◦ F = F1 ◦ h holds, it results the following assertion. Corolar 2.1. If (1) is CA-equivalent with (5), then (5) is also CA-equivalent with (1). Theorem 2.1. [3] Equations (1) and (5) are CA-equivalent if and only if the algebras E(·) and E0(∗) are continuously isomorphic. The continuous linear mapping h : E → E is an automorphism for (1) if and only if it is a continuous automorphism of algebra E(·).

If Φt and Ψt denote the ﬂows of equations (1) and (5), respectively, then the two equations are CA-equivalent if and only if there exists a continuous invertible linear mapping h : E0 → E such that

h ◦ Ψt = Φt ◦ h. Remark 2.1. The existence of a CA-equivalence h : E0 → E allows to identify the spaces E0 and E. Then, Theorem 2.1 assures us that there exists a 1-to-1 correspondence between the set of classes of CA-equivalent HQDEs on E and the set of classes 18 Ilie Burdujan of isomorphic commutative algebras deﬁned on E. Accordingly, there exists a correspondence between certain qualitative properties of a HQDE (1) and the properties invariant up to an isomorphism of the associated algebra.

Remark 2.2. The binary CA-equivalence relation defined by Definition 2.2 on the set of HQDEs on a fixed Banach space is an equivalence relation (i.e. it is reflex- ive, symmetric and transitive). That is why, in what follows, we shall use the term equivalence instead of CA-equivalence.

Theorem 2.1 assures us that there exists a correspondence between the aﬃnely invariant properties of any HQDE and the properties invariant up to an isomorphism of the associated algebra. This correspondence is not completely identiﬁed. Some of its components are presented in the next Proposition.

Proposition 2.2. The following assertions hold, for any HQDE (1) on the Banach space E: 1. the set N(E) of all nilpotents of order two of E(·) is in a bijective correspondence with the set of all steady state solutions of (1), 2. the set I(E) of all idempotents of E(·) is in a bijective correspondence with the set of ray (from or to 0) solutions of (1), 3. E(·) is a nilalgebra if and only if all solutions of (1) are polynomials, 4. E2 = E · E is a proper ideal of E if and only if (1) has at least a linear prime integral, 5. if E(·) is a power-associative algebra then all solutions of (1) are rational functions.

In fact, any structural property of E(·) (i.e., a property which is invariant up to an isomorphism) enforces the existence of a property of Eq. (1) (and conversely). For example, if E(·) has no idempotent element then the zero solution of (1) is not asymptotically stable; if (1) has a Liapunov function then E has no idempotent element. NOTE. These results were already proved, in the ﬁnite dimensional case, by Kinyon&Sagle, Rorhl,¨ Walcher (see [6], [9], [14]).

3. DERIVATIONS AND AUTOMORPHISMS OF A HQDS Deﬁnition 3.1. A derivative of F is any continuous linear transformation D : E → E satisfying DF(X) = F0(X) · DX f or all X ∈ E, (8) dF(X + sY) where F0(X) · Y = lim f or all X, Y ∈ E. Any derivative of the vector s→0 ds form F is also named a derivation of HQDE (1). Automorphisms and derivations of homogeneous quadratic diﬀerential systems 19

Proposition 3.1. D is a derivation of HQDE (1) if and only if it is a derivation of E(·). Proof. Indeed, the equality G(X + sY, X + sY) − G(X, y) F0(X) · Y = lim = 2G(X, Y) s→0 s holds for all X, Y ∈ E. It implies that D satisﬁes to Leibniz rule.

We denote by Der F the set of all derivatives of F (i.e. the set of all derivations of (1)) and by Der E the so-called derivation algebra of E(·). Proposition 1.2 asserts that Der F = Der E. Consequently, Der F is a Lie subalgebra of g`(E). Deﬁnition 3.2. An automorphism of F is a continuous invertible linear transformation ϕ : E → E satisfying to condition F ◦ ϕ = ϕ ◦ F. Any automorphism of F is also called an automorphism of HQDE (1). Proposition 3.2. ϕ is an automorphism of HQDE (1) if and only if it is an automorphism of E(·). We denote by Aut F the set of all automorphisms of F (i.e. the set of all automorphisms of (1)) and by Aut E the set of all automorphisms of algebra E(·). Proposition 1.3 asserts that Aut F = Aut E. Consequently, Aut F is a Lie subgroup of GL(E). Any solution of (1) is analytic. Let us consider its ﬂow Φt. Then, for each ϕ ∈ Aut F we have Φt ◦ ϕ = ϕ ◦ Φt . Some features of the complex connections between Aut E and Der E are presented in the next Proposition and Example as well. Proposition 3.3. [10] Let D be a derivation of a real non-associative algebra E(·). Then: (i) exp tD for t ∈ R is a uniparametric group of automorphisms of E, (ii) the Lie algebra of Aut E is Der E.

Example. Algebra having in basis B = (e1, e2, e3) the multiplication table

2 e1 = 0 e1 · e2 = e1 e1 · e3 = 0 2 2 e2 = ae2 e2 · e3 = a3 e3 = e2 with a ∈ R \{2} has the derivation algebra     α 0 β      Der E =  0 0 0  | α, β ∈ R ,     0 0 0  20 Ilie Burdujan and the automorphism group     α 0 β      Aut E =  0 1 0  | α, β ∈ R, α , 0 .     0 0 ±1 

4. THE CLASSIFICATION OF REAL 3-DIMENSIONAL NN-ALGEBRAS WITH DERIVATIONS In [2], [3] it was proved that any real 3-dimensional NN-algebra A(·) has dim Der A ∈ {0, 1} and that there exist NN-algebras having a derivation. Conse- quently, the problem to classify, up to an isomorphism, the real 3-dimensional NN- algebras having derivations is consistent. Let A(·) be a real 3-dimensional NN-algebra with a nonzero derivation. Then there exists a derivation D with the eigenvalues {0, ±i} ([1], [2]). Accordingly, there exists a basis B = (e1, e2, e3) such that

2 2 De1 = 0, De2 = −e3, De3 = e2, e2 = e3, e2 · e3 = 0. In fact, the natural vector space decomposition A = ker D ⊕ Im D exists. Since ker D is a 1-dimensional NN-algebra, it contains an idempotent element e1. Then, 2 2 2 the equalities De2 = −2e2 · e3 = 0 imply e2 = e3 = εe1 with ε , 0. We can choose e2 and e3 such that ε = ±1. Supposing now that e1 · e2 = ae1 + be2 + ce3 and applying D to it, it follows e1 · e3 = −ce2 + be3. Applying again D to e1 · e3 it results that e1 · e2 = be2 + ce3. Therefore, the multiplication table of the algebra, in basis B, is

2 e1 = e1 e1 · e2 = be2 + ce3 e1 · e3 = −ce2 + be3 T 2 2 e2 = εe1 e3 = εe1 e2 · e3 = 0, where either ε = ±1, b2 + c2 , 0 or ε = 1, b = c = 0 (these conditions assure that A is an NN-algebra). We shall denote by A(b, c, ε) the algebra having, in a basis B, the multiplication table T. In order to classify, up to an isomorphism, the real 3-dimensional NN-algebras we need to identify the lattices of their subalgebras. First at all, we determine the idempotents of such algebras, because they identify all the 1-dimensional subalgebras. We denote by I(A) the set of all idempotents of A(·). The condition 2 (xe1 + ye2 + ze3) = xe1 + ye2 + ze3 is equivalent to the system   x2 + ε(y2 + z2) = x  (2bx − 1)y − 2cxz = 0 (9)   2cxy + (2bx − 1)z = 0. Automorphisms and derivations of homogeneous quadratic diﬀerential systems 21

Obviously, this last homogeneous system has the null solution, which is not of any interest in the problem of idempotents; since e1 is an idempotent, the system (4) has necessarily the solution x = 1, y = z = 0. It remains to look for the solutions with x < {0, 1} and y2 + z2 , 0. In this case, necessarily

2bx − 1 −2cx ∆ = = (2bx − 1)2 + 4c2 x2 = 0, 2cx 2bx − 1 1 i.e. b , 0, x = , c = 0. This time, system (4) has solutions only when k = 2b 2b − 1 ε > 0, namely: 4b2 1 √ √ x = , y = k cos θ, z = k sin θ for θ ∈ [0, 2π). 2b Therefore, the following four classes of real 3-dimensional NN-algebras arise naturally: C1. A(b, c, ε) with c , 0, C2. A(b, 0, ε) with b , 0, and k ≤ 0, C3. A(b, 0, ε) with b , 0, and k > 0, C4. A(0, 0, 1). The following assertions are readily provable:

• if A belongs to class C1, then I(A) = {e1} and Le1 has the eigenvalues {1, b ± ic},

• if A belongs to class C2, then I(A) = {e1} and Le1 has the eigenvalues {1, b, b}, • if A belongs to class C3, then 1 √ I(A) = {e } ∪ {e(b) = e + k(e cos θ + e sin θ) | θ ∈ [0, 2π), b , 0}; 1 2b 1 2 3 1 1 − b L has the eigenvalues {1, b, b} and L has the eigenvalues 1, , , e1 e(b) 2 2b

• algebra A(0, 0, ε) has I(A) = {e1} and Le1 has the eigenvalues {1, 0, 0}. Taking into account the idempotents and their spectra it results that the four classes C1, C2, C3, C4 provide a partition of the set of all real 3-dimensional NN-algebras with a derivation. Anyone of these classes can be separately analysed for being possibly partitioned in subclasses of non-isomorphic algebras. Proposition 4.1. The set of algebras belonging to class C1 have the properties: (i) Algebras A(b, c, ε) and A(b, −c, ε), (with c , 0), are always isomorphic, (ii) Algebras A(b, c, ε) and A(b0, c0, ε), with c > 0 and c0 > 0, are isomorphic if and only if b = b0 and c = c0. (iii) Any two algebras A(b, c, 1) and A(b0, c0, −1) with c > 0 and c0 > 0 are non- isomorphic. 22 Ilie Burdujan

Proof. (i) It is enough to consider the multiplication table of A(b, c, ε) in basis B0 = ( f1 = e1, f2 = e2, f3 = −e3). 0 0 (ii) If T : A(b, c, ε) → A(b , c , ε) is an isomorphism, then necessarily T(e1) = f1. As 0 0 e1 and f1 must have the same spectrum it results b = b and c = c. (iii) Let us suppose that A(b, c, 1) has, in basis B, the multiplication table T while 0 0 0 A(b , c , −1) has, in basis B = ( f1, f2, f3), the multiplication table 2 0 0 0 0 f1 = f1 f1 · f2 = b f2 + c f3 f1 · f3 = −c f2 + b f3 T’ 2 2 f2 = − f1 f3 = − f1 f2 · f3 = 0. 0 0 If T : A(b, c, ε) → A(b , c , −ε) is an isomorphism, then necessarily T(e1) = f1 and 0 0 2 2 a = a, b = b. If T(e2) = α f1 +β f2 +γ f3, then T(e2) = T(e1) = (T(e2)) is equivalent to   α2 − β2 − γ2 = 1  bαβ − cαγ = 0   cαβ + bαγ = 0. The last two equations imply αβ = αγ = 0. Since the first equation excludes the possibility α = 0, the condition α , 0 implies β = γ = 0 and T(αe1 − e2) = 0 (what contradicts the injectivity of T). In a similar way it is proved the following Proposition. Proposition 4.2. The set of algebras belonging to classes C2 and C3 have the properties: (i) Algebras A(b, 0, ε) and A(b0, 0, ε) with bb0 , 0 are isomorphic if and only if b = b0. (ii) Any two algebras A(b, 0, 1) and A(b0, 0, −1) with bb0 , 0 are not isomorphic. Therefore, these results give the classification up to an isomorphism of the real 3-dimensional NN-algebras which have a nonzero derivation. More exactly, it was obtained the following result. Theorem 4.1. If A(·) is a real 3-dimensional commutative NN-algebra which has a nonzero derivation, then there exists a basis in A such that its multiplication table has the form corresponding to one of the following nine classes of non-isomorphic algebras: 1◦ A(b, c, 1) with b, c ∈ R and c > 0, 2◦ A(b, c, −1) with b, c ∈ R and c > 0, ◦ 1 3 A(b, 0, 1) with b ∈ R, b , 0 and b ≤ 2, ◦ 1 4 A(b, 0, −1) with b ∈ R \{1} and b ≥ 2, ◦ 1 5 A(b, 0, 1) with b ∈ R \{1} and b > 2, ◦ 1 6 A(b, 0, −1) with b ∈ R, b , 0 and b < 2, Automorphisms and derivations of homogeneous quadratic differential systems 23

7◦ A(1, 0, 1), 8◦ A(1, 0, −1), 9◦ A(0, 0, 1). Proposition 4.3. Every algebra A(b, c, ε), with b2 + c2 , 0, is simple.

Proof. Let I ⊂ A be a nonzero ideal and v = xe1 + ye2 + ze3 ∈ I, v , 0; then the elements v · e1 = xe1 + (by − cz)e2 + (cy + bz)e3, v · e2 = εye1 + bxe2 + cxe3, v · e3 = εze1 − cxe2 + bxe3, 2 2 2 2 2 belong to I. Consequently, if ∆ = x(b + c )[x − ε(y + z )] is nonzero then v · e1, v · e2 and v · e3 are linearly independent and, necessarily, I ≡ A. If x = 0, then the equalities v · e2 = εye1, v · e3 = εze1 and v , 0 imply e1, e1 · e2, e1 · e3 ∈ I, i.e. I ≡ A. In the case when x , 0 and ax2 − ε(y2 + z2) = 0, the linear independence of vectors v · e2, v · e3 and (v · e2) · e3 = εcxe1 + εy(−ce2 + be3)(∈ I) is equivalent 2 2 2 to condition ∆1 = εx(b + c )[cx − εyz] , 0; similarly, the linear independence of vectors v · e2, v · e3 and (v · e3) · e2 = −εcxe1 + εz(be2 + ce3)(∈ I) is equivalent 2 2 2 2 2 to condition ∆2 = −εx(b + c )[cx + εyz] , 0. Therefore, if ∆1 + ∆2 , 0 then 2 2 2 I ≡ A. If ∆1 + ∆2 = 0, then cx = εyz = 0 imply necessarily c = 0 and b , 0. Thus, the complementary cases y = 0 and, respectively, y , 0 must be considered. 1 When y = 0 (necessarily, c = 0 and b , 0), then e = v · e ∈ I, e · e ∈ I and 2 bx 2 1 2 1 e = (v · e ) · e ∈ I, e · e ∈ I, i.e. I ≡ A. In its turn, y , 0 implies z = 0, 1 εbx 2 2 1 3 1 1 i.e. v = xe + ye . Then e = v · e ∈ I, e · e ∈ I and e = (v · e ) · e ∈ I, 1 2 3 bx 3 1 3 1 εbx 3 3 e1 · e3 ∈ I, i.e. I ≡ A.

2 NOTE. If b = c = 0 then A = Re1 is a nontrivial ideal. Proposition 4.4. The algebras A(1, 0, ε) are unitary and power-associative as well.

According to Theorem 4.2.2 [3], Der A = RD. Therefore, {etD | t ∈ R} is a uniparametric group of automorphisms for A(·); it is isomorphic to the matrix group     1 0 0       0 cos θ − sin θ  θ ∈ [0, 2π) . Certainly, Aut A do not contain other uni-     0 sin θ cos θ  parametric subgroup of automorphisms. The problem is to establish whether other new automorphisms of A(·) exist or do not exist. The answer is contained in the next Proposition. Proposition 4.5. Aut A coincides with the matrix group     1 0 0       0 cos θ − sin θ  θ ∈ [0, 2π) .     0 sin θ cos θ  24 Ilie Burdujan

Recall that the trace form of algebra A(·) is the bilinear form g : A×A → R deﬁned by

g(v, w) = trace Lv·w for all v, w ∈ A.

Let us consider v = x1e1 + x2e2 + x3e3 and w = y1e1 + y2e2 + y3e3 in A. Since   x εx εx  1 2 3   bx − cx bx −cx  Lv =  2 3 1 1  , cx2 + bx3 cx1 bx1 v · w = [x1y1 + ε(x2y2 + x3y3)]e1 + [b(x1y2 + x2y1) − c(x1y3 + x3y1)]e2 + [c(x1y2 + x2y1) + b(x1y3 + x3y1)]e3 and trace Lv = (2b + 1)x1, it results that

g(v, w) = (2b + 1)[x1y1 + ε(x2y2 + x3y3)]. This bilinear form vanishes identically when b = −1/2 and it is nondegenerate whenever b , −1/2. In what follows we try to solve the HQDSs corresponding to these NN-algebras. The before presented results assures that, for a HQDS with three unknown functions whose associated algebra is an NN-algebra with a nonzero derivation, which belongs to class C1, there exists a change of variables (i.e., of unknown functions) such that the system becomes  dx  1 = x2 + ε(x2 + x2)  dt 1 2 3   dx2  = 2bx1 x2 − 2cx1 x3 (10)  dt  dx  3 = 2cx x + 2bx x dt 1 2 1 3 with ε = ±1 and b, c ∈ R and c > 0. The next equation is obtained adding the last two equations, after their multiplication with x2 and respectively x3,

d(x2 + x2) 2 3 = 4bx (x2 + x2). (11) dt 1 2 3

2 2 x2 Using the change of unknown functions y2 = x + x , z2 = , the system (5) 2 3 x3 becomes  dx  1 = x2 + εy  dt 1 2   dy2  = 4bx1y2 (12)  dt  dz  2 = −2cx (1 + z2). dt 1 2

This last system allows us to determine the following independent prime integrals for Automorphisms and derivations of homogeneous quadratic diﬀerential systems 25

(5): ! x2 2bε 1 − 2b f (x , x , x ) = ln 2b 1 + − · ln x2 + x2 , 1 1 2 3 x2 + x2 1 − 2b 2b 2 3 2 3 2 2 x2 f2(x1, x2, x3) = c · ln x + x + 2b · arctg 2 3 x3 if b < {0, 1/2}. In the case b = 0, c > 0, ε = 1 the system (5) has the following two independent prime integrals: 2 2 f1(x1, x2, x3) = x2 + x3, 1 x1 f2(t, x1, x2, x3) = q · arctg q − t. 2 2 2 2 x2 + x3 x2 + x3 If b = 0, c > 0, ε = −1 then the system (5) has the following two independent prime integrals: f (x , x , x ) = x2 + x2, 1 1 2 3 2 3 q 2 2 1 x1 − x2 + x3 f2(t, x1, x2, x3) = 2 2 · ln q − t. 2(x2 + x3) 2 2 x1 + x2 + x3 In case b = 1/2, c > 0, ε = 1, system (5) has the following prime integrals:

2 2 x2 f1(x1, x2, x3) = c ln (x + x ) + arctg , 2 3 x3 2 x1 2 2 2 2 x2 + x3 f2(x1, x2, x3) = (x2 + x3) e . Finally, in the case b = 1/2, c > 0, ε = −1 the system (5) has the following two independent prime integrals:

2 2 x2 f1(x1, x2, x3) = c ln(x + x ) + arctg , 2 3 x3 2 x1 2 2 f2(x1, x2, x3) = 2 2 + ln(x2 + x3). x2 + x3

The HQDS corresponding to an algebra of type C2 or C3 has the form  dx  1 = x2 + ε(x2 + x2)  dt 1 2 3   dx2  = 2bx1 x2 (13)  dt  dx  3 = 2bx x . dt 1 3 It has the following two independent prime integrals:

x2 f1(x1, x2, x3) = , hx3 i 2 2 2 −1/b f2(x1, x2, x3) = (1 − 2b)x1 + ε(x2 + x3) x3 . 26 Ilie Burdujan

Let us solve the Cauchy problem with the initial data x1(0) = α, x2(0) = β, x3(0) = γ. Considering the complex function z = x2 + ix3, the last two equations in (5) leads to dz = 2(b + ic)x z. dt 1 A straightforward computation gives us

2bθ(t) x2(t) = ke cos (2cθ(t) + ϕ0) 2bθ(t) x3(t) = ke sin (2cθ(t) + ϕ0)

γ R t where ϕ = arctg , k2 = β2 + γ2 and θ(t) = x (τ) dτ. 0 β 0 1 In its turn, the equation (6) gives

2 2 2 4bθ(t) x2 + x3 = k e .

Therefore, the first equation in (5) may be transformed into the following integro- differential equation: dx 1 = x2 + εk2e4bθ(t). (14) dt 1 Differentiating (14), it follows

2 d x1 dx1 3 − 2(1 + 2b)x1 + 4bx = 0. (15) dt2 dt 1

In the case b , 0, this nonlinear equation can be analysed by the method of Lie symmetries. System (5) can be solved in the case b = 0. Indeed, for ε = 1, the Cauchy problem, with the initial data x1(0) = α, x2(0) = β, x3(0) = γ, has the solution  ksin kt + αcos kt  x (t) = k  1 kcos kt − αsin kt  x (t) = k cos (2cθ(t) + ϕ ) (16)  2 0 x3(t) = k sin (2cθ(t) + ϕ0) 1 where θ(t) = − ln |a sin kt − k cos kt|. In the case ε = −1, the solution is k   k sinh kt + α cosh kt  x (t) = k  1 k cosh kt − α sinh kt  (17)  x2(t) = k cos (2cθ(t) + ϕ0)  x3(t) = k sin (2cθ(t) + ϕ0), Automorphisms and derivations of homogeneous quadratic diﬀerential systems 27 where

kx kx ln tanh − 1 ln tanh + 1 α 2 α 2 θ(t) = − 1 + − 1 − + k k − α k k + α !2 kx kx ln k tanh − 2a tanh + k α2 2 2 (k2 + α2) ln k +( + k) − . k (k + α)(k − α) k(k2 − α2)

According to Proposition 1.3 [7], the orbit etD(P) of P is a solution to HQDS (5) if and only if P is a nonzero solution of equation P2 = DP. Let us ﬁnd the solution of HQDS (5) that are orbits of the Lie group Aut A. If 2 P = α e1 + β e2 + γ e3, then the equation P = DP becomes   α2 + ε(β2 + γ2) = 0  2bαβ − 2cαγ = −γ (18)   2cαβ + 2bαγ = β. Equation (18) has a nonzero solution when ε = −1, only. Any such a solution must have α , 0. Therefore, the last two equations in (18) can be written in the form ( 2bα β − (2cα − 1)γ = 0 (2cα − 1)β + 2bα γ = 0. This homogeneous system, with the unknowns β and γ, has a nonzero solution if and only if 2bα −(2cα − 1) = 0 ⇔ 4b2α2 + (2cα − 1)2 = 0. 2cα − 1 2bα

2 1 Therefore, the solutions P to DP = P in A(·) lie in the plane x1 = −2c on the circle

1 1 with center (−2c, 0, 0) and radius R = −2c . More exactly, they are 1 P(ω) = − e + R[e cos ω + e sin ω]. 2c 1 2 2 Then, the solution which is the trajectory through P(ω) is 1 X (t) = etDP(ω) = − e + R[e cos (t + ω) + e sin (t + ω)]. ω 2c 1 2 3

It is a periodic solution of the least period 2π. Moreover, Xω(t) is isolated in the set of all periodic solutions with period 2π. Following Kinyon& Sagle [6], let us denote by S 1(c) and S 2(c) the sets S 1(c) = tD {v |trace Lv = c} and S 2(c) = {v |trace Lv2 = c}, respectively. Then, the solution e P 28 Ilie Burdujan is on the plane S 1(c), where c = trace LP. Moreover, any periodic solution lies on S 1(c) ∩ S 2(0), what agrees with Proposition 5.23 [6]. d2 x dx NOTE. Equation (15) is of the form 1 + f (x ) 1 + g(x ) = 0 which contains dt2 1 dt 1 as particular cases van der Pol’s equation and Duﬃng’s equation, as well. We shall study it in a forthcoming paper.

References

[1] B G., Osborn M. J., The derivation algebra of a real division algebra, Amer. J. of Math., 103, 6(1981), 1135–1150. [2] B I., On derivation algebra of a real algebra without nilpotents of order two, Ital. J. of Pure and Applied Math., 8(2000), 137–154. [3] B I., Quadratic differential systems, 2008, Ed. PIM-Ias¸i. (in Romanian) [4] B I., Infinitesimal groups associated with quadratic dynamical systems, ROMAI Journal, 1, 1(2005), 37–42. [5] D T., Classifications and Analysis of Two-dimensional Real Homogeneous Quadratic Differ- ential Equation Systems, J. Diff. Eqs., 32(1979), 311–334. [6] K K. M., S A. A., Quadratic Dynamical Systems and Algebras, J. of Diff. Eqs, 117(1995), 67–127 . [7] K K. M., S A. A., Automorphisms and Derivations of Differential Equations and Alge- bras, Rocky Mountain J. of Math., 24, 1(1994), 135–153. [8] M L., Quadratic Differential Equations and Non-associative Algebras, in ”Contributions to the Theory of Nonlinear Oscillations”, Annals of Mathematics Studies, 45, Princeton University Press, Princeton, N. Y., 1960. [9] R¨ H., Algebras and differential equations, Nagoya Math. J., 68(1977), 59–122. [10] S, A. A., W, R. E., Introduction to L groups and L algebras, Academic Press, New York and London, 1973. [11] S,D, Algebraic particular integrals, integrability and the problem of the center, Trans. A.M.S., 338(1979), 799–841. [12] S,D, Basic Algebro-Geometric concepts in the study of planar polynomial vector fields, Publicacions Mathematiques,` 41(1997), 269–295. [13] V, N. I.,S˘ı, K. S., Geometrical Classification of Quadratic differential systems , Dif- ferentialnye Uravnenje, 13, 5(1977), 803–814. (in Russian) [14] W, S., Algebras and differential equations, Hadronic Press, Palm Harbor, 1991. ROMAI J., 6, 1(2010), 29–40

ON RADII OF STARLIKENESS AND CLOSE-TO-CONVEXITY OF A SUBCLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS Adriana C˘ata¸s Faculty of Sciences, University of Oradea, Romania [email protected] Abstract By making use of a multiplier transformation, a subclass of p-valent functions in the open unit disc is introduced. The main results of the present paper provide various interesting properties of functions belonging to the new subclass. Some of these properties include, for example, several coeﬃcient inequalities and distortion bounds for the function class which is considered here. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided.

Keywords: analytic function, multiplier transformations, coeﬃcient inequalities, distortion bounds, radii of starlikeness and close-to-convexity. 2000 MSC: 30C45.

1. PRELIMINARIES Let H be the class of analytic functions in the open unit disc

U = {z ∈ C : |z| < 1}

n and H[a, n] be the subclass of H consisting of functions of the form f (z) = a+anz + n+1 an+1z + ··· . Let A(p, n) denote the class of normalized functions f (z) of the form X∞ p k f (z) = z + akz , (p, n ∈ N := {1, 2, 3,... }) (1) k=p+n which are analytic in the open unit disc. In particular, we set

A(p, 1) := Ap and A(1, 1) := A = A1.

For two functions f (z) given by (1) and g(z) given by

X∞ p k g(z) = z + bkz , (p, n ∈ N) (2) k=p+n

29 30 Adriana C˘ata¸s the Hadamard product (or convolution) ( f ∗ g)(z) is defined, as usual, by X∞ p k ( f ∗ g)(z):= z + akbkz := (g ∗ f )(z). (3) k=p+n Definition 1.1. [3] Let f ∈ A(p, n). For δ ∈ R, δ ≥ 0, l ≥ 0, we define the multiplier transformations Ip(δ, l) on A(p, n) by the following infinite series " # X∞ k + l δ I (δ, l) f (z):= zp + a zk. (4) p p + l k k=p+n It follows from (4) that

0 (p + l)Ip(δ + 1, l) f (z) = l · Ip(δ, l) f (z) + z(Ip(δ, l) f (z)) . (5) If f is given by (1) then we have

δ Ip(δ, l) f (z) = ( f ∗ ϕp,l)(z), (6) where " # X∞ k + l δ ϕδ (z) = zp + zk. (7) p,l p + l k=p+n Let T(n, p) denote the class of functions f (z) of the form X∞ p k f (z) = z − akz , ak ≥ 0, p, n ∈ N, (8) k=n+p which are analytic in the open unit disc. Let A(n) be the class of functions of the form X∞ k f (z) = z − akz , ak ≥ 0, n ∈ N, k=n+1 ∗ which are analytic in the open unit disc. Let Sn(α) denote the subclass of A(n) consisting of functions which satisfy ! z f 0(z) Re > α, z ∈ U, 0 ≤ α < 1. f (z)

∗ A function f (z), in Sn(α) is said to be starlike of order α in U. A function f (z) ∈ A(n) is said to be convex of order α if it satisﬁes ! z f 00(z) Re 1 + > α, z ∈ U, 0 ≤ α < 1. f 0(z) On radii of starlikeness and close-to-convexity of a subclass of analytic functions... 31

Let Cn(α) be the subclass of A(n) consisting of all convex of order α functions [2]. ∗ We denote by Sn(p, α) and Cn(p, α) the classes of p-valently starlike functions of order α in U, 0 ≤ α α, z ∈ U, 0 ≤ α α, z ∈ U, 0 ≤ α < p . (10) n f 0(z) ∗ An interesting uniﬁcation of the function classes Sn(p, α) and Cn(p, α) is provided by the class Tn(p, α, γ) of functions f (z) ∈ T(n, p), which also satisfy the following inequality ! z f 0(z) + γz2 f 00(z) Re > α, z ∈ U, 0 ≤ α < p, 0 ≤ γ ≤ 1. (11) γz f 0(z) + (1 − γ) f (z)

The class Tn(p, α, γ) was investigated by Alintas¸et al. [1]. 2. COEFFICIENT INEQUALITIES In this section we will define a new class of p-valently starlike functions by using the multiplier transformations Ip(m, l), m ∈ N, l ≥ 0 as in (4) and we will establish certain coefficient inequalities relating to the new introduced class. Definition 2.1. Let 0 ≤ α < p, 0 ≤ γ ≤ 1, m ∈ N, l ≥ 0, p ∈ N∗. A function f m belonging to T(n, p) is said to be in the class Tl (n, p, α, γ) if and only if ( 0 0 ) (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z)) Re > α, z ∈ U. (12) (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z)) m Remark 2.1. The class Tl (n, p, α, γ) is a generalization of the subclasses 0 ∗ ∗ 1 i) T0(1, 1, α, 0) ≡ T (α) ≡ S1(α) and T0(1, 1, α, 0) ≡ C(α) ≡ C1(α) defined and studied by Silverman [6]; 0 1 ii) T0(n, 1, α, 0) and T0(n, 1, α, 0) studied by Chatterjea [4] and Srivastava et al. [7]; m iii) T0 (n, p, α, γ) studied by Kamali [5]. Theorem 2.1. Let the function f be defined by (8). Then f belongs to the class m Tl (n, p, α, γ) if and only if X∞ cp,k(m, l)(k − α)[p + l + γ(k − p)]ak ≤ (p + l)(p − α). (13) k=n+p 32 Adriana C˘ata¸s where " # k + l m c (m, l) = . (14) p,k p + l The result is sharp and the extremal functions are

p (p + l)(p − α) k fk(z) = z − · z , k ≥ n + p. (15) cp,k(m, l)(k − α)[p + l + γ(k − p)]

Proof. Assume that the inequality (4) holds and let |z| = 1. Then we have

0 0 (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z)) − p = (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z))

X∞ " #m " # k + l p + l + γ(k − p) k−p (k − p)akz p + l p + l k=n+p

= ∞ " # " # X k + l m p + l + γ(k − p) 1 − a zk−p p + l p + l k k=n+p " # " # X∞ k + l m p + l + γ(k − p) ka − p p + l p + l k k=n+p ≤ p + " # " # ≤ p − α. X∞ k + l m p + l + γ(k − p) 1 − a p + l p + l k k=n+p Consequently, by the maximum modulus theorem one obtains

m f (z) ∈ Tl (n, p, α, γ).

m Conversely, suppose that f (z) ∈ Tl (n, p, α, γ). Then from (4) we ﬁnd that  " # " #   X∞ m   p k + l p + l + γ(k − p) k   pz − kakz   p + l p + l   k=n+p  Re  " # " #  > α.  X∞ m   p k + l p + l + γ(k − p) k   z − akz   p + l p + l  k=n+p

Choose values of z on the real axis such that

0 0 (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z)) (1 − γ)z(Ip(m, l) f (z)) + γz(Ip(m + 1, l) f (z)) On radii of starlikeness and close-to-convexity of a subclass of analytic functions... 33 is real. Letting z → 1− through real values, we obtain

 X∞ " #m " #   k + l p + l + γ(k − p)   p − kak   p + l p + l   k=n+p  Re ≥ α  X∞ " #m " #   k + l p + l + γ(k − p)   1 − ak   p + l p + l  k=n+p or, equivalently " # " # X∞ k + l m p + l + γ(k − p) p − ka p + l p + l k k=n+p    ∞ " #m " #   X k + l p + l + γ(k − p)  ≥ α 1 − ak  p + l p + l   k=n+p  which gives (27).

m Theorem 2.2. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then

X∞ (p + l)(p − α) a ≤ (16) k c (m, l)(p + l + γn)(n + p − α) k=n+p p,n+p and X∞ (p + l)(p − α)(n + p) ka ≤ . (17) k c (m, l)(p + l + γn)(n + p − α) k=n+p p,n+p The equality in (16) and (17) is attained for the function f given by (15). Proof. By using Theorem 2.1, we ﬁnd from (4) that X∞ (p + l + γn)(n + p − α)cp,n+p(m, l) ak k=n+p

X∞ ≤ cp,k(m, l)(k − α)[p + l + γ(k − p)]}ak ≤ (p + l)(p − α), k=n+p which immediately yields the ﬁrst assertion (16) of Theorem 2.2. On the other hand, taking into account the inequality (4), we also have X∞ (p + l + γn)cp,n+p(m, l) (k − α)ak ≤ (p + l)(p − α) k=n+p 34 Adriana C˘ata¸s that is X∞ (p + l + γn)cp,n+p(m, l) kak k=n+p X∞ ≤ (p + l)(p − α) + α(p + l + γn)cp,n+p(m, l) ak k=n+p which, in view of the coeﬃcient inequality (16), can be put in the form X∞ (p + l + γn)cp,n+p(m, l) kak k=n+p

(p + l)(p − α) ≤ (p + l)(p − α) + α(p + l + γn)cp,n+p(m, l) cp,n+p(m, l)(p + l + γn)(n + p − α) and this completes the proof of (17).

3. DISTORTION THEOREMS m Theorem 3.1. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then we have

p (p + l)(p − α) n+p |Ip(i, l) f (z)| ≥ |z| − · |z| (18) cp,k(m − i, l)(n + p − α)(p + l + γn) and p (p + l)(p − α) n+p |Ip(i, l) f (z)| ≤ |z| + · |z| (19) cp,k(m − i, l)(n + p − α)(p + l + γn) for z ∈ U, where 0 ≤ i ≤ m and cp,k(m − i, l) is given by (14). The equalities in (5) and (6) are attained for the function f given by

(p − α)(p + l)m+1 f (z) = zp − zn+p. (20) n+p (p + n + l)m(n + p − α)(p + l + γn)

m m−i Proof. Note that f ∈ Tl (n, p, α, γ) if and only if Ip(i, l) f (z) ∈ Tl (n, p, α, γ), where X∞ p k Ip(i, l) f (z) = z − cp,k(i, l)akz . (21) k=n+p

By Theorem 2.1, we know that X∞ cp,k(m − i, l)(n + p − α)(p + l + γn) cp,k(i, l)ak ≤ k=n+p On radii of starlikeness and close-to-convexity of a subclass of analytic functions... 35

X∞ ≤ cp,k(m, l)(k − α)[p + l + γ(k − p)]ak ≤ (p + l)(p − α) k=n+p that is X∞ (p + l)(p − α) c (i, l)a ≤ . (22) p,k k c (m − i, l)(n + p − α)(p + l + γn) k=n+p p,k The assertions of (5) and (6) of Theorem 3.1 follow immediately. Finally, we note that the equalities (5) and (6) are attained for the function f deﬁned by

p (p + l)(p − α) n+p Ip(i, l) f (z) = z − z . (23) cp,k(m − i, l)(n + p − α)(p + l + γn) This completes the proof of Theorem 3.1.

m Corollary 3.1. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then we have (p + l)(p − α) | f (z)| ≥ |z|p − |z|n+p (24) cp,k(m, l)(n + p − α)(p + l + γn) and (p + l)(p − α) | f (z)| ≤ |z|p + |z|n+p (25) cp,k(m, l)(n + p − α)(p + l + γn) for z ∈ U. The equalities in (24) and (25) are attained for the function fn+p given in (7).

m Corollary 3.2. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then we have (p + l)(p − α)(n + p) | f 0(z)| ≥ p|z|p−1 − |z|n+p−1 (26) cp,k(m, l)(n + p − α)(p + l + γn) and (p + l)(p − α)(n + p) | f 0(z)| ≤ p|z|p−1 + |z|n+p−1 (27) cp,k(m, l)(n + p − α)(p + l + γn) for z ∈ U. The equalities in (26) and (27) are attained for the function fn+p given in (7).

m Corollary 3.3. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then the unit disc is mapped onto a domain that contains the disc

cp,k(m, l)(n + p − α)(p + l + γn) − (p + l)(p − α) |w| < . (28) cp,k(m, l)(n + p − α)(p + l + γn)

The result is sharp with the extremal function fn+p given in (7). 36 Adriana C˘ata¸s 4. CLOSURE THEOREMS

Let the functions fi be deﬁned for i = 1, 2,..., s, by X∞ p k fi(z) = z − ak,iz , ak,i ≥ 0, n, p ∈ N, z ∈ U. (29) k=n+p

m We shall prove the following results for the closure of the class Tl (n, p, α, γ). m Theorem 4.1. Let the functions fi defined by (8) be in the class Tl (n, p, α, γ), for every i = 1, 2,..., s. Then the functions h defined by Xs h(z) = di fi(z), di ≥ 0 (30) i=1 m is also in the same class Tl (n, p, α, γ) if Xm di = 1. (31) i=1 Proof. According to the definition of h, we can write   X∞ Xs  p   k h(z) = z −  diak,i z . k=n+p i=1

m Further, since fi are in the class Tl (n, p, α, γ) for every i = 1, 2,..., s we get X∞ cp,k(m, l)(k − α)[p + l + γ(k − p)]ak,i ≤ (p + l)(p − α), k=n+p where cp,k(m, l) is given by (28). Hence we can see that   X∞ Xs    cp,k(m, l)(k − α)[p + l + γ(k − p)]  diak,i = k=n+p i=1   Xs  X∞    = di  cp,k(m, l)(k − α)[p + l + γ(k − p)]ak,i ≤ i=1 k=n+p Xs ≤ (p + l)(p − α) di = (p + l)(p − α), i=1 m which implies that h is in the class Tl (n, p, α, γ). On radii of starlikeness and close-to-convexity of a subclass of analytic functions... 37

m Corollary 4.1. The class Tl (n, p, α, γ) is closed under a convex linear combination. m Proof. Let the function fi, i = 1, 2, be in the class Tl (n, p, α, γ). It is suﬃcient to show that the function h deﬁned by

h(z) = µ f1(z) + (1 − µ) f2(z), 0 ≤ µ ≤ 1 (32) m is in the class Tl (n, p, α, γ). But, taking s = 2, d1 = µ and d2 = 1 − µ in Theorem m 4.1, one obtains that Tl (n, p, α, γ) is a convex set. Remark 4.1. As a consequence of Theorem 4.1, there exist the extreme points of the class T j(n, γ, α, λ).

Theorem 4.2. Let fk be deﬁned as in (15), k ≥ n + p and p fn+p−1 = z , z ∈ U. (33) m Then f is in the class Tl (n, p, α, γ) if and only if it can be expressed in the form X∞ f (z) = µk fk(z) (34) k=n+p−1 X∞ where µk ≥ 0 (k ≥ n + p − 1) and µk = 1. k=n+p−1 Proof. Suppose that f can be expressed as in (34), namely

X∞ (p + l)(p − α) f (z) = zp − µ · zk. k c (m, l)(k − α)[p + l + γ(k − p)] k=n+p p,k Then it follows that X∞ cp,k(m, l)(k − α)[p + l + γ(k − p)]× k=n+p (p + l)(p − α) × = cp,k(m, l)(k − α)[p + l + γ(k − p)] X∞ = (p + l)(p − α) µk ≤ (p + l)(p − α) k=n+p m and by Theorem 2.1, f belongs to the class Tl (n, p, α, γ). m Conversely, assume that the function f belongs to the class Tl (n, p, α, γ). Then ∞ X cp,k(m, l)(k − α)[p + l + γ(k − p)] a ≤ 1. (p + l)(p − α) k k=n+p 38 Adriana C˘ata¸s

We denote by

cp,k(m, l)(k − α)[p + l + γ(k − p)] µ = a , k ≥ n + p k (p + l)(p − α) k and X∞ µn+p−1 = 1 − µk. k=n+p We notice that f can be expressed in the form (34). This completes the proof of Theorem 4.2.

m Corollary 4.2. The extrem points of the class Tl (n, p, α, γ) are the functions fk, k ≥ n + p, n, p ∈ N given by Theorem 4.2.

5. RADII OF CLOSE-TO-CONVEXITY, STARLIKENESS AND CONVEXITY m Theorem 5.1. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then f is close-to-convex of order δ (0 ≤ δ < p), hence univalent, in the disc |z| < r1, where

" # 1 (p − δ)cp,k(m, l)(k − α)[p + l + γ(k − p)] k−p r1 = inf , k ≥ n + p. (35) k (p + l)(p − α)k

The result is sharp with the extremal function f given by (15).

f 0(z) Proof. It is suﬃcient to show that zp−1 − p ≤ p − δ, |z| < r1 where r1 is given by (35). From (8) we have

f 0(z) X∞ − p ≤ ka |z|k−p. zp−1 k k=n+p

f 0(z) Thus zp−1 − p ≤ p − δ if ! X∞ k a |z|k−p < 1. (36) p − δ k k=n+p

But Theorem 2.1 conﬁrms that X∞ cp,k(m, l)(k − α)[p + l + γ(k − p)] a ≤ 1, (37) (p + l)(p − α) k k=n+p where cp,k(m, l) is given by (28). On radii of starlikeness and close-to-convexity of a subclass of analytic functions... 39

Hence (36) will be true if ! k cp,k(m, l)(k − α)[p + l + γ(k − p)] |z|k−p < . p − δ (p + l)(p − α) That is " # 1 (p − δ)cp,k(m, l)(k − α)[p + l + γ(k − p)] k−p |z| < . (38) (p + l)(p − α)k Theorem 5.1 follows easily from (38). m Theorem 5.2. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then f is starlike of order δ, 0 ≤ δ < p, hence univalent, in the disc |z| < r2 where " # 1 (p − δ)cp,k(m, l)(k − α)[p + l + γ(k − p)] k−p r2 = inf , k ≥ n + p. (39) k (p + l)(p − α)(k − δ) The result is sharp with the extremal function f given by (15).

z f 0(z) Proof. It is suﬃcient to show that f (z) − p < p − δ, |z| < r2 where r2 is given in (39). If f is given by (8), then X∞ (k − p)a |z|k−p k z f 0(z) k=n+p − p ≤ . f (z) X∞ k−p 1 − ak|z| k=n+p Thus, X∞ k − δ a |z|k−p < 1. (40) p − δ k k=n+p Hence, by using (37), the inequality (40) will be true if

" # 1 (p − δ)cp,k(m, l)(k − α)[p + l + γ(k − p)] k−p |z| < . (41) (p + l)(p − α)(k − δ) and this completes the proof of Theorem 5.2. m Corollary 5.1. Let the function f deﬁned by (8) be in the class Tl (n, p, α, γ). Then f is convex of order δ, 0 ≤ δ < p (hence f is univalent), in the disc |z| < r3 where " # 1 p(p − δ)cp,k(m, l)(k − α)[p + l + γ(k − p)] k−p r3 ≤ inf , k ≥ n + p. (42) k k(p + l)(p − α)(k − δ) and cp,k(m, l) is given by (14), k ≥ n + p. 40 Adriana C˘ata¸s References

[1] Alintas¸, O., Irmak, H., Srivastava, H.M., Fractional calculus and certain starlike functions with negative coefficietns, Comput. Math. Appl., 30(2)(1995), 9-15. [2] Alintas¸, O., Owa, S., Neighborhoods of certain analytic functions with negative coefficient, In- ternat. J. Math. and Math. Sci., 19(1996), 797-800. [3] Catas¸,˘ A., On certain class of p-valent functions defined by new multiplier transformations, Pro- ceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [4] Chatterjea, S.K., On starlike functions, J. Pure Math., 1(1981), 23-26. [5] Kamali, M., Neighborhoods of a new class of p-valently functions with negative coefficients, Math. Ineq. Appl., 9, 4(2006), 661-670. [6] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1975), 109-116. [7] Srivastava, H.M., Owa, S., Chatterjea, S.K., A note on certain classes of starlike functions, Rend. Sem. Mat. Univ. Padova, 77(1987), 115-124. ROMAI J., 6, 1(2010), 41–45

PROXIMAL POINT METHODS FOR VARIATIONAL INEQUALITIES INVOLVING REGULAR MAPPINGS Corina L. Chiriac Department of Mathematics, Bioterra University, Bucharest, Romania [email protected] Abstract In this paper we consider the following general version of the proximal point algorithm for solving a variational inequality, namely: ﬁnd x ∈ C such that hF(x), u − xi ≥ 0 for all u ∈ C,where F : X → X∗, X is a Banach space with its dual X∗ and C ⊂ X a ∗ nonempty, closed, convex set. First, choose any sequence of functions fn : X → X that are Lipschitz continuous. Then pick an initial element x0 and ﬁnd xn+1 ∈ C such that

fn(xn+1 − xn) + F(xn+1) + NC(xn+1) 3 0 for n = 0, 1, 2, ...

where NC is the normal cone mapping of C. We prove that if the Lipschitz constant of fn is bounded by half the reciprocal of the modulus of regularity of F + NC , then there exists a neighborhood V of x (x being a solution of the variational inequality) such that for each initial point x0 ∈ V one can ﬁnd a sequence {xn} generated by the algorithm which is linearly convergent to x.

Keywords: variational inequality, proximal point algorithm, regular mappings. 2000 MSC: 47J20, 47J25, 58C30.

1. INTRODUCTION In this paper we study the convergence of a general version of the proximal point algorithm for solving the variational inequality problem: ﬁnd x ∈ C such

hF(x), u − xi ≥ 0 (1) for all u ∈ C, where F : X → X∗, X is a Banach space with its dual X∗ and C ⊂ X ∗ a nonempty closed and convex set. Choose a sequence of functions fn : X → X with fn(0) = 0 and consider the following algorithm: given x0 ﬁnd a sequence xn by applying the iteration

fn(xn+1 − xn) + F(xn+1) + NC(xn+1) 3 0 (2) for n = 0, 1, 2, ... . We prove in this work that if x is a solution of (1) and the mapping T = F + NC is

41 42 Corina L. Chiriac metrically regular at x for 0 and with locally closed graph near (x, 0), then, for any sequence of functions fn that are Lipschitz continuous in a neighborhood U of the origin, the same for all n, and whose Lipschitz constants ln have supremum that is bounded by half the reciprocal of the modulus of regularity of T, there exists a neighborhood V of x such that for each initial point x0 ∈ V one can ﬁnd a sequencee xn satisfying (2) which is linearly convergent to x in the norm of X. If fn(x) = αn x and fn : X → X, we obtain the classical proximal point algorithm applied for solving variational inequality

αn(xn+1 − xn) + T(xn+1) 3 0 (3) for n = 0, 1, 2, ... proposed by Rockafellar [6] for the case when X is a Hilbert space and F is a monotone mapping. In particular, Rockafellar (see [6], Proposition 3) showed that when xn+1 is an approximate solution of (3) and F is maximal monotone, then for a sequence of positive scalars αn the iteration (3) produces a sequence xn which is weakly convergent to a solution to (1) for any starting point x0 ∈ C. In the last thirty years a number of authors have studied generalizations of the proximal point algorithm with applications to speciﬁc variational inequalities. We mention here the papers by Solodov and Svaiter [7], Auslender and Teboulle [1], Ka- plan and Tichatschke [5]. In this paper we consider the proximal point method, by employing recent de- velopments on regularity properties of mappings most of which can be found in the paper [4]. In Section 2 we present some background material on metric regularity. Section 3 gives a statement and a proof of the convergence result.

2. METRIC REGULARITY Let X and Y be Banach spaces, let G be a set-valued mapping G : X → 2Y and let (x, y) ∈ gphG. Here gphG = {(x, y) ∈ X × Y|y ∈ G(x)} is the graph of G. We denote by d(x, K) the distance from a point x to a set K, that is, d(x, K) = in fy∈K kx − yk. −1 Br(z) denotes the closed ball of radius r centered at z and G is the inverse of G deﬁned as x ∈ G−1(y) ⇔ y ∈ G(x).

Deﬁnition 2.1. The mapping G is said to be metrically regular at x for y if there exists a constant k > 0 such that

d(x, G−1(y)) ≤ kd(y, G(x)), for all (x, y) close to (x, y). (4)

The inﬁmum of k for which (2.1) holds is the regularity modulus denoted reg G(x|y). The case when G is not metrically regular at x for y corresponds to regG(x|y) = ∞. Proximal point methods for variational inequalities involving regular mappings 43

An important result in the theory of metric regularity is the Lyusternik-Graves theorem that says that the metric regularity is stable under perturbations of order higher than one. Deﬁnition 2.2. A set K ⊂ X is locally closed at z ∈ K if there exists a > 0 such that the set K ∩ Ba(z) is closed. Theorem 2.1. (Lyusternik-Graves [3]) Consider a mapping G : X → 2Y and any (x, y) ∈ gphG at which gphG is locally closed. Consider also a function g : X → Y which is Lipschitz continuous near x with a Lipschitz constant δ. If regG(x|y) < k < ∞ and δ < k−1, then reg(g + G)(x|g(x) + y) ≤ (k−1 − δ)−1. In the proof of our main result we used the following set-valued generalization of the Banach ﬁxed point theorem.

Theorem 2.2. [2] Let (X, ρ) be a complete metric space, and consider a set-valued mapping Φ : X → 2X, a point x ∈ X, and nonnnegative scalars α and θ be such that 0 ≤ θ < 1, the sets Φ(x) ∩ Ba(x) are closed for all x ∈ Ba(x) and the following conditions hold: (i) d(x, Φ(x)) < α(1 − γθ); (ii) e(Φ(u) ∩ Ba(x), Φ(v)) ≤ θρ(u, v) for all u, v ∈ Ba(x). Then Φ has a ﬁxed point in Ba(x), that is, there exists x ∈ Ba(x) such that x ∈ Φ(x).

In the above, e(A, B) = supx∈Ad(x, B). 3. CONVERGENCE OF THE PROXIMAL POINT ALGORITHM Here we present the proof of our main result. We observe that the variational inequality problem (1.1) is equivalent to the problem: ﬁnding x ∈ X such that

0 ∈ F(x) + NC(x) X∗ where NC : X → 2 is the normal cone mapping, given by ∗ 0 0 NC(x) = y ∈ X | for every x ∈ C, hx − x, yi ≤ 0 if x ∈ C; NC(x) = ∅ if x < C.

Theorem 3.1. Consider a mapping T = F + NC and let x be a solution of the variational inequality (1.1). Let gphT be locally closed at (x, 0) and let T be metrically ∗ regular at x for 0. Choose a sequence of functions fn : X → X with fn(0) = 0 which are Lipschitz continuous in a neighborhood U of 0, the same for all n, with Lipschitz constants ln satisfying 1 sup ln < . (5) n 2regT(x|0) 44 Corina L. Chiriac

Then there exists a neighborhood V of x such that for any x0 ∈ V there exists a sequence xn generated by the proximal point algorithm (2) which is linearly convergent to x.

Proof. Let l = supnln, then from (5) there exists k > regT(x|0) such that kl < 0.5. Then one can choose λ which satisﬁes ((kl)−1 − 1)−1 < λ < 1. Let r be such that the mapping T is metrically regular at x for 0 with a constant k and neighborhoods Br(x), B2lr(0) and B2r(0) ⊂ U. Let x0 ∈ Br(x) ∩ C. For any x ∈ Br(x) we have

k− f0(x − x0)k = k f0(x − x0) − f0(0)k ≤ l0 kx0 − xk ≤ 2rl0 ≤ 2lr. −1 We will show that the mapping Φ0(x) = T (− f0(x − x0)) satisﬁes the assumptions of the ﬁxed-point result in Theorem 2.2. First, by using the assumptions that T is metrically regular at x for 0, 0 ∈ T(x) and f0(0) = 0, we have −1 d(x, Φ0(x)) = d(x, T (− f0(x − x0))) ≤ kd(− f0(x − x0), T(x)) ≤ k k f0(0) − f0(x − x0)k ≤ kl0 kx0 − xk ≤ kl0r < r(1 − kl0).

For any u, v ∈ Br(x), by the metric regularity of T, −1 e(Φ0(u) ∩ Br(x) ∩ C, Φ0(v)) = sup d(x, T (− f0(v − x0))) ≤ −1 x∈T (− f0(u−x0))∩Br(x) sup kd(− f0(v − x0), T(x)) ≤ k k− f0(u − x0) − (− f0(v − x0))k ≤ −1 x∈T (− f0(u−x0))∩Br(x) kl0 ku − vk.

Hence there exists a ﬁxed point x1 ∈ Φ0(x1) ∩ Br(x) ∩ C,

x1 ∈ Br(x) ∩ C and 0 ∈ f0(x1 − x0) + T(x1). (6)

If x1 = x there is nothing more to prove. Assume x1 , x. For any x ∈ Br(x), we have k− f1(x − x1)k ≤ 2l1r ≤ 2lr. Now, we set

r1 = λ kx1 − xk . (7)

−1 Since λ < 1 we have r1 < r. Consider the mapping Φ1(x) = T (− f1(x − x1)). By (7) and the metric regularity of T −1 d(x, Φ1(x)) = d(x, T (− f1(x − x1))) ≤ kd(− f1(x − x1), T(x)) ≤ k k f1(0) − f1(x − x1)k ≤ kl1 kx1 − xk < r1(1 − kl1).

For any u, v ∈ Br1 (x), by the metric regularity of T, we obtain

−1 e(Φ1(u) ∩ Br(x) ∩ C, Φ1(v)) = sup d(x, T (− f1(v − x1))) ≤ −1 x∈T (− f1(u−x1))∩Br(x)

≤ sup kd(− f1(v − x1), T(x)) ≤ −1 x∈T (− f1(u−x1))∩Br(x) Proximal point methods for variational inequalities involving regular mappings 45

≤ k k− f1(u − x1) − (− f1(v − x1))k ≤ kl1 ku − vk .

Hence, by Theorem 2.2, there exists x2 ∈ Φ1(x1)∩Br(x)∩C which by (7), satisﬁes

kx2 − xk ≤ λ kx1 − xk .

By induction we deduce that if xn ∈ Br(x) ∩ C, then k− fn(x − xn)k ≤ 2lr. −1 For rn = λ kxn − xk by applying Theorem 2.2 to Φn(x) = T (− fn(x − xn)), we obtain the existence of xn+1 ∈ Brn (x) ∩ C such that

0 ∈ fn(xn+1 − xn) + F(xn+1) + NC(xn+1).

Thus we established that

kxn+1 − xk ≤ λ kxn − xk for all n. (8)

Since λ < 1, the sequence xn converges linearly to x. The proximal point algorithm (2) represents an iteratively applied perturbation of the mapping which is small enough to preserve the metric regularity of T. The possibility for choosing the sequence fn gives more freedom to enhance the convergence and T, for example, is metrically regular in the case: F is a continuous mapping and NC a closed, convex cone.

References

[1] A. Auslender, M. Teboulle, Lagrangian duality and related multiplier methods for variational inequality problems, SIAM J. Optim., 10(2000), 1097-1115. [2] A. L. Dontchev, W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Mth. Soc., 121(1994), 481-489. [3] A. L. Dontchev, A. S. Lewis, R. T. Rockafellar, The radius of metric regularity, Trans. AMS., 355(2002), 493-517. [4] A. L. Dontchev, R. T. Rockafellar, Regularity and conditioning of solution mappings in variational analysis, Set-Valued Anal., 12(2004), 79-109. [5] A. Kaplan, R. Tichatschke, Proximal-based regularization methods and succesive approximation of variational inequalities in Hilbert spaces. Well-posedness in optimization and related topics, (Warsaw, 2001), Control Cybernet., 31(2002), 521-544. [6] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Op- tim., 14(1976), 877-898. [7] M. V. Solodov, B. F. Svaiter, A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, Set-Valued Anal., 7(1999), 323-345.

ROMAI J., 6, 1(2010), 47–56

SET-NORM CONTINUITY OF SET MULTIFUNCTIONS Anca Croitoru Faculty of Mathematics, ”Al. I. Cuza” University, Ia¸si,Romania [email protected] Abstract In this paper, we present diﬀerent types of continuous set multifunctions with respect to a set norm (such as uniformly autocontinuous or autocontinuous from above), their relationships with non-additive set multifunctions and some properties of atoms and pseudo-atoms for null-null-additive set multifunctions.

Keywords: set-norm, autocontinuous from above, uniformly autocontinuous, null-additive, null-null- additive, sn-continuous, sn-exhaustive, atom, pseudo-atom. 2000 MSC: 28B20, 28E10.

1. INTRODUCTION Non-additive measures have been lately studied by many authors (see for example Asahina [1], Choquet [2], Dempster [4], Denneberg [5], Dobrakov [6], Drewnowski [7], Li [15], Liginal and Ow [16], Pap [17], Precupanu [18], Shafer [20], Sugeno [21], Suzuki [22], Wu Congxin and Wu Cong [23]) due to their applications in statis- tics, economy, theory of games, human decision making, medicine. In non-additive measure theory, some continuity conditions are used to prove important results with respect to non-additive measures (for example, Theorem of Egoroﬀ in Li [15]). In Precupanu and Croitoru [19], Gavrilut¸[10], Gavrilut¸and Croitoru [11, 12], Croitoru et al. [3] we extended some classical measure or integral concepts to set multifunctions, proposing a framework for the set-valued case. In this paper, we introduce and study diﬀerent types of set-norm continuous set multifunctions, such as uniformly autocontinuous or autocontinuous from above. We also establish some properties of atoms and pseudo-atoms for null-null-additive set multifunctions.

2. PRELIMINARIES

In the sequel, X will be a real linear space and P0(X) the family of non-empty subsets of X. On P0(X) we consider an order relation denoted by ” ≤ ” and we shall write (P0(X), ≤). We write E < F if E ≤ F and E , F, for E, F ∈ P0(X). For convenience, the notation F ≥ E will be used instead of E ≤ F.

47 48 Anca Croitoru

If X is a normed space, then P f (X) is the family of non-empty closed subsets of X and Pb f (X) is the family of non-empty closed bounded subsets of X. For every M, N ∈ P0(X), we denote h(M, N) = max {e(M, N), e(N, M)}, where e(M, N) = supd(x, N) is the excess of M over N and d(x, N) is the distance x∈M from x to N. It is known that h becomes an extended metric on P f (X) (i.e. it is a metric which can also take the value +∞) and h becomes a metric (called Pompeiu- Hausdorﬀ) on Pb f (X) (Hu and Papageorgiou [13]). ∗ We denote N = N\{0}, R+ = [0, +∞) and R+ = [0, +∞].

Example 2.1. I. The usual set inclusion ”⊆” is an order relation on P0(X) and we write (P0(X), ⊆).

II. Let (X, ≤) be a real ordered vector space. For every E, F ∈ P0(X) and α ∈ R let E + F = {x + y|x ∈ E, y ∈ F},

αE = {αx|x ∈ E}, E − F = {x − y|x ∈ E, y ∈ F}. Let P be the cone of positive elements of X, that is P = {x ∈ X|x ≥ 0}. For every E, F ∈ P0(X), we set E ≺ F if and only if E ⊆ F − P and F ⊆ E + P. The relation ” ≺ ” is reﬂexive and transitive. For instance, let E ∈ P0(R). Then E {0} if and only if E ⊆ [0, +∞). If X = R, then the relation ”≺” is an order relation on Pkc(R) - the family of nonempty compact convex subsets of R.

Deﬁnition 2.1. A function | · | : P0(X) → [0, +∞] is called a set-norm on P0(X) if it satisﬁes the conditions: (i) |E| = 0 ⇔ E = {0}, for E ∈ P0(X). (ii) |αE| = |α| · |E|, ∀α ∈ R, ∀E ∈ P0(X). (iii) |E + F| ≤ |E| + |F|, ∀E, F ∈ P0(X).

Deﬁnition 2.2. A set-norm | · | on (P0(X), ≤) is called monotone if |E| ≤ |F| for every sets E, F ∈ P0(X), so that E ≤ F.

Example 2.2. Let (X, k · k) be a real normed space and |E|h = h(E, {0}) = sup kxk, for x∈E every E ∈ P0(X), where h is the Pompeiu-Hausdorﬀ metric. Then the function | · |h is a monotone set-norm on (P0(X), ⊆), called the set-norm induced by h. We now recall some well-known results. In the sequel, T is a nonempty set and C is a ring of subsets of T. Set-norm continuity of set multifunctions 49

Deﬁnition 2.3. Let ν : C → R+ be a set function. (i) A set A ∈ C is said to be an atom of ν if ν(A) > 0 and for every B ∈ C, with B ⊆ A, we have either ν(B) = 0 or ν(A\B) = 0. (ii) A set A ∈ C is called a pseudo-atom of ν if ν(A) > 0 and B ∈ C, B ⊆ A implies either ν(B) = 0 or ν(B) = ν(A).

Deﬁnition 2.4. A set function ν : C → R+, so that ν(∅) = 0, is said to be: (i) monotone if ν(A) ≤ ν(B), for every A, B ∈ C, with A ⊆ B. (ii) null-monotone if for every A, B ∈ C, A ⊆ B and ν(B) = 0 ⇒ ν(A) = 0. (iii) a submeasure (in the sense of Drewnowski [7]) if ν is monotone and subadditive, that is, ν(A ∪ B) ≤ ν(A) + ν(B), for every A, B ∈ C, with A ∩ B = ∅. (iv) ﬁnitely additive if ν(A∪B) = ν(A)+ν(B), for every A, B ∈ C, so that A∩B = ∅. (v) exhaustive if lim ν(An) = 0, for every sequence of pairwise disjoint sets n→∞ (An)n∈N ⊂ C. (vi) order-continuous (shortly o-continuous) if lim ν(An) = 0, for every sequence n→∞ ∞ T ∗ of sets (An)n∈N∗ ⊂ C, so that An & ∅ (i.e. An ⊇ An+1, and An = ∅, (∀) n ∈ N ). n=1 (vii) uniformly autocontinuous if for every ε > 0, there is δ(ε) > 0, so that for every B ∈ C, with ν(B) < δ(ε), we have ν(A ∪ B) < ν(A) + ε, for every A ∈ C. (viii) null-additive if ν(A ∪ B) = ν(A), whenever A, B ∈ C and ν(B) = 0. (ix) null-null-additive if ν(A ∪ B) = 0, whenever A, B ∈ C and ν(A) = ν(B) = 0.

3. SET-NORM CONTINUOUS SET MULTIFUNCTIONS In this section, we present diﬀerent types of set-norm continuity for set multifunctions and relationships among them. In the sequel, T is a nonempty set and C is a ring of subsets of T. Deﬁnition 3.1. (Gavrilut¸[8,9]) Let µ : C → P0(X) be a set multifunction. µ is said to be: (i) monotone if µ(A) ≤ µ(B), for every A, B ∈ C, with A ⊆ B. (ii) null-monotone (shortly, n-mon) if for every A, B ∈ C, so that A ⊆ B we have

µ(B) = {0} ⇒ µ(A) = {0}.

(iii) a multisubmeasure (shortly, msm) if µ is monotone, µ(∅) = {0} and µ(A∪ B) ≤ µ(A)+µ(B), for every A, B ∈ C, with A∩ B = ∅ (or, equivalently, for every A, B ∈ C). (iv) a multimeasure if µ(∅) = {0} and µ(A ∪ B) = µ(A) + µ(B), for every A, B ∈ C with A ∩ B = ∅. (v) null-additive (shortly, n-add) if for every A, B ∈ C,

µ(B) = {0} ⇒ µ(A ∪ B) = µ(A). 50 Anca Croitoru

(vi) null-null-additive (shortly, n-n-add) if for every A, B ∈ C,

µ(A) = µ(B) = {0} ⇒ µ(A ∪ B) = {0}.

Remarks 3.1. I. Let ν : C → R+ be a set function with ν(∅) = 0 and the set multifunction µ : C → (P0(R), ⊆) defined by µ(A) = [0, ν(A)], for every A ∈ C. Then µ is monotone (null-additive, null-null-additive, respectively) if and only if ν monotone (null-additive, null-null-additive, respectively). Moreover, µ is a multimeasure (a multisubmeasure, respectively) if and only if ν is finitely additive (a submeasure in the sense of Drewnowski [7], respectively). II. In Definition 3.1, some known notions are extended to the set-valued case. The difficulty arises here since we have to consider an order relation on P0(X) and many classical measure theory proof methods fail. For instance, if µ : C → (P0(R), ⊆) is single-valued and monotone, then µ reduces in fact to a constant function defined by µ(A) = {µ(∅)}, for every A ∈ C. So, the definitions in set-valued case do not reduce to the usual single-valued case.

III. The deﬁnition of a P0(X)-valued (or even P f (X)-valued) multimeasure (or multisubmeasure) can not be reduced to that of the single-valued case because P0(X) (and also P f (X)) is not a linear space: indeed, P0(X) is not a group with respect to the addition ”+” deﬁned by M + N = {x + y|x ∈ M, y ∈ N}, for every M, N ∈ P0(X).

IV. If ν1, ν2 are two finite measures defined on an algebra C, so that ν1 ≤ ν2 and ν2 is a probability measure, then one obtains a particular multimeasure µ : C → P0([0, 1]), defined by µ(A) = [ν1(A), ν2(A)], for every A ∈ C, which is the simplest example of a probability multimeasure. We recall that a multimeasure M : C → P0([0, 1]) is said to be a probability multimeasure if 1 ∈ M(T). These probability multimeasures are used in control, robotics and decision theory (in Bayesian estimation).

In Gavrilut¸[8,9] and Croitoru et al. [3], some types of continuous (with respect to the Pompeiu-Hausdorﬀ metric) set multifunctions were introduced. We now extend and study these continuity deﬁnitions to the set-norm case.

Deﬁnition 3.2. Let |·| be a set-norm on P0(X). If µ : C → P0(X) is a set multifunction, then µ is said to be: (i) set-norm autocontinuous from above (shortly, sn-ac-ab) if for every (Bn)n∈N ⊂ C so that lim |µ(Bn)| = 0, we have n→∞

lim |µ(A ∪ Bn| = |µ(A)|, ∀A ∈ C. n→∞ (ii) set-norm uniformly autocontinuous (shortly, sn-u-ac) if for every ε > 0, there is δ(ε) = δ > 0, so that for every A ∈ C and every B ∈ C, with |µ(B)| < δ, we have |µ(A ∪ B)| < |µ(A)| + ε. Set-norm continuity of set multifunctions 51

Theorem 3.1. Let | · | be a monotone set-norm on (P(X), ⊆) and µ : C → P0(X) a set multifunction. Then the following statements hold: I. If µ is a multisubmeasure, then µ is sn-u-ac. II. If µ is sn-u-ac, then µ is sn-ac-ab. III. If µ is sn-ac-ab, then µ is null-null-additive. IV. If µ is a multisubmeasure, then µ is null-additive. V. If µ is null-additive, then µ is null-null-additive and null-monotone. So, the following schema is working: msm =⇒ n-mon ⇓ u ⇑ sn-u-ac n-add ⇓ ⇓ sn-ac-ab =⇒ n-n-add Proof. I. Let ε > 0 and B ∈ C such that |µ(B)| < ε. Since µ is monotone, it results |µ(A)| ≤ |µ(A ∪ B)|, for all A ∈ C. Then |µ(A ∪ B)| ≤ |µ(A) + µ(B)| ≤ |µ(A)| + |µ(B)| < |µ(A)| + ε, which proves that µ is sn-u-ac. II. Let A ∈ C and (Bn)n∈N ⊂ C, so that |µ(Bn)| → 0. Since µ is sn-u-ac, for every ε > 0, there is d > 0, so that for every A ∈ C and every B ∈ C, with |µ(B)| < δ, we have (1) |µ(A ∪ B)| < |µ(A)| + ε.

Since |µ(Bn)| → 0, there is n0 ∈ N, such that |µ(Bn)| < d, for every n ∈ N, n ≥ n0. From (1) it follows |µ(A ∪ Bn)| < |µ(A)| + ε, for every natural n ≥ n0. By the monotonicity of µ, we have

(2) |µ(A)| ≤ |µ(A ∪ Bn)|, ∀n ≥ n0. From (1) and (2) it follows that µ is sn-ac-ab. III. Let A, B ∈ C, such that µ(A) = µ(B) = {0} and let Bn = B, for every n ∈ N. Then |µ(Bn)| → 0. Since µ is sn-ac-ab, we have lim |µ(A ∪ Bn)| = |µ(A)| = 0. This n→∞ implies |µ(A ∪ B)| = 0, so µ(A ∪ B) = {0} and thus µ is null-null-additive. IV and V result straightforward. The following examples show that the converses of the statements in Theorem 3.1 are false.

Example 3.1. I. Let T = N, C = P(N) and µ : C → (P0(R), | · |h) deﬁned for every A ∈ C by µ(A) = {0} if A is ﬁnite and µ(A) = [1, ∞), if A is countable. Then µ is sn-u-ac and it is not a multisubmeasure.

II. Let T = {a, b}, C = P(T) and µ : C → P f (R) deﬁned by µ(T) = [0, 1], µ({a}) = 1 µ({b}) = [0, 3 ] and µ(∅) = {0}. Then µ is null-additive, but it is not a multisubmeasure. 52 Anca Croitoru

III. Let ν : C → R0 be a set function with ν(∅) = 0 and µ : C → (P0(R+), | · |h) deﬁned by µ(A) = {ν(A)}, for every A ∈ C. Then µ is sn-ac-ab (sn-u-ac, respectively) if and only if ν is sn-ac-ab (sn-u-ac, respectively).

IV. Let ν : C → R+ be a set function with ν(∅) = 0 and µ : C → (P0(R), |·|h) deﬁned by µ(A) = [0, ν(A)], for every A ∈ C. Then µ is sn-ac-ab (sn-u-ac, respectively) if and only if ν is sn-ac-ab (sn-u-ac, respectively).

V. Let T = [0, 1], C the Borel σ-algebra on T, ł : C → R+ the Lebesgue measure π and µ : C → (P0(R+), | · |h) deﬁned by µ(A) = {ν(A)}, where ν(A) = tg( 2 ł(A)), for every A ∈ C. According to Example 4-[14], ν is ac-ab, but it is not u-ac. From the above III, it results that µ is sn-ac-ab, but it is not sn-u-ac.

VI. Let T = {a, b}, C = P(T) and µ : C → P0(R) deﬁned by µ(T) = [0, 1], 1 µ({b}) = [0, 2 ] and µ({a}) = µ(∅) = {0}. Then µ is null-monotone and null-null- additive, but it is not a multisubmeasure and it is not null-additive.

VII. Let T = {a, b}, C = P(T) and µ : C → P0(R) deﬁned by µ(A) = {1, 2, 3} if A = T and µ(A) = {0} otherwise. Then µ is null-monotone, but µ is not null-null- additive and not null-additive.

VIII. Let T = {a, b}, C = P(T) and µ : C → P0(R) deﬁned by µ(A) = {1, 2} if A = {a} or A = {b}, µ(∅) = {3} and µ({a, b}) = {0}. Then µ is null-null-additive, but not null-monotone.

Deﬁnition 3.3. Let | · | be a monotone set-norm on (P0(X), ≤). A set multifunction µ : C → P0(X) is said to be: (i) set-norm exhaustive (shortly, sn-exhaustive) if lim |µ(An)| = 0, for every pair- n→∞ wise disjoint sequence of sets (An)n∈N∗ ⊂ C. (ii) set-norm continuous (shortly, sn-continuous) if lim |µ(An)| = 0, for every se- n→∞ quence of sets (An)n∈N∗ ⊂ C such that An & ∅.

Remarks 3.2. Let | · | be a monotone set-norm on (P0(X), ≤). If C is ﬁnite, then every set multifunction µ : C → P0(X) with µ(∅) = {0} is sn-exhaustive and sn-continuous.

Theorem 3.2. Let | · | be a monotone set-norm on (P0(X), ≤). If C is a σ-ring, µ : C → P0(X) is monotone, sn-continuous and µ(∅) = {0}, then µ is sn-exhaustive. S∞ Proof. Let (An)n∈N∗ be a sequence of mutually disjoint sets of C and let Bn = Ak, k=n ∗ ∗ for every n ∈ N . Then Bn ∈ C, for every n ∈ N and Bn & ∅. Since µ is sn-continuous, it results |µ(Bn)| → 0, which implies |µ(An)| → 0. So, µ is sn- exhaustive. Set-norm continuity of set multifunctions 53 4. ATOMS AND PSEUDO-ATOMS This section contains several properties of atoms and pseudo-atoms for null-null- additive set multifunctions.

Deﬁnition 4.1. ([10,11,12]) Let µ : C → P0(X) be a set multifunction. (i) A set A ∈ C is said to be an atom of µ if µ(A) , {0} and for every B ∈ C, with B ⊆ A, we have either µ(B) = {0} or µ(A\B) = {0}. (ii) A set A ∈ C is called a pseudo-atom of µ if µ(A) , {0} and for every B ∈ C, with B ⊆ A, we have either µ(B) = {0} or µ(B) = µ(A). (iii) µ is said to be non-atomic (non-pseudo-atomic, respectively) if it has no atoms (no pseudo-atoms, respectively).

Remarks 4.1. Let µ : C → P0(X) be a set multifunction, with µ(∅) = {0}. I. If µ is monotone, then µ is non-atomic (non-pseudo-atomic, respectively) if for every A ∈ C, with µ(A) ) {0}, there is B ∈ C so that B ⊆ A, µ(B) ) {0} and µ(A\B) ) {0} (µ(A) ) µ(B), respectively). II. If µ is null-monotone, then A ∈ C is an atom of µ if and only if A is an atom of µ. III. If µ is null-additive, then every atom of µ is a pseudo-atom of µ (as we shall see in Examples 4.1-I, the converse is not valid).

Example 4.1. I. Let T = {a, b, c}, C = P(T) and µ : C → P0(R) defined by µ(A) = [0, 1] if A , ∅ and µ(A) = {0} if A = ∅. Then µ is null-additive, A = {a, b} is a pseudo-atom of µ, but not an atom of µ. II. Let T = 2N = {0, 2, 4,...}, C = P(T) and µ : C → P0(R) defined for every A ∈ C by:  {0}, if A = ∅ µ(A) =  1 2 A ∪ {0}, if A , ∅ 1 x where 2 A = { 2 | x ∈ A}. µ is a multisubmeasure. If A ∈ C, with cardA = 1 and A , {0} or A ∈ C, A = {0, 2n}, n ∈ N∗, then A is an atom of µ (and a pseudo-atom of µ, too, according to Remark 4.1-III). By cardA we mean the cardinal of A. If A ∈ C, with cardA ≥ 2 and there exist a, b ∈ A such that a , b and ab , 0, then A is not a pseudo-atom of µ (and not an atom of µ, according to Remark 4.1-III). III. Let C = P(N) and µ : C → P f (R) defined for every A ∈ C by  {0}, if A is finite µ(A) =  {0} ∪ [nA, +∞), if A is infinite and nA = min A.

Then µ is monotone and non-pseudo-atomic.

Remarks 4.2. Let µ : C → P0(X) be a set multifunction, with µ(∅) = {0}. 54 Anca Croitoru

I. If A ∈ C is a pseudo-atom of µ and B ∈ C, B ⊆ A is such that µ(B) ) {0}, then B is a pseudo-atom of µ and µ(B) = µ(A). II. Suppose µ is null-monotone. If A ∈ C is an atom of µ and B ∈ C, B ⊆ A is such that µ(B) ) {0}, then B is an atom of µ and µ(A\B) = {0}.

Theorem 4.1. Suppose µ : C → P0(X) is monotone, so that µ(∅) = {0} and A, B ∈ C are pseudo-atoms of µ. Then the following statements hold: I. µ(A) , µ(B) ⇒ µ(A ∩ B) = {0}. II. Suppose µ is null-null-additive. If µ(A ∩ B) = {0}, then A\B and B\A are pseudo-atoms of µ and µ(A\B) = µ(A), µ(B\A) = µ(B). Proof. I) Suppose µ(A ∩ B) ) {0}. According to Remark 4.2-I, we have µ(A ∩ B) = µ(A) = µ(B), which is false. II. Let us prove that µ(A\B) ) {0}. Suppose on the contrary that µ(A\B) = {0}. Since µ is null-null-additive, we have µ(A) = µ((A\B)∪(A∩ B)) = {0}, which is false. So, µ(A\B) ) {0} and from Remark 4.2-I, it results that A\B is a pseudo-atom of µ and µ(A\B) = µ(A). Analogously, B\A is a pseudo-atom of µ and µ(B\A) = µ(B).

Theorem 4.2. Suppose µ : C → P0(X) is monotone and null-null-additive, so that µ(∅) = {0} and A, B ∈ C are pseudo-atoms of µ. Then there exist pairwise disjoint sets E1, E2, E3 ∈ C, with A ∪ B = E1 ∪ E2 ∪ E3, such that, for every i ∈ {1, 2, 3}, either Ei is a pseudo-atom of µ or µ(Ei) = {0}.

Proof. Let E1 = A ∩ B, E2 = A\B, E3 = B\A. We have the following cases: (i) µ(E1) = {0}. According to Theorem 4.1-II, E2 and E3 are pseudo-atoms of µ and µ(E2) = µ(A), µ(E3) = µ(B). (ii) µ(E1) ) {0}, µ(E2) ) {0}, µ(E3) ) {0}. By Remark 4.2-I, E1 is a pseudo-atom of µ and µ(E1) = µ(A) = µ(B). Analogously, E2 and E3 are pseudo-atoms of µ. (iii) µ(E1) ) {0}, µ(E2) = {0}, µ(E3) ) {0}. From Remark 4.2-I, it results that E1 is a pseudo-atom of µ and µ(E1) = µ(A) = µ(B). Analogously, E3 is a pseudo-atom of µ and µ(E3) = µ(B). The last two cases are similar to (iii). (iv) µ(E1) ) {0}, µ(E2) ) {0}, µ(E3) = {0}. (v) µ(E1) ) {0}, µ(E2) = µ(E3) = {0}.

Remarks 4.3. Let µ : C → P0(X) be monotone, null-null-additive, such that µ(∅) = {0}. I. By Theorem 4.2, the union of two pseudo-atoms A, B of µ either is a pseudo- atom of µ or is equal to an union of two pairwise disjoint pseudo-atoms E1, E2 of µ. In the latter case, we set E1 = A, E2 = B\A. II. By induction, it is easy to show that for any sequence of pseudo-atoms of µ, n m (Ai)i=1, n ∈ N ∪ {+∞}, there is a sequence (E j) j=1, m ∈ N ∪ {+∞}, m ≤ n, of pairwise Sn Sm disjoint pseudo-atoms of µ, such that Ai = E j. i=1 j=1 Set-norm continuity of set multifunctions 55

Theorem 4.3. Let | · | be a monotone set-norm on (P0(X), ⊆). Suppose C is a σ-ring and µ : C → P0(X) is monotone, null-null-additive, sn-continuous, µ(∅) = {0} and µ has atoms. Then there exists a ﬁnite or countable family of pairwise disjoint atoms of n µ, (Bi)i=1, n ∈ N ∪ {+∞}, satisfying the conditions: (i) |µ(Bi−1)| ≥ |µ(Bi)|, ∀1 0, ∃n0 ∈ N , such that |µ( Bk)| < ε. k≥n0 (iii) for every atom A of µ, there is Bk such that µ(A 4 Bk) = {0}.

∞ Proof. We remark that (ii) is verified for every infinite sequence (Bi)i=1 of pair- S ∗ wise disjoint sets of C. We have Bk ∈ C, for every n ∈ N and since µ is sn- Sk≥n autocontinuous, it follows lim |µ( Bk)| = 0. n→∞ k≥n ∗ ∞ Suppose, for some m ∈ N , there is an infinite sequence (Bk)k=1 of pairwise disjoint 1 S 1 ∗ atoms of µ satisfying |µ(Bk)| ≥ m . Then |µ( Bk)| ≥ |µ(Bk)| ≥ m , for all n ∈ N , k≥n n a contradiction. So we obtain the sequence (Bk)k=1 as follows. First we choose a p1 maximal finite (possibly empty) sequence (Bk)k=1 of pairwise disjoint atoms of µ p2 such that |µ(Bk)| ≥ 1. Next, we choose a maximal finite sequence (Bk) of atoms k=p1+1 1 of µ which are pairwise disjoint with all Bk, 1 ≤ k ≤ p1 and so that 2 ≤ |µ(Bk)| < 1. pi By induction we obtain at the i-th step a maximal finite sequence (Bk) of k=pi−1+1 pairwise disjoint atoms of µ which are disjoint of every Bk, 1 ≤ k ≤ pi−1 and so that 1 1 i−1 ≤ |µ(Bk)| < i . Then we rearrange the sets Bk by non-increasing of |µ(Bk)| to satisfy (i). By Remark 4.2-II, if E, F are any two atoms of µ, then either µ(E ∩ F) = {0} or µ(E4F) = {0}. If an atom A of µ is so that µ(A ∩ Bk) = {0} for every k, then pi ∗ 1 1 (Bk) is not maximal at the i-th step for i ∈ N , ≤ |µ(A)| < . k=pi−1+1 i−1 i So there is an element Bk that satisfies (iii).

Theorem 4.4. Let C be a σ-ring and µ : C → P f (X) is monotone, sn-continuous, null-null-additive, µ(∅) = {0} and µ has atoms. Then there exists a ﬁnite or inﬁnite family (Bk) of pairwise disjoint atoms of µ, such that for every A ∈ C and every ε > 0, there are: a subsequence (B ) of (B ), i ∈ N and F, E ∈ C such that A = S S ki k 0 ( Bki \F) ∪ E, |µ( Bki )| ≤ ε, µ(F) = {0}, and E contains no atoms of µ. i i≥i0

n Proof. Let (Bi)i=1, n ∈ N ∪ {+∞}, be the family as in the proof of Theorem 4.3, and let I be the set of all indices i such that there is an atom C ⊂ A, C ∈ C so that S S µ(C 4 Bi) = {0}. Then the subsequence (Bi)i∈I and the sets F = Bi\A, E = A\ Bi i∈I i∈I satisfy the conclusion of the theorem.

Acknowledgement. The author is grateful to prof. dr. Ilie Burdujan and the Referees for valuable suggestions in the improvement of this paper. 56 Anca Croitoru References [1] Asahina, S., Uchino, K., Murofushi, T., – Relationship among continuity conditions and null- additivity conditions in non-additive measure theory, Fuzzy Sets and Systems, 157(2006), 691- 698. [2] Choquet, G., – Theory of capacities, Ann. Inst. Fourier (Grenoble), 5(1953-1954), 131-292. [3] Croitoru, A., Gavrilut¸, A., Mastorakis, N.E., Gavrilut¸, G., – On diﬀerent types of non-additive set multifunctions, WSEAS Transactions on Mathematics, 8(2009), 246-257. [4] Dempster, A.P., – Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist., 38(1967), 325-339. [5] Denneberg, D., – Non-additive Measure and Integral, Kluwer Academic, Dor- drecht/Boston/London, 1994. [6] Dobrakov, I., - On submeasures, I, Dissertationes Math., 112(1974), 5-35. [7] Drewnowski, L., – Topological rings of sets, continuous set functions. Integration, I, II, III, Bull. Acad. Polon. Sci., Ser.´ Math. Astron. Phys., 20(1972), 269-286. [8] Gavrilut¸, A.,- Regularity and o-continuity for multisubmeasures, An. S¸t. Univ. Ias¸i, 50(2004), 393-406. [9] Gavrilut¸, A.,– Regularity and autocontinuity of set multifunctions, Fuzzy Sets and Systems, 161(2010), 681-693. [10] Gavrilut¸, A., – Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions, Fuzzy Sets and Systems, 160(2009), 1308-1317. [11] Gavrilut¸, A., Croitoru, A., – Non-atomicity for fuzzy and non-fuzzy multivalued set functions, Fuzzy Sets and Systems, 160(2009), 2106-2116. [12] Gavrilut¸, A., Croitoru, A., – Pseudo-atoms and Darboux property for set multifunctions, submitted for publication. [13] Hu, S., Papageorgiou, N. S., – Handbook of Multivalued Analysis, vol. I, Kluwer Acad. Publ., Dordrecht, 1997. [14] Jiang, Q., Suzuki, H., Wang, Z., Klir, G. – Exhaustivity and absolute continuity of fuzzy measures, Fuzzy Sets and Systems, 96(1998), 231-238. [15] Li, J., – On Egoroﬀ theorem on fuzzy measure spaces, Fuzzy Sets and Systems, 135(2003), 367- 375. [16] Liginlal D., Ow T.T., - Modelling attitude to risk in human decision process: An application of fuzzy measures, Fuzzy Sets and Systems, 157(2006), 3040-3054. [17] Pap, E., – Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995. [18] Precupanu, A.M., - On the set valued additive and subadditive set functions, An. S¸t. Univ. Ias¸i, 29(1984), 41-48. [19] Precupanu, A.M., Croitoru, A. – A Gould type integral with respect to a multimeasure I., An. S¸t. Univ. ”Al.I. Cuza” Ias¸i, s. I-a, Matematica,˘ XLVIII, 1(2002), 165-200. [20] Shafer, G., – A Mathematical Theory of Evidence, Princeton University Press, Princeton, N.J., 1976. [21] Sugeno, M., - Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974. [22] Suzuki, H., - Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41(1991), 329-342. [23] Wu, Congxin, Wu, Cong, – A note on the range of null-additive fuzzy and non-fuzzy measure, Fuzzy Sets and Systems, 110(2000), 145-148. ROMAI J., 6, 1(2010), 57–78

A MATHEMATICAL MODEL FOR CLONAL EXPANSION OF ANTIGEN SPECIFIC T CELLS Marina Dolfin, Demetrio Criaco Department of Mathematics, University of Messina, Italy mdolfi[email protected], [email protected] Abstract In this paper a macroscopic phenomenological model for the T cell mediated immune response due to a single type of antigen challenge is developed in the framework of the thermodynamic theory of fluid mixtures. Proliferation events accounting for the generation of T cell clones are considered, determining a non conservative balance for mass and momentum densities for the mixture as a whole in a way analogous to some cases in tumor modelling and growth of tissues.

Keywords: mixture theory, T cell clonal expansion, proliferative events. 2000 MSC: 97U99.

1. INTRODUCTION In this paper, we elaborate a mathematical model on the activation and clonal expansion of T cells during the immune response to a single type of antigen challenge. In constructing our model, we use an approach based on the phenomenology of the problem under consideration and we obtain the field equations governing the dynamics and the interactions of Antigen Non Experienced T cells (naive T cells) and Antigen Experienced T cells (T helper cells) in presence of Antigen Presenting Cells (dendritic cells), i.e. cells bearing the antigen [1]-[4], taking into account that the interactions among these three populations of cells are due to the presence of chemicals (cytokines in our case) which act as soluble mediators. At the same time the populations of cells secrete cytokines in a feedback effect [5]-[10]. The field equations are mathematically constructed in the framework of the thermodynamic of reacting fluid mixtures adapted to the case in which proliferative events (i.e. events which do not satisfy the law of conservation of mass for the mixture as a whole) occur ([11] for analogies in the case of growth of soft tissues). We take into consideration two fundamental phases: the activation and the clonal expansion. Basing on biological considerations, we assume that the proliferation of T helper cells, generating clones, which happens during the clonal expansion phase is not a conservative event, i.e. it does not satisfy the mass density conservation of the mixture of biological fluids as a whole, at least during the acute phase of inflammation (about 24-48 hours after the encounter with the antigen). Our model is based on the fundamental picture of sensing and response mechanism of the above introduced three populations of cells in the presence of a set of

57 58 Marina Dolfin, Demetrio Criaco cytokines which act as mediators via signaling inducing genetic pathways inside the cells and determining the genetic mutation of naive T cells into T helper cells. In fact the dynamics of the introduced populations of cells involve genetic mutations (naive T cells mutating into T helper cells) and proliferation events (see eq.(16)) due to the generation of clones of T helper cells, then it cannot be simply related to fundamental mechanistic properties of individual cell response (see [12] for analogies in the case of bacterial populations or [13] for analogies in the case of dynamics of leukocytes in tissue inflammatory response). The macroscopic dynamics is fundamentally related to the biological phenomena of motility of cells [13, 14] (which results in avoiding overcrowding [15]) and chemotactic response of the cells [16] together with the diffusion of the chemicals (cytokines in our case) [17]-[18]. The motility of cells is introduced via a flux in the balance equations of the three population of cells [13], the diffusion of the chemicals (cytokines) is of the Fick’s type and the chemotactic effect is introduced via a force in the balance equations of momenta of the cell populations [16]. In particular, in section 2 we illustrate the modelled aspects of the dynamics of the activation of naive T cells and the clonal expansion of T helper cells during the immune response to a single type of antigen challenge by adopting a phenomenological point of view which uses observations at the meso- and microscopic level. In section 3 we derive explicit expressions for the balance equations of mass density for all the constituents of the mixture, by phenomenologically specifying the fluxes and productions of mass densities. In section 5 we derive explicit expressions for the balance equations of momentum densities for all the constituents of the mixture by phenomenologically specifying the production of momentum densities and by defining appropriate equations of state for the involved stresses. In sections 4 and 6 we derive explicit expressions for the balance equations for mass and momentum densities for the mixture of biological fluids as a whole, by applying the thermodynamical theory of reacting mixtures of fluids, in the particular case in which proliferative events occur [11]. In section 7, a quasi-linear system of PDEs with terms of the second order is obtained governing the dynamics of activation of naive T cells and clonal expansion of T helper cells during the immune response to a single type of antigen challenge, by summarizing the balance equations for the mass and momentum densities obtained in the previous sections. A particular case in which the motility of the cells and the diffusion of the chemicals are not taken into account is considered, obtaining an associated quasi-linear system of PDEs of the first order . In a forthcoming paper, the hyperbolicity of this last system of equations is proved and the propagation of non linear waves in the general case is studied by using an asymptotic perturbative method. A mathematical model for clonal expansion of antigen specific T cells 59 2. PHENOMENOLOGY OF SOME ASPECTS OF THE DYNAMICS OF T CELL MEDIATED IMMUNE RESPONSE In this section we present some phenomenological aspects regarding the activation and clonal expansion of T cells during the immune response to a single type of antigen challenge. The schematization which follows is the result of a deep analysis of a part of the huge biomedical and biomathematical literature on the topic of T cell mediated immune response[1]-[18]. The immune system is a complex system of cells and molecules distributed throughout living bodies and providing mainly a basic defense against bacteria, virus, fungi and other pathogenic agents (referred to as antigens). It offers also a defense against pathologically transformed cells. In the immune system response to antigens, an important involved population of cells is that of lymphocytes. Lymphocytes are divided into two groups: T cells and B cells. The mathematical modellization of this paper regards only some particular dynamics involving T cells. T cells are produced in the thymus and they are antigen specific (bearing specific antigen receptors) but they do not have specific functionalities; in this state they are called naive T cells. Due to the first encounter with the antigen, naive T cells take specific functionalities and are called in general Antigen Experienced (or Antigen activated) T cells. Antigen Experienced T cells divide mainly into three subgroups, depending on their different functionalities: the T helper cells, the cytotoxic T cells and the suppressor T cells. Our modellization regards a particular T helper cell dynamics which happens mainly in lymph nodes. In the activation and clonal expansion of T cells an important role is played by some soluble mediators (chemicals) which are collectively called cytokines and which induce signal transductions inside the nucleus of the cell, determining the genetic mutation of Antigen Not Experienced cells (naive T cells) into Antigen Expe- rienced cells ( T helper cells in our model); in modelling the dynamics of activation of naive T cells, we use a macroscopic approach considering only the macroscopic phenomenological effect (the changing of naive T cells into T helper cells in this case) of dynamics which happen at a meso- and microscopic level. For instance, the transduction signals induced by the cytokines happen at the mesoscopic level, but we consider only the macroscopic output, i.e. the production of T helper cells and their clones and the related decrement of naive T cells. The most important difference of functionalities between Antigen Not Experi- enced T cells (naive T cells) and Antigen Experienced ones (T helper in our case), consists in the fact that the activated ones may divide mitotically (generating clones) due to a second encounter with the same type of antigen; this phenomenon is called clonal expansion [1]-[4]. We will consider this phenomenon in our model as a proliferative event [11]. In our initial efforts in the modellization of this complex dynamics we consider an unbounded tissue medium. 60 Marina Dolfin, Demetrio Criaco 2.1. SOME CYTOKINES RELATED TO PARTICULAR FUNCTIONALITIES OF T CELLS In this section, we illustrate some aspects of the important connection among a specified set of cytokines and some fundamental functionalities of naive T cells and T helper cells [20]-[25]. The functional activation of naive T cells into T helper cells follows the first encounter with the antigen. The presence of T helper cells is connected to the secretion of some specific cytokines. Some T helper cells are mainly connected to the secretion of the cytokines Il-2, IFN-γ and TNF-α while some others T helper cells are mainly connected to the secretion of cytokines IL-4, IL-5, IL-13 and IL-10 [4]. We kindly refer the reader to a forthcoming paper [26], where a specific differentiation of the T helper set into T helper cell of type 1 (Th1) and T helper cells of type 2 (Th2) is modelled. The presence of specific cytokines in the lymphatic micro-environment is also certainly connected to the functional activation of naive T cells into T helper cells. In particular we have chosen to collect seven types of cytokines (IL-2, INF-γ, INF-α, IL-4, IL-5, IL-13 and IL-10) under the name of set of cytokines due to the role played in the activation and clonal expansion of T cells. The effect of these cytokines is taken into account in our model by using phenomenologically derived explicit expressions for the production of mass densities in the balance equations of mass and momentum densities for the populations cells depending on the specified set of cytokines.

2.2. APOPTOSIS Apoptosis (or cell death) is the fate of many of activated T lymphocytes. Activated lymphocytes die because of a monotonous, repeated stimulations due to the presence of antigens and/or concentrations of cytokines [4]. This phenomenon is called activation induced cell death and is taken into account in our model (see section 3.4). Antigen Not Experienced lymphocytes die by apoptosis unless they are rescued from it by their speciﬁc antigen. This phenomenon is called programmed cell death. We consider also this phenomenon in our model (see section 3.3).

2.3. CLONAL EXPANSION OF ACTIVATED T CELLS Clonal expansion of T cells is a dynamics involved in the so called secondary immune response, which is the immune response due to a second encounter with a specific type of antigen. The activated T helper cells produced in the first encounter due to the activation, produce clones due to a second encounter with the specific type of antigen. The phase of proliferation of activated T cells as clones is also called clonal expansion phase [10]. A mathematical model for clonal expansion of antigen specific T cells 61 2.4. CHEMOTAXIS Here we introduce the biological phenomenon of chemotaxis, which is taken into account in our model via interactions forces acting on T helper and dendritic cells (see sections 5.3 and 5.4) and determined by the presence of the specific cytokines. The response of a biological population to a stimulus of the environment is called taxis (from the Greek ”to arrange”). There are lots of kind of ”taxis”; the type of stimulus determines the prefix on the word taxis. For instance geotaxis is the response to the gravitational force and acts on flies and birds while in the aerotaxis the stimulus is determined by oxygen and acts on bacteria. Specifically, a chemical directed movement is called chemotaxis and it describes the influence of chemical substances present in the environment on the movement of mobile species (such as bacteria or lymphocytes as in our case). To make an example, when a bacterial infection invades the body it may be at- tacked by movement of cells towards the source as a result of chemotaxis [18]. In this case one speaks of positive chemotaxis and the chemical is called a chemoattractant. Regarding to the specific dynamics involved in the T cell mediated immune response, it is for instance experimentally proved that lymphocytes cells move toward a region of bacterial inflammation by moving up a chemical gradient [13]. Phisically, the chemotactic interaction can be introduced as a force that tends to aggregate the cells, driving them along the direction of the chemoattractor chemical gradient [16], i.e.

fc = χ∇c, (1) where c is the concentration of the chemoattractant and χ is the chemotactic sensitivity coeﬃcient [17, 18] which may, in general, depend on the concentrations of the mobile specie and of the chemoattractant; in this paper it is taken as constant to en- light the fundamental behaviors. The bulk force per unit mass fc is then introduced in the balance equation for the momentum density of the cells and accounts for cell- cell interaction via the chemotactic signaling [16]. In our model we introduce the chemotactic interactions following [16], by introducing appropriate forces into the balance equations of momentum densities for the populations of activated T helper and dendritic cells (see section 5). In our case the role of chemoattractor is played by the speciﬁed cytokines.

3. BALANCE EQUATIONS OF MASS DENSITIES FOR THE CONSTITUENTS OF THE MIXTURE OF BIOLOGICAL FLUIDS We model the populations of cells (naive T, T helper and dendritic cells) and the chemicals (the set of cytokines) as an homogeneous mixture of biological fluids. Following Muller [27], we suppose that each point is occupied by ”particles of all constituents”. The equations of balance of mass and momentum densities in regular 62 Marina Dolfin, Demetrio Criaco points of the mixture differ from the corresponding equations of balance in a single body fluid by a production term [27, 28]: in fact mass of a constituent may be produced by chemical reactions and momentum may be produced by interaction forces and by the production due to the chemical reactions. In our model, mass of a constituent may be produced by genetic mutations of naive T cells into T helper cells, in a sense analogous to chemical reactions among the constituents. Moreover, in the thermodynamic theory of mixtures of fluids, the production densities of mass of the constituents are constrained by the requirement that the sum of the productions densities of all the constituents is zero [27, 28], in order to achieve the conservation of mass density for the mixture as a whole. This requirement still applies in our model, except that for the net generation of clones of T helper cells which is the net number of clones produced during the clonal expansion phase; this generation of clones represents a proliferative event in which the mass density of the mixture is not conserved (see [11] for analogies in growth phenomena in soft tissues and [29] and [30] for analogies in tumor modeling). By analyzing the biological phenomenon under consideration, one may deduce that this proliferative phenomenon regards only the acute phase of the immune response, while for large values of time the number of lymphocytes is kept constant [1, 4]. We take into consideration this particular dynamics into our model by introducing a proliferation rate function for the T helper cell clones with a special dependance on time in order to mimic the dependance on time of the immune response (see sections 3.4 and 4); in particular the proliferation rate function is assumed to be the solution of the logistic equation of Verhulst. As a result of this assumption, the mass density of the mixture as a whole is conserved for large values of time, although it is not conserved for the first phase of the immune response to the antigen challenge (the acute phase of the immunological response is of the order of 24-48 hours [4]). In this section we present explicit expressions for the balance equations of mass densities for all the constituents of the mixture modeling some aspects of the dynamics of activation of naive T cells and subsequent clonal expansion of T helper cells.

3.1. CONSTITUENTS OF THE MIXTURE OF BIOLOGICAL FLUIDS The constituents of the mixture of biological ﬂuids under consideration are:

1 naive T cells of mass density ρT and concentration cT ,

2 antigen activated T helper cells of mass density ρTh and concentration cTh ,

3 dendritic cells of mass density ρd and concentration cd, 4 a set of cytokines (IL-2, IFN-γ, INF-α, IL-4, IL-5, IL-13, IL-10) of mass density ρ1 and concentration c1. A mathematical model for clonal expansion of antigen speciﬁc T cells 63

The density of the mixture of ﬂuids is given by [27, 28]

ρ = ρT + ρTh + ρd + ρ1 (2) where ρ ρ ρ ρ c = T , c = Th , c = d , c = 1 , with c + c + c + c = 1. (3) T ρ Th ρ d ρ 1 ρ T Th d 1 3.2. LOCAL FORM OF THE GENERAL BALANCE EQUATION OF MASS DENSITY The local form of the general balance equation for a generic biological specie of mass density % is assumed analogous to the usual balance equation of mass density in the thermodynamics of continuum media [27] as ∂% + ∇ · (%v + J ) = τ + s (4) ∂t % % % where v% is the velocity of the specie, J% is the local flux of the specie at any point in the tissue space, τ% is the local production of the specie (or local net generation rate [13]) and s is the supply from outside. Supply is different from production because it may be controlled from the exterior [27]. In eq.(4) the differential operator ∂ k nabla is ∇ = ( ∂xk ) where x , (k = 1, 2, 3) represent the spatial coordinates (i.e. the components of the position vector x in Eulerian coordinates in a cartesian reference frame) and t is time. By using biological considerations [1, 4], we assume that no supply from outside is present in our case. In the following we continue to call the local net generation rate with the name production of mass density in analogy to the usual terminology of continuum mechanics. In the next sections, we deduce phenomenologically explicit expressions characterizing the fluxes and productions of mass densities for the three populations of cells (T naive, T helper and dendritic cells) and the chemical mediators represented by the set of cytokines.

3.3. NAIVE T CELL BALANCE EQUATION OF MASS DENSITY Regarding the naive T cells, we model the biological phenomenon of random motility of cells (which results in avoiding overcrowding [15]), with a diﬀusion-like ﬂux [12, 13, 31, 14]

JT = −r∇ρT (5) where JT have units of viable cell biomass/volume and r is the so called random motility coefficient (analogous to a molecular diffusion coefficient) (see [12] for the case of dynamics of bacterial populations and [13] for the dynamics of leukocyte 64 Marina Dolfin, Demetrio Criaco in tissue inflammatory response). Data fit with experimental results show that the random motility coefficient is of the order of 10−5cm2/s [12]. The production of mass density of naive T cells (τT ) is due to the following four contributions

activation of naive T cells mutating into T helper cells (see section 2), generation of newly borne naive T cells in the thymus (idem), programmed cell death of naive T cells (idem).

The first contribution is modeled as a negative term inducing a decrement in the number of naive T cells, linearly depending on the actual value of the density of naive T cells via a parameter depending on the concentration of cytokines (because of the fact that the cytokines belonging to the introduced set induce the activation of naive T cells into T helper cells); this term is given by −h(c1)ρT where h(c1) is the activation rate factor of naive T cells into T helper cells. Assuming a linear relation regarding the dependance of h on c1, one finally writes the explicit phenomenological form of the first contribution as

˜ −hc1ρT (6) where h˜ is the constant activation rate of naive T cells into T helper cells. In the following we continue to call h˜ by the name h. More complicated choices are also possible however and do not aﬀect the model qualitatively. The second contribution can be modelled via a ﬁrst order term depending on the actual value of the mass density of naive T cells ρT as

k0ρT (7) where k0 is the constant growth rate of naive T cells produced by the thymus. The third contribution can be modeled as a negative ﬁrst order term inducing a decrement in the number of naive T cells due to the apoptosis, depending on the actual value of the mass density of naive T cells

−kapρT (8) where kap is the constant apoptotic rate of naive T cells. All these terms have units of viable cell biomass/volume·time [13]. By summing up the three contributions (6), (7) and (8), one obtains the production density of naive T cells as

τT = (k0 − kap − hc1)ρT . (9) All the introduced constants are positive. (For some experimental data on coefficients see [32]). By plugging in the general form of the balance of mass density (4), A mathematical model for clonal expansion of antigen specific T cells 65 the phenomenologically assumed flux (5) and production of mass density (9), one obtains the balance equation of mass density for naive T cells as ∂ρ T + ∇ · (ρ v ) − r4ρ = (k − k − hc )ρ (10) ∂t T T T 0 ap 1 T where vT is the velocity of naive T cells and remembering that the random motility coefficient r is constant.

3.4. TH CELL BALANCE EQUATION OF MASS DENSITY Regarding the T helper cells, we model again the phenomenon of random motility of cells (which results in avoiding overcrowding [15]) with a diﬀusion-like ﬂux [12, 13, 31, 14]

JTh = −r∇ρTh (11) where, basing on biological considerations [17], the same random motility coeﬃcient r has been assumed as for naive T cells. To phenomenologically determine the production of mass density in the case of T helper cells, one has to consider ﬁrstly the phase of activation (i.e. the phase of genetic mutation of naive T cells into T helper cells) and then the phase of clonal expansion (generation of clones of T helper cells) (see section 2).

The production of mass density of T helper cells (τTh ) is due to the following three contributions

genetic mutation of naive T cells into T helper cells (see section 2), activation induced cell death of T helper cells (idem), clonal expansion of T helper cells produced during the activation phase (idem)

The ﬁrst two contributions are determined during the activation phase. The ﬁrst contribution is modelled as a term equal and opposite in sign to the term (6)

hc1ρT . (12) The second contribution is modeled as a term linearly depending on the actual value of density of T helper cells

−hap(c1)ρTh , (13) where the induced cell death rate kap depends on the concentration of the considered set of cytokines (see section 2 for the biological based motivation of this assumption). Assuming a linear relation regarding the dependance of hap on c1, this last term takes the form 66 Marina Dolﬁn, Demetrio Criaco

˜ hapc1ρT1 , (14) where hãp is the constant induced cell death coefficient characterizing the apoptosis of T helper cells. More complicated choices are also possible however and do not affect the model qualitatively. In the following we will continue to call hãp by the name hap. By summing up the two terms (12) and (14), one obtains the production of mass density of T helper cells during the activation phase

(τ ) = hc ρ − h c ρ . (15) Th activ. |{z}1 T |ap {z1 T }h genetic mutation cell death During the subsequent phase of proliferation called clonal expansion phase, the T helper cells (produced during the activation phase) proliferate by generation of clones so that the total production of mass of T helper cells is given by

activ. and proli f. z }| {

τTh = α(t)(τTh )activ., (16) where α is the proliferation rate factor of T helper cells characterizing the clonal expansion phase. By biological considerations, we assume that the proliferation rate factor is function of time. Remark 3.1. The function of time representing the proliferation rate factor may be characterized by the property that α(t) → 1 when t → ∞ in order to mimic the fact that the peak of the T cells differentiation happens within 24-48 hours after the antigen intrusion and then decreases disappearing within 5-10 days [1, 4]. To mimic this observed phenomenon, one may choose the function α(t) to be the analytical dα α α0k solution of the logistic equation of Verhulst = r(1 − )α, i.e. α(t) = −rt dt k α0+(k−α0)e where r is the intrinsic growth rate factor and k is the saturation level. One may assumes the saturation level k = 1 in order to mimic the fact that the total number of lymphocytes in the organism is kept constant for large values of time with respect to the time interval of the acute phase of the immune response. Then we define the net generation of clones of T helper cells as the difference between the total number of T helper cell clones which results from the activation and clonal expansion phases and the initial set of the same cells, i.e. the T helper cells produced during the activation phase

(τTh )proli f. = α(t)(τTh )activ. − (τTh )activ. = [α(t) − 1](hc1ρT − hapc1ρTh ) (17) and we classify it as a proliferation event. Eq.(17) will be used in the following. By plugging in the general form of the balance equation of mass density (4) the phenomenologically derived ﬂux (11) and production of mass density (16), one obtains the balance equation of mass density for T helper cells as A mathematical model for clonal expansion of antigen speciﬁc T cells 67

∂ρ Th + ∇ · (ρ v ) − r4ρ = α(t)(hc ρ − h c ρ ) (18) ∂t Th Th Th 1 T ap 1 Th where vTh is the velocity of the T helper cells and remembering that the random motility coeﬃcient r is kept constant.

3.5. DENDRITIC CELL BALANCE EQUATION OF MASS DENSITY Regarding the dendritic cells, we model again the random motility of cells (which results in avoiding overcrowding [15]) with a diﬀusion-like ﬂux [12, 13, 31, 14]

Jd = −r∇ρd (19) where for simplicity and not aﬀecting the model qualitatively, the same random motility coeﬃcient r has been assumed as for naive T cells and T helper cells. By analyzing the phenomenology of the dynamics of activation and clonal expansion of T cells with a special regard to the Antigen Presenting Cells (dendritic cells in this case) [1, 4] one may deduce that the production of mass density of dendritic cells is zero

τd = 0. (20) By plugging into the general form of the balance of mass density (4) the phenomenologically derived ﬂux (19) and production of mass density (20), one obtains the balance of mass density for dendritic cells as ∂ρ d + ∇ · (ρ v ) − r4ρ = 0 (21) ∂t d d d where vd is the velocity of dendritic cells and remembering again that the random motility coeﬃcient r is kept constant.

3.6. SET OF CYTOKINES BALANCE EQUATION OF MASS DENSITY Chemoattractant (cytokines in this case) diffusion is assumed to follow Fick’s law, with diffusion coefficient D [13],

J1 = −D∇ρ1. (22) In general D could be function of the concentration of the attractant [21, 22], however we will assume it constant in order to elucidate the most fundamental behavior. The diffusion coefficient has unit of area/time. In [12] one can find interesting observations about the effect of the diffusion coefficient with respect to the chemotactic response; in fact the larger the diffusion coefficient, the faster the gradient of the attractant will decay and the smaller the chemotactic response. 68 Marina Dolfin, Demetrio Criaco

The production of mass density of the set of cytokines (τ1) is due to the following three contributions

secretion of cytokines of the considered set by the T helper cells (see section 2),

secretion of cytokines of the considered set by the dendritic cells (idem),

consuption of cytokines of the considered set due to degradation processes.

The ﬁrst two contributions to the production of mass density of the set of cytokines can be modeled as the sum of two terms both depending on the actual value of the mass density of the set of cytokines via two functions of the concentration of T helper and dendritic cells respectively

µ(cTh )ρ1 + ν(cd)ρ1, (23) where µ(cTh ) is the production rate of the set of cytokines due to the production by the T helper cells and ν(cd) is the production rate of the set of cytokines due to the production by the dendritic cells. Assuming a linear relation regarding the dependance of µ(cTh ) on cTh and the dependance of ν(cd) on cd, the term (23) takes the form

(˜µcTh + ν˜cd)ρ1, (24) whereµ ˜ is the constant production rate of the set of cytokines by the T helper cells andν ˜ is the constant production rate of the set of cytokines by the dendritic cells. More complicated choices are also possible however and do not aﬀect the model qualitatively. In the following we continue to call the quantitiesµ ˜ andν ˜ with the name µ and the name ν respectively. The third contribution to the production of the considered set of cytokines, i.e. the consumption due to degradation processes, can be modelled via the following ﬁrst order term [16] 1 − ρ , (25) γ 1 where γ is the half-life rate of the set of cytokines. By summing up the contributions (24) and (25), one obtains the production density of the set of cytokines as

1 τ = (µc + ν c − )ρ . (26) 1 Th 1 d γ 1 By plugging into the general form of the balance of mass density (4), the phenomenologically derived ﬂux (22) and production of mass density (26), one obtains the balance of mass density for the set of cytokines as A mathematical model for clonal expansion of antigen speciﬁc T cells 69

∂ρ 1 1 + ∇ · (ρ v ) − D4ρ = (µc + νc − )ρ , (27) ∂t 1 1 1 Th d γ 1 where v1 is the velocity of the set of cytokine and remembering that the diﬀusion coeﬃcient D is kept constant in this case.

4. FLUID MIXTURE BALANCE EQUATION OF MASS DENSITY We define the baricentral velocity of the mixture [27, 28] as 1 v = (ρ v + ρ v + ρ v + v ). (28) ρ T T Th Th d d 1 The sum of all the productions of mass density of the constituents given by eqs. (9), (16), (20) and (26) is 1 (k − k − hc )ρ + (µc + νc − )ρ + α(t)(hc ρ − h c ρ ) (29) 0 ap 1 T Th d γ 1 1 T ap 1 Th which, by simple algebraic calculations, can be rewritten as 1 (k − k − h c )ρ + (µc + νc − )ρ + [α(t) − 1](hc ρ − h c ρ ). (30) 0 ap ap 1 T Th d γ 1 1 T ap 1 Th As already said, the net generation of clones of T helper cells (17) represents a proliferative event (i.e. not conservative of the mass density, at least regarding to the acute phase of the immune response), so that by substracting the quantity (17) from eq. (30), one obtains the sum of productions of mass densities of the components of the fluid mixture subjected to the requirement to be zero because of the conservation of mass density of the mixture as a whole 1 (k − k − h c )ρ + (µc + νc − )ρ = 0. (31) 0 ap ap 1 T Th d γ 1 By summing up the balances of mass for all the constituents (10), (18), (21) and (27) and by taking into account the requirement (31), one obtains the balance equation of mass density for the mixture as ∂ρ + ∇ · (ρv) − r4ρ + (r − D)4ρ = [α(t) − 1](hc ρ − h c ρ ), (32) ∂t 1 1 T ap 1 Th where relation (2) for the density of the mixture has been used. The mass density of the mixture as a whole is not conserved in this case (see [11] for analogies in growth of soft tissues and [29] and [30] for analogies in tumor modeling). Remembering the observations made in Remark 3.1, one notices that due to the property of the function α(t) (in particular that for t → ∞ it is α = 1), for large values of t the production of mass density of the mixture as a whole tends to zero and 70 Marina Dolfin, Demetrio Criaco the conservation of mass is achieved. This fact mimics well the phenomenology of the T cell mediated immune response (see section 2).

5. BALANCE EQUATIONS OF MOMENTUM DENSITIES FOR THE CONSTITUENTS OF THE MIXTURE OF BIOLOGICAL FLUIDS In this section we present explicit expressions for the balance equations of momentum densities for all the constituents of the mixture modeling some aspects of the dynamics of activation of naive T cells and clonal expansion of T helper cells. In the thermodynamic theory of mixtures of ﬂuids, the productions of momentum densities of mass of all the constituents are constrained by the requirement that the sum of the productions of momentum densities of all constituents is zero in order to a achieve the conservation of momentum density for the mixture as a whole [27, 28]. This requirement still applies in our model, except that for the production of momentum due to the net generation of clones of T helper cells. This generation of clones represents a proliferative event which determines that the momentum density of the mixture as a whole is not conserved, although only regarding the acute phase of the immune response to the antigen challenge.

5.1. LOCAL FORM OF THE GENERAL BALANCE EQUATION OF MOMENTUM DENSITY Following [16], we assume the general form of the balance equation of momentum for the generic biological ﬂuid of density %, in a way analogous to the balance equation of momentum density for a constituent of a reacting mixture of ﬂuids [27, 28]

∂%v% + ∇ · [%v ⊗ v − t ] = m + ρf, (33) ∂t % % % where v% is the velocity of the constituent, f is the external force per unit mass, t% is the stress tensor related to the considered constituent and m is the production of momentum density which is given by the sum of two terms

m = I + τ%U. (34) In eq.(34), I is the interaction force exerted on the constituent by the other constituents (chemotactic force in our model), U is the velocity of the produced mass density and τ%U is the production of momentum density that accompanies the production of mass τ%. In the case of the constituents of the mixture modeling the dynamics of activation and clonal expansion of T cells mediated immune response, from biological considerations, one can assume that no external forces are present [17], i.e. the force f = 0 in eq.(33) is zero for all the constituents. A mathematical model for clonal expansion of antigen speciﬁc T cells 71

The production of momentum density of the mixture as a whole is constrained by the requirement that the sum of the productions of momentum density of all the constituents is zero [27]; it expresses the conservation of momentum of the mixture as a whole. This requirement still applies to the case of our model, with the exception of the production of momentum density due to the net generation of clones of T helper cells due to the clonal expansion phase (see the introduction to section 2). We remark here, following [16], that the first and second terms on the left hand side of eq.(33), are only formally identical to the usual material acceleration of a continuum but this does not mean that cells behave like particles, the Galilean inertia being negligible in this context. In particular the non linear term on the left hand side of eq.(33) accounts for persistence in cell motion, i.e. for the cells ”inertia” in changing their direction [33]. To construct our model, in the following sections we write explicit phenomenological expressions for the quantity m for each constituent of the mixture of fluids and we assume specific constitutive relations for the stress tensor of each constituent of the mixture of biological fluids introduced in section 3.1.

5.2. NAIVE T CELL BALANCE EQUATION OF MOMENTUM DENSITY Basing on biological considerations [17], we may neglect the chemotactic attraction on naive T cells due to the presence of the chemoattractors (cytokines) (in literature naive T cells are also called ”resting” T cells [1, 4]). No other interaction forces, except the chemotactic one are taken into account in our model, so in the case of naive T cells the interaction force due to the other constituents is assumed to be zero

IT = 0. (35)

The production of momentum density for naive T cells due to the production of mass density τT (see eq.(9)) is given by

mT = τT vT = (k0 − kap − hc1)ρT vT , (36) where vT is the velocity of the produced mass in this case. By using eqs.(35) and (36), the general form of the balance equation of momentum density (33) in the case of naive T cells takes the form

∂(ρ v ) T T + ∇ · [ρ v ⊗ v − t ] = (k − k − hc )ρ v , (37) ∂t T T T T 0 ap 1 T T where tT is the stress related to naive T cells. 72 Marina Dolﬁn, Demetrio Criaco 5.3. T HELPER CELL BALANCE EQUATION OF MOMENTUM DENSITY The chemotactic attraction acting on the T helper cells is introduced as a force that tends to aggregate the cells, driving them along the direction of the chemical gradient [16], which is set of cytokines (see section 2) in this case

ITh = χhρTh ∇c1, (38) where χh is the chemotactic sensitivity coeﬃcient measuring the strength of T helper cell response to chemotactic signaling [16].

The production density of mass τTh given by eq.(16) determines a production of momentum density for the T helper cells given by

mTh = τTh vTh = α(t)(hρT − hapρTh )c1vTh , (39) where vTh is the velocity of the produced mass (see eq.(34)) in this case. The production of momentum density for the T helper cells is given by the sum of eqs. (38) and (39)

α(t)(hρT − hapρTh )c1vTh + χhρTh ∇c1. (40) By using eq.(40), the general form of the balance of momentum density (33) in the case of T helper cells takes the form:

∂(ρ vT ) Th h + ∇ · [ρ v ⊗ v − t ] = α(t)(hρ − h ρ )c v + χ ρ ∇c (41) ∂t Th Th Th Th T ap Th 1 Th h Th 1

where tTh is the stress tensor related to the T helper cells. 5.4. DENDRITIC CELL BALANCE EQUATION OF MOMENTUM DENSITY The set of cytokines acts as chemoattractor on the dendritic cells [1, 4]; the chemotactic interaction is introduced also for the dendritic cells as a force that tends to aggregate the cells, driving them along the direction of the chemical gradient [16] of the chemoattractor (cytokines)

Id = χdρd∇c1. (42) The production of momentum density for dendritic cells due to the production of density of mass τd (see eq.(20)) is zero

md = τdvd = 0. (43) A mathematical model for clonal expansion of antigen speciﬁc T cells 73

By using eqs.(42) and (43), the general form of the balance equation (33) in the case of the dendritic cells takes the form ∂(ρ v ) d d + ∇ · [ρ v ⊗ v − t ] = χ ρ ∇c . (44) ∂t d d d d d d 1 5.5. SET OF CYTOKINES BALANCE EQUATION OF MOMENTUM DENSITY We may assume that no interactions force are exerted on the set of cytokines, so that

Ic1 = 0. (45) The production of density of momentum for the set of cytokines is due to the production density of mass balance τ1 (see eq.(26)) and is given by 1 m1 = τ1v1 = (µ1cTh + ν1cd − )ρ1v1, (46) γ1 where v1 is the velocity of the produced mass in this case. By using eqs.(45) and (46), the general form of balance of density of momentum (33) in the case of the set of cytokines takes the form

∂(ρ1v1) 1 + ∇ · [ρ1v1 ⊗ v1 − t1] = (µ1cTh + ν1cd − )ρ1v1. (47) ∂t γ1 6. BALANCE EQUATION OF MOMENTUM DENSITY OF THE MIXTURE The total net generation of clones of T helper cells (17), determines a contribution to the production of momentum density of the mixture given by

τρvTh = [α(t) − 1](hρT − hapρTh )c1vTh . (48) The sum of all the productions of momentum densities of the constituents given by eqs. (36, (40), (43) and (46) is

(k0 − kap − hc1)ρT vT + α(t)(hρT − hapρTh )c1vTh + χhρTh ∇c1+ 1 +χ ρ ∇c + (µc + νc − )ρ v , (49) d d 1 Th d γ 1 1 which, by simple algebraic calculations, can be rewritten as

1 (k − k − hc )ρ v + χ ρ ∇c + χ ρ ∇c + (µc + νc − )ρ v + 0 ap 1 T T h Th 1 d d 1 T1 d γ 1 1 74 Marina Dolﬁn, Demetrio Criaco

+[α(t) − 1](hρT − hapρTh )c1vTh . (50)

As already said, the net generation of clones of T helper cells (17) represents a proliferative event (i.e. not conservative also of the momentum density, at least regarding to the acute phase of the immune response), so that by substracting the quantity (48) from eq. (50), one obtains the sum of productions of momentum densities of the components of the ﬂuid mixture subjected to the requirement to be zero because of the conservation of momentum density of the mixture as a whole [27, 28]

1 (k − k − hc )ρ v + χ ρ ∇c + χ ρ ∇c + (µc + νc − )ρ v = 0. (51) 0 ap 1 T T h Th 1 d d 1 T1 d γ 1 1

By summing up equations (37), (41), (44) and (47), and by taking into account the requirement (51), one obtains the balance equation for the momentum density of the mixture of biological ﬂuids as

∂(ρv) + ∇ · [ρv ⊗ v − t] = [α(t) − 1](hρ − h ρ )c v . (52) ∂t T ap Th 1 Th

To derive eq. (52), the following deﬁnition for the mixture stress t has been used [27, 28]

t = tT + tTh + td + th − ρT uT ⊗ uT − ρTh uTh ⊗ uTh − ρdud ⊗ ud − ρ1u1 ⊗ u1, (53) where

uT = vT − v , uTh = vTh − v , ud = vd − v , u1 = v1 − v (54) are the partial velocities.

7. MATRIX FORM OF THE BALANCE EQUATIONS

Among the ﬁve introduced mass densities (ρ, ρT , ρTh , ρd, ρ1) only 5 − 1 = 4 are independent. Then, we choose the following state space C of independent ﬁelds

C = (ρ, ρTh , ρd, ρ1, v, vTh , vd, v1). (55)

To determine of these fields we need the appropriate number of field equations [28]. They are based on the balance equations of mass densities and momentum densities of the constituents; these equations have been deduced in section 3 and they form the A mathematical model for clonal expansion of antigen specific T cells 75 following system

 ρ  ∂ρ + ∇ · (ρv) − r4ρ + (r − D)4ρ = [α(t) − 1](h 1 ρ − h c ρ ) ,  ∂t 1 ρ T ap 1 T1    ∂ρT ρ  h + ∇ · (ρ v ) − r4ρ = α(t)(hρ − h ρ ) 1 ,  ∂t Th Th Th T ap Th ρ    ∂ρ  d + ∇ · (ρ v ) − r4ρ = 0,  ∂t d d d   ρ  ∂ρ1 Th ρd 1  + ∇ · (ρ1v1) − D4ρ1 = (µ + ν − )ρ1,  ∂t ρ ρ γ   ∂(ρv) + ∇ · [ρv ⊗ v − t] = [α(t) − 1](hρ − h ρ ) ρ1 v ,  ∂t T ap Th ρ Th    ∂(ρT vT ) ρ ρ  h h + ∇ · [ρ v ⊗ v − t ] = α(t)(hρ − h ρ ) 1 v + χ ρ ∇ 1 ,  ∂t Th Th Th Th T ap Th ρ Th h Th ρ    ∂(ρdvd) ρ1  + ∇ · [ρ vd ⊗ vd − td] = χ ρ ∇ ,  ∂t d d d ρ   ρ  ∂ρ1v1 Th ρd 1 ∂t + ∇ · [ρ1v1 ⊗ v1 − t1] = (µ ρ + ν ρ − γ )ρ1v1, (56) where the relation deﬁning the density of the mixture (2) and the deﬁnitions (53) and (54) have been considered.

7.1. CONSTITUTIVE ASSUMPTIONS Constitutive equations for the stress tensor of each constituent of the mixture are needed in order to close the system of equations (56). In our model, we assume that the ﬂuids modeling the populations of cells and the chemicals are non-viscous and simple [27, 28]; i.e. the following relations hold

tT = −pT (ρT , T)I, tTh = −pTh (ρTh , T)I, td = −pd(ρd, T)I, t1 = −p1(ρ1, T)I, (57) where I is the identity matrix and T is the absolute temperature. Each ﬂuid constituent of the mixture is simple in the sense that the partial pressure of each constituent depends only on its own density, and on T [28]. Regarding to our model the phenomenon under consideration is assumed isothermal so the dependance on T is not considered. By substituting the constitutive equations (57) into the expression for the stress tensor of the mixture (53), the following equation is obtained

t = −pI − (ρT uT ⊗ uT + ρTh uTh ⊗ uTh + ρdud ⊗ ud + ρ1u1 ⊗ u1), (58) where p = pT + pTh + pd + p1 is the scalar pressure of the mixture. In the case of isothermal processes in non-viscous fluids, the following equation of state may be 76 Marina Dolfin, Demetrio Criaco assumed [34] ∂F p = ρ2 (59) ∂ρ where F is the free energy and T the absolute temperature ([34]). By assuming a linear dependance of the free energy on the mass density for each constituent of the fluid mixture, we obtain the following equations of state for the partial pressures

p = pˆ ρ2 , p = pˆ ρ2 , p = pˆ ρ2, p = pˆ ρ2, (60) T T T Th Th Th d d d 1 1 1

∂F ∂F ∂F ∂F where the quantitiesp ˆT = ∂ρ , pˆTh = ∂ρ , pˆd = ∂ρ , pˆ1 = ∂ρ , are positive constants T Th d 1 [17, 18]. Because of the low involved velocities of the cells and the chemicals [17, 18], we disregard all the quadratic terms in the velocities in the balance equations of momentum densities; from a modellization point of view this means that we do not take into account from now on in our model the eﬀect of persistence in cell motion [35] (see section 5). The system (56), together with (57) and (60) (and neglecting the inertial terms), takes the form

 ρ  ∂ρ + ∇ · (ρv) − r4ρ + (r − D)4ρ = [α(t) − 1](hρ − h ρ ) 1 ,  ∂t 1 T ap Th ρ    ∂ρT ρ  h + ∇ · (ρ v ) − r4ρ = α(t)(hρ − h ρ ) 1 ,  ∂t Th Th Th T ap Th ρ    ∂ρ  d + ∇ · (ρ v ) − r4ρ = 0,  ∂t d d d   ρ  ∂ρ1 Th ρd 1  + ∇ · (ρ1v1) − D4ρ1 = (µ1 + ν1 − )ρ1,  ∂t ρ ρ γ (61)   ∂(ρv) + 2p ˆρ∇ρ = [α(t) − 1](hρ − h ρ ) ρ1 v ,  ∂t T ap Th ρ Th    ∂ρT vT ρ ρ  h h + 2p ˆ ρ ∇ρ − χ ρ ∇ 1 = α(t)(hρ − h ρ ) 1 v ,  ∂t Th Th Th h Th ρ T ap Th ρ Th    ∂ρdvd ρ1  + 2p ˆdρ ∇ρ − ρ χ ∇ = 0,  ∂t d d d d ρ   ρ  ∂ρ1v1 T1 ρd 1 ∂t + 2p ˆ1ρ1∇ρ1 = (µ ρ + ν ρ − γ )ρ1v1, Eqs.(61) give a system of 16 quasi-linear second order PDEs for the mass densities of the mixture of biological ﬂuids considered, the T helper cells, the dendritic cells and the set of cytokines, together with the related velocities. We introduce

T U = (ρ, ρTh , ρd, ρ1, v, vTh , vd, v1) , (62) where again the relation deﬁning the density of the mixture (2) has to be considered. Then, the system of equations (61) can be written in the following matrix form A mathematical model for clonal expansion of antigen speciﬁc T cells 77

∂U ∂2U Aα(U) + rHk(U) + B(U, x0) = 0, (63) ∂xα ∂(xk)2 where α = 0, 1, 2, 3 and the xk(k = 1, 2, 3), x0 = t represent, respectively, the spatial coordinates (i.e. the components of the position vector x in Eulerian coordinates in a cartesian reference frame) and time, Aα(U) (with α = 0, 1, 2, 3), Hk(U)( with k = 1, 2, 3) are appropriate 16 × 16 square matrices and B(U, x0) is the appropriate column vector. The terms containing derivatives of the second order is multiplied by the random motility coefficient r which is a very little (r 1) parameter (see section 3.3) and the Einstein summation convention over repeated indices is understood. If one does not consider the terms of the second order into the system of PDEs (63) (corresponding in disregarding the effects of random motility of cells and diffusion of the cytokines), a quasi-linear system of equations of the first order is obtained; in a forthcoming paper [19] the hyperbolicity in time direction for this particular case is proved and the propagation of non linear waves in the general case represented by the system PDEs (63) is studied by using a perturbative method.

References

[1] G.R. Burmester, A.Pezzutto, J.Wirth, Color Atlas of Immunology, Verlag, Germany, 2003. [2] A.S. Perelson,G. Weisbuch, Immunology for physicists, Reviews of Modern Physics, 69, 4(1997), 1219-1267. [3] A. Lanzavecchia, F. Sallusto, M.L. Dustin, Dynamics of T lymphocyte responses: intermediates, effectors, and memory cells, Science, 290(2000), 92-97. [4] G. Pinchuk, Immunology, Schaum’s Outline Series, Mc Graw-Hill, 2002. [5] B. Kohler, Mathematically modeling dynamics of T cell responses: Predictions concerning the generation of memory cells, Journal of Theoretical Biology, 245(2007), 669-676. [6] A.S. Perelson, Modelling viral and immune system dynamics, Nature Rev. Immunol., 2(2002), 8-36. [7] K.M. Murphy, W. Ouyang, J.D. Farrar, J.F. Yang, S. Ranganath, H. Asnagli, M. Afkarian, T.L. Murphy, Signaling and transcription in T helper development, Annual Review of Immunology, 18(2000), 451-494. [8] K. Nelms, A.D. Keegan, J. Zamorano, J.J. Ryan, W.E. Paul, The IL-4 receptor: signaling mechanisms and biologic functions, Annu Rev Immunol., 17(1999), 701-38. [9] L. Mariani, M. Lohning,¨ A. Radbruch, T. Hofer,¨ Transcriptional control networks of cell differentiation: insights from helper T Lymphocytes, Prog Biophys Mol Biol., 86(2004), 45-76. [10] L. H. Glimcher, K. M Murphy, Lineage commitment in the immune system: the T helper lymphocyte grows up, Genes Dev., 14(2000), 1693-1711. [11] J.D. Humphrey, K.R. Rajagopal, A constrained mixture model for growth and remodeling of soft tissues, Mathematical Models and Methods in Applied Sciences, 12, 3(2002), 407-430. [12] R.M. Ford, D.A. Lauffenburger, Analysis of chemotactic becterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients, Bull. of Math. Biol., 53, 5(1991), 721-749. 78 Marina Dolfin, Demetrio Criaco

[13] D.A. Lauffenburger, K.H. Keller, Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, J. Theor. Biol., 81(1979), 475-503. [14] R.T. Tranquillo, D.A. Lauffenburger, SH. Zigmond, A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations, The J. of Cell Biology, 106(1988), 303-309. [15] K. Pointer, T. Hillen, Volume filling and quorum sensing in models for chemosensitive movement, Canadian Applied Mathematics Quaterly, 10, 4(2003), 280-301. [16] A. Tosin, D. Ambrosi, L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks, Bulletin of Mathematical Biology, (on-line), 2006, 1-20. [17] J.D. Murray, Mathematical Biology I: An introduction, Springer, 2002. [18] J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2002. [19] M.Dolfin, L.Restuccia, accepted communication to Simai Congress, 2010. [20] R.A. Seder, W.E. Paul, Acquisition of lymphokine-producing phenotype by CD4+ T cells, Annu. Rev. Immunology, 12(1994), 635-673. [21] E.F. Keller, L.A. Segel, Model for chemotaxis, J. Theor. Biol., 30(1971), 225-234. [22] E.F. Keller, L.A. Segel, Traveling ands of chemotactic bacteria: a theoretical analysis., J. Theor. Biol., 30(1971), 235-248. [23] F. Zanlungo, S. Rambaldi, G. Turchetti, An automata based microscopic model inspired by the clonal expansion, Mathematical modeling of Biological Systems II, A. Deutsch et al. (Eds.), Birkhauser, Boston, 2008. [24] V.A.A. Jansen, H. Korthals Altes, G.A. Funk, D. Wodarz, Contrasting B cell- and T cell-based protective vaccines, J. of Theoretical Biology, 234(2005), 39-48. [25] R.J. De Boer, A.S. Perelson, T cell repertoires and competitive exclusion, Journal of Theoretical Biology, 169(1994), 375-390. [26] M. Dolfin, D. Criaco, A mathematical model on activation and clonal expansion of T helper cells (preprint). [27] I. Muller, Thermodynamics, Pitman Advanced Publishing Program, 1985. [28] Muller, T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37(1998), 84-92. [29] A. Georgescu, L. Palese, G. Raguso, Biomatematica - Modelli dinamica e biforcazione, Cacucci Editore, 2009. [30] J.S. Bertram, The molecular biology of cancer (Review), Molecular Aspects of Medicine, 21(2001), 167-223. [31] D.A. Lauffenburger, Ph.D Thesis, University of Minnesota, 1979. [32] R.J. De Boer, D. Homann, A.S. Perelson, Different dynamics of CD4+ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection., J. Immunol., 171(2003), 3928-3935. [33] P. Friedl, K. Wolf, Tumour-cell invasion and migration: diversity and escape mechanisms, Nature Rev. Cancer, 3(2003), 362-374. [34] G.Carini, 101 Lezioni di Istituzioni di Fisica Matematica, Mediterranean Press, 1989. [35] N. Bellomo, E. De Angelis, L. Preziosi, Multiscale Modeling and Mathematical Problems related to Tumor volution and Medical Therapy, J. of Theoretical Biology, 5, 2(2003), 111-136. ROMAI J., 6, 1(2010), 79–84

NEW PROPERTIES FOR A CLASS OF MULTIVARIATE OPERATORS Loredana-Florentina Galea Agora University, Oradea, Romania [email protected] Abstract In this paper we are concerned with a class of multivariate operators. Relying on the results of the weakly Picard operator’s theory, our aim is to study the good and special weakly Picard properties for these operators.

Keywords: Picard operators, weakly Picard operators, good Picard operators, special Picard operators. 2000 MSC: 47H10, 41A10.

1. INTRODUCTION The class of multivariate operators was introduced in 2008 by C. Bacot¸iu [2]. In this paper our aim is to study the good and special weakly Picard properties for these operators. The notions of good and special weakly Picard operator are defined by I.A. Rus in 2003, [5], and good and special convergence of type M of the sequence of successive approximation in metric spaces was introduced by L. D’Apuzzo in 1976, [1]. Let (X, d) be a metric space and A : X → X an operator. In this paper we use the following notations: P(X) = {Y ⊂ X | Y , ∅}; FA = {x ∈ X | A(x) = x} - the fixed point set of A; I(A) = {Y ∈ P(X) | A(Y) ⊂ Y}- the family of nonvoid invariant subsets of A; 0 1 n+1 n A := 1X, A := A, ..., A := A ◦ A , n ∈ N. In what follows we present the basic features of the theory while in the next section we state the main results. Definition 1.1. (I.A. Rus - [6], [7], [5]) Let (X, d) be a metric space. 1) An operator A : X → X is a weakly Picard operator (briefly WPO) if the sequence m of successive approximations (A (x0))m∈N converges for all x0 ∈ X and the limit (which may depend on x0) is a fixed point of A. ∗ 2) If the operator A : X → X is WPO and FA = {x }, then by definition the operator A is a Picard operator (briefly PO). 3) If the operator A : X → X is WPO, then the operator A∞ defined by A∞ : X → X, A∞ (x) := lim Am (x) can be considered. m→∞

79 80 Loredana-Florentina Galea

The basic result in the WPO’s theory is the following:

Theorem 1.1. (Characterization theorem [6], [7], [5])S An operator A : X → X is WPO if and only if there exists a partition of X, X = Xλ, such that: λ∈Λ (a) Xλ ∈ I (A) , ∀λ ∈ Λ;

(b) A|Xλ : Xλ → Xλ is PO, ∀λ ∈ Λ. Deﬁnition 1.2. Let (X, d) be a metric space and A : X → X a WPO. P∞ 1) A : X → X is a good WPO, if the series d Am−1 (x) , Am (x) converges, for all m=1 x ∈ X (see [7]). When the sequence d Am−1 (x) , Am (x) is strictly decreasing m∈N∗ for all x ∈ X, A is a good WPO of type M (see [1]). P∞ 2) A : X → X is a special WPO, if the series d (Am (x) , A∞ (x)) converges, for all m=1 m ∞ x ∈ X(see [7]). When the sequence (d (A (x) , A (x)))m∈N∗ is strictly decreasing for all x ∈ X, A is a special WPO of type M (see [1]). Theorem 1.2. (S. Mure¸san,L.F. Galea - [3]) Let (X, d) be a metric space and A : X → X a WPO. If A is a special WPO then A is a good WPO. In [2], C. Bacot¸iu introduced the class of multivariate approximation operators and investigated some properties for these operators. We present the basic results regarding this class of multivariate approximation operators. For a ﬁxed integer p > 1, we consider the domain D := [0, 1] × [0, 1] × ... × [0, 1] = [0, 1]p | {z } p Let us denote:

<0> α := (0, 0, ..., 0) = 0Rp ,

<1> <1> <1> α1 := (1, 0, ..., 0) , α2 := (0, 1, ..., 0) , ..., αp := (0, 0, ..., 1) .

That is, with the notation M1 := {i1 : 1 6 i1 6 p} ⊂ N, αi , (i1 ∈ M1) represents the p 1 vector from R which has 1 on the position i1 and 0 in the rest. Generally, for all k ∈ 1, p and for all 1 6 i1 < i2 < ... < ik 6 p we deﬁne αi ,i ,...,i p 1 2 k as the vector from R which has 1 on the positions i1, i2, ..., ik and 0 in the rest. < k > is the number of positions on which appears the number 1 and i1, i2, ..., ik indicate these positions. The rest of the components are equal to 0. We denote k Mk := {(i1, i2, ..., ik) : 1 6 i1 < i2 < ... < ik 6 p} ⊂ N n o n o and deﬁne the set: ν := α<0> ∪ α : k = 1, p and (i , ..., i ) ∈ M . D i1,...,ik 1 k k New properties for a class of multivariate operators 81 ! p Remark 1.1. Card (M ) = , ∀ k ∈ 0, p and k k ! p P p p Card (νD) = = 2 := N. k=0 k p For any m1, m2, ..., mp ∈ N , consider the p-net

k ∆mk := 0 = xk,mk,0 < xk,mk,1 < ... < xk,mk,mk = 1 and the following system of real functions with positive values:

0 6 ψk,mk,i ∈ C [0, 1] , ∀ i = 0, mk, ∀ k = 1, p. We assume that the following conditions are satisﬁed mPk ( ) 1) ψk,mk,i x = 1, ∀ x ∈ [0, 1] , ∀ k = 1, p ; i=0 mPk ( ) 2) xk,mk,iψk,mk,i x = x, ∀ x ∈ [0, 1] , ∀ k = 1, p ; i=0 ( ) ( ) 3) ψk,mk,0 0 = ψk,mk,mk 1 = 1 , ∀ k = 1, p.

Deﬁnition 1.3. (C. Bacot¸iu - [2]) The operators Lm1,...,mp : C (D) → C (D) deﬁned m mP1 mPk Pp ( ) ( ) ( ) by Lm1,...,mp f x1, ..., xp := ...... ψ1,m1,i1 x1 ...ψk,mk,ik xk ... i1=0 ik ip

...ψp,mp,ip xp · f x1,m1,i1 , ..., xk,mk,ik , ...xp,mp,ip for any f ∈ C (D) and for any x1, ..., xp ∈ D are called multivariate operators. For these operators we have the following properties regarding the sets ( ) Xλ := f ∈ C (D) : f (α) = f α := λ , ∀α = α ∈ νD for any i1,...,ik ω α i1,...,ik i1,...,ik N λ = (λ1, ..., λN) ∈ R , (P1) Xλ is a closed subset of C (D); (P2) X ∈ I L , for all λ; λ Sm1,...,mp (P3) C (D) = Xλ is a partition of C (D); λ∈R (P4) In the following notations: n o K := {0, 1, ..., m1} × {0, 1, ..., m2} × ... × 0, 1, ..., mp n o p ∂K := (0, 0, ..., 0) , (m1, 0, ...0) , ..., m1, m2, ..., mp ⊂ R P um1,...,mp x1, ..., xp := ψm1,1,i1 (x1) ...ψmp,p,ip xp (i1,..,ip)∈∂K 82 Loredana-Florentina Galea n o

σm1,...,mp := inf um1,...,mp x1, ..., xp : x1, ..., xp ∈ D

Lm ,...,m : Xλ → Xλ is a contraction, having: 1 p Xλ ( ) ( ) ( ) 1 Lm1,...,mp f − Lm1,...,mp g C(D) 6 1 − σm1,...,mp k f − gkC(D) for all f, g ∈ Xλ. Theorem 1.3. (C. Bacot¸iu - [2]) The multivariate operators L are WPO, for m1,...,mp p σm1,...,mp , 0 and for m1, ..., mp ∈ N , we have: P P L∞ ( f ) x , ..., x = C0 + C1 x + C2 x x + m1,...,mp 1 p 0 i1 i1 i1,i2 i1 i2 i1∈M1 (i1,i2)∈M2 P +... + Ck x ...x + .... + C p x x ...x i1,...,ik i1 ik 1,2,...,p 1 2 p (i1,...,ik)∈Mk for all f ∈ C (D) and for all x , ..., x ∈ D, where C0 and Ck , ∀ k ∈ 1, p, 1 p 0 i1,...,ik ∀ (i1, ..., ik) ∈ Mk are real numbers which depend on f , and there have the following expressions: 0 <0> C0 = f α , P P Ck := (−1)k f α<0> + (−1)k−1 f α<1> + (−1)k−2 f α<2> + ... i1,...,ik iS 1 iS 1 ,iS 2 S 1=1 16S 1 ~~+ (−1)0 f α . iS 1 ,...,iS l i1,...,ik 16S 1~~

0 P 1 P 2 = 1 − σm ,...,m f x1, ..., xp − C + C x1 + C xi xi 1 p 0 i1 i1,i2 1 2 i1∈M1 (i1,i2)∈M2 ! P k p + C xi xi ...xi + ... + C x1 x2...xp 6 i1,...,ik 1 2 k 1,2,...,p (i1,...,ik)∈Mk New properties for a class of multivariate operators 83

6 1 − σm1,...,mp · C ( ) where C = max diam (Im f ) , (p + 1) max | f (x)| . x∈[0,1]p By induction, for n ∈ N∗, we have: n ( ) ∞ ( ) Lm1,m2,...,mp f x1, ..., xp − Lm1,m2,...,mp f x1, ..., xp =

1 n−1 ( ) 1 ∞ ( ) Lm1,m2,...,mp Lm1,m2,...,mp f x1, ..., xp − Lm1,m2,...,mp Lm1,m2,...,mp f x1, ..., xp 6 ( ) n 6 1 − σm1,m2,...,mp · C, ∀ x ∈ D, C = max diam (Im f ) , (p + 1) max | f (x)| x∈[0,1]p P∞ n ( ) ∞ ( ) Hence Lm1,...,mp f x1, ..., xp − Lm1,...,mp f x1, ..., xp 6 m=1 n 1− 1−σm1,m2,...,mp 6 C · 1 − σm1,m2,...,mp · , ∀ x ∈ D with σm1,m2,...,mp ( ) C = max diam (Im f ) , (p + 1) max | f (x)| . x∈[0,1]p

~~Thus, we obtained the following result.~~

~~Theorem 2.1. The multivariate operators Lm1,...,mp are special weakly Picard operators of type M on C (D).~~

~~On the other hand, we have: n ( ) n−1 ( ) Lm1,m2,...,mp f x1, ..., xp − Lm1,m2,...,mp f x1, ..., xp =~~

~~ 1 n−1 ( ) 1 n−2 ( ) Lm1,m2,...,mp Lm1,m2,...,mp f x1, ..., xp − Lm1,m2,...,mp Lm1,m2,...,mp f x1, ..., xp 6~~

n−1 6 1 − σm1,m2,...,mp · C1, ∀ x ∈ D, C1 = diam (Im f ) P∞ So, Ln ( f ) x , ..., x − Ln−1 ( f ) x , ..., x 6 C 1 , ∀ x ∈ D, ∀ f ∈ m1,...,mp 1 p m1,...,mp 1 p 1 σ m=1 m1,...,mp C (D). Relying on the above results, we have the following theorem.

~~Theorem 2.2. The operators Lm1,m2,...,mp are good weakly Picard operators of type M on C (D). 84 Loredana-Florentina Galea References~~

[1] L. D’Apuzzo, On the notion of good and special convergence of the method of succesive approximations, Ann. Istit. Univ. Navale Napoli, 45/46(1976/1977), 123-138. (in Italian) [2] C. Bacot¸iu, Picard Operators and Applications, Napoca Star, Cluj-Napoca, 2008. [3] S. Mures¸an, L.F. Galea, On good and special weakly Picard operators (submitted). [4] I. M. Olaru, On some integral equation with deviating argument, Studia Univ. Babes-Bolyai, Cluj-Napoca, Seria Mathematica 1, 4(2005), 65-73. [5] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, 2001. (in Roma- nian) [6] I. A. Rus, Weakly Picard Operators and Applications, Seminar on Fixed Point Theory Cluj- Napoca, 2(2001), 41-58. [7] I. A. Rus, Picard operators and applications, Sci. Math.Japon., 58, 1(2003), 191-219. ROMAI J., 6, 1(2010), 85–94

STUDY OF THE LYAPUNOV STABILITY IN A BIOLOGICAL MODEL Raluca-Mihaela Georgescu Faculty of Mathematics and Computer Science, University of Pite¸sti,Romania [email protected] Abstract A biological mathematical model with three parameters, which describes a predator- prey relationship is analyzed. In this paper only the case of all three parameters strictly positive is considered. Numerically it is found that the only candidates for the Lyapunov asymptotically stable or unstable sets governing the phase portrait are the equilibrium points, so that their type was investigated. The corresponding global dynamic bifurcation diagram is carried out. Acknowledgement. This work was supported by the Grant 11/5.06.2009 within the framework of the Russian Foundation for Basic Research - Romanian Academy collaboration.

~~Keywords: dynamical system, Lyapunov stability, normal form. 2000 MSC: 37N25, 34D20.~~

1. INTRODUCTION This paper deals with a particular family of planar vector fields which represents a generalization of the Lotka-Volterra system. The considered model is obtained by certain parametrization of time from a particular case of a model proposed by Bazykin and Khibnik, [2]   x2(1 − x)  x˙ = − xy,  n + x  (1)   y˙ = −γy(m − x), where x and y represent the population numbers of the prey and the predator, respectively, and m, n, γ are nonnegative parameters describing the behaviour of isolated populations and their interaction. The presence of the three parameters is the source of a rich dynamics. The work is structured as follows: in Section 2 we present the model, in Section 3 we study the equilibrium points and find conditions for nonhyperbolicity for these. In Section 4 we deduce the normal forms for the cases when the equilibria are nonhyperbolic. For E2 the computations for finding the normal form are very complicated and we are forced to present only the algorithm, for a particular choice of the parameters. Then, in Section 5, conclusions concerning the dynamics are presented and we represent the global dynamic bifurcation diagram by using the WINPP program.

85 86 Raluca-Mihaela Georgescu 2. THE MATHEMATICAL MODEL We consider the Cauchy problem consisting of the system of ordinary diﬀerential equations (s.o.d.e.) (1) with initial conditions x(0) = x0, y(0) = y0. By multiplying the system (1) with n + x, we obtain an equivalent form of the (1) ( (n + x) · x˙ = x2(1 − x) − xy(n + x), (2) (n + x) · y˙ = −γy(m − x)(n + x).

Introducing a new time τ through the relation dt = (n + x)dτ, (2) becomes ( x˙ = x(x − yn − x2 − xy), (3) y˙ = −γy(m − x)(n + x), where, this time, the dot over quantities stands for the diﬀerentiation with respect to τ. This is the s.o.d.e. we are concerned with herein. Due to physical reasons, the phase space must be the ﬁrst quadrant (without axes of coordinates). However, for mathematical reasons we consider, in addition, the origin, and the half-axes.

3. THE EQUILIBRIUM POINTS We remark that, from a biological point of view the equilibrium having both coordinates non zero are of interest. However, we study all the equilibrium points in the phase space speciﬁed above. These are: O(0, 0), E1(1, 0) and E2(m, (m(1 − m)/(n + m)), that belongs to the closure of the ﬁrst quadrant only if m ∈ [0, 1]. At m = 0, E2 coincides with O, and at m = 1, E2 coincides with E1. The number and the multiplicity of the equilibrium points depend on the values of the parameters m, n and γ. By the Lyapunov-Perron linearization principle, the Lyapunov asymptotic stability of a hyperbolic equilibrium point, say (x∗, y∗), depends on the eigenvalues of the matrix of the system linearized around the point [4]. More exactly, the attractivity of (x∗, y∗) corresponds to the eigenvalues of the matrix ! 2x − yn − 3x2 − 2xy −x(n + x) A = . (4) γy(n − m + 2x) −γ(m − x)(n + x) (x∗,y∗) In the following we analyze the nature of the equilibrium points and this attractivity for all possible nonnegative values of the parameters m, n and γ. Since in our case the asymptotic stability (resp. the instability) implies attractivity (resp. repulsivity) and, conversely, for the sake of simplicity in the sequel we refer only to attractivity and repulsivity. ! 0 0 For the equilibrium point O, A becomes A = , which has the eigen- 0 −γmn values λ1 = 0, λ2 = −γmn < 0. Thus O is a saddle-node. Study of the Lyapunov stability in a biological model 87

For the equilibrium point E1, the matrix A takes the form ! −1 −(n + 1) A = , 0 −γ(m − 1)(n + 1) and has the eigenvalues λ1 = −1 < 0, λ2 = −γ(m − 1)(n + 1). Therefore, if m < 1, E1 is a saddle, a saddle-node for m = 1 and an attractive node for m > 1. For the equilibrium point E2, the matrix A takes the form  2   −m(m + 2mn − n)   −m(m + n)  A =  m + n  . (5) γm(1 − m) 0

In the study of E2 we consider only m ∈ (0, 1). The eigenvalues of (5) are the roots of the characteristic equation λ2 − tr A λ + det A = 0, (6) −m(m2 + 2mn − n) with det A = γm2(1 − m)(n + m) > 0, tr A = , and ∆ = tr2 A − m + n 4 det A. Since det A > 0 it follows that E2 is an attractive (repulsive) focus or an attractive (repulsive) node, depending on the sign of tr A and ∆, a saddle-node if det A = 0 and a Hopf (degenerated or nondegenerated) singularity if tr A = 0. Due to the Hartman-Grobman theorem, we are interested only in nonhyperbolic equilibria: the saddle-nodes O(0, 0) and E1(1, 0) for m = 1 and the Hopf singularity 2 E2(m, (m(1 − m)/(n + m)) for m + 2mn − n = 0. In order to see whether A is a degenerated or a nondegenerated singularity we have to derive the normal form of (3) at A [1].

~~4. NORMAL FORMS FOR THE NONHYPERBOLIC SINGULARITIES Propozit¸ia4.1. The normal form of the (3) at the point O(0, 0) is ( 2 3 n˙1 = n1 + O(n ), 3 (7) n˙2 = −γmnn2 − γ(m − n)n1n2 + O(n ),~~

T where n = (n1, n2) , therefore, O is a nondegenerated saddle-node equilibrium. Proof. First, we write the system (3) in the equivalent form ( x˙ = x2 − nxy − x3 − x2y, (8) y˙ = −γmny + γ(n − m)xy + γx2y,

The eigenvalues of the matrix deﬁning the linear terms in the system (8) are λ1 = 0, λ2 = −γmn. In order to reduce the second order nonresonant terms in (8) we deter- T T mine the transformation x = n + h(n), where x = (x, y) and n = (n1, n2) , suggested 88 Raluca-Mihaela Georgescu by Table 1, found by applying the normal form method.

~~m1 m2 Xm,1 Xm,2 Λm,1 Λm,2 hm,1 hm,2 2 0 1 0 0 γmn - 0 1 1 −n γ(n − m) −γmn 0 1/γm - 0 2 0 0 −2γmn −γmn 0 0~~

~~Table 1.~~

Here Λm,1, Λm,2 are the eigenvalues of the associated Lie operator, while Xm is the second order homogenous vector polynomial in (8). We ﬁnd the transformation ( x = n + [1/γm]n n , 1 1 2 (9) y = n2,

2 carrying (8) into (7). We have the coeﬃcient of the n1 nonzero, therefore, by [1], the equilibrium point O corresponding the dynamical system generated by a s.o.d.e. of the form (7) is a nondegenerated saddle-node.

Propozit¸ia4.2. The normal form of the (3) at the point E1(1, 0) is ( 2 3 n˙1 = −n1 − (2 + γ + 2γn + 3n + γn )n1n2 + O(n ), 2 2 3 (10) n˙2 = −γ(n + 1) n2 + O(n ), therefore, E1(1, 0) is a nondegenerated saddle-node equilibrium.

Proof. First, we translate the point E1 at the origin by means of the change u1 = T x − 1, u2 = y. Let u = (u1, u2) . Then, in u, (3) reads ( 2 3 2 u˙1 = −u1 − (n + 1)u2 − 2u1 − (n + 2)u1u2 − u1 − u1u2, 2 (11) u˙2 = γ(n + 1)u1u2 + γu1u2. The eigenvalues of the matrix deﬁning the linear terms in the system (11) are λ = −1, λ = 0 and the corresponding eigenvectors read u = (1, 0)T and u = 1 2 ! λ1 ! ! λ2 u 1 n + 1 v (n+1, −1)T . Thus, with the change of coordinates 1 = 1 , (11) u2 0 −1 v2 achieves the form ( 2 2 3 v˙1 = −v1 − 2v1 − Bv1v2 − Cv2 + O(v ), 2 3 (12) v˙2 = −Dv1v2 − Ev2 + O(v ), where B = 2 + 3n + γ + 2γn + γn2, C = (1 + n) γn2 + 2 γn + n + γ , D = γ(n + 1) and E = γ (1 + n)2, such that the matrix deﬁning the linear part is diagonal. In order to Study of the Lyapunov stability in a biological model 89 reduce the second order nonresonant terms in (12) we determine the transformation T T v = n + h(n), where v = (v1, v2) and n = (n1, n2) , suggested by Table 2, found by applying the normal form method.

~~m1 m2 Xm,1 Xm,2 Λm,1 Λm,2 hm,1 hm,2 2 0 -2 0 -1 -2 2 0 1 1 -B -D 0 -1 - D 0 2 -C -E 1 0 -C 0~~

~~Table 2.~~

Here Λm,1, Λm,2 are the eigenvalues of the associated Lie operator, while Xm is the second order homogenous vector polynomial in (12). We ﬁnd the transformation ( 2 2 2 v1 = n1 + 2n1 − (1 + n) γn + 2 γn + n + γ n2, v2 = n2 + γ(n + 1)n1n2, taking (12) into (10). We have γ(n + 1) , 0, therefore, by [1], the equilibrium point E1 corresponding the dynamical system generated by a s.o.d.e. of the form (10) is a nondegenerated saddle-node.

The Hopf (nodegenerated or degenerated) singularity appears at m2 −2mn−n = 0, 2 therefore n = m /(1 − 2m). Thus, E2 becomes E2(m, 1 − 2m). Now, the calculus become very complicated, so, in order to show how to compute the normal form, we ﬁnd the normal form for a particular choice of the parameters, namely γ = 1, m = 2/5 and n = 4/5, thus we study E2(2/5, 1/5). In this case, the system (3) becomes  !  4  x˙ = x x − n − x2 − xy ,  5 ! ! (13)  2 4  y˙ = −y − x + x .  5 5

Propozit¸ia4.3. The normal form of (13) at E2(2/5, 1/5) is ! √ ! ! w˙ 0 −6 2/25 w 1 = √ 1 − (14) w˙2 6 2/25 0 w2  ! √ !   2 2  1 w1 77 2 −w2  4 −(w1 + w2) − −  + O(w ), 4 w2 72 w1 and, thus, E2 is a nondegenerated Hopf singularity for this choice of parameters. 90 Raluca-Mihaela Georgescu

Proof. First, we translate the point E2 at the origin with the aid of the change u1 = T x − 2/5, u2 = y − 1/5. Let u = (u1, u2) . Then, in u, (13) reads  12 2 8  2 3 2  u˙1 = − u2 − u1 − − u1u2 − u1 − u1u2, 25 5 5 (15)  6 1 6  u˙ = u + u2 + u u + u2u . 2 25 1 5 1 5 1 2 1 2 The eigenvalues√ of the matrix√ deﬁning the linear terms in (15) are λ = λ = 6 2i/25 and, let u = (i 2, 1)T be an eigenvector corresponding to the 1 2 λ1 √ positive eigenvalue. We have u = (0, 1)T + i( 2, 0)T . Thus, with the change of the ! λ1 ! √ ! ! u1 v1 2 0 1 1 1 coordinates = PMC , where P = and MC = , u2 v2 0 1 2 −i i ! √ √ ! ! u 1 2 2 v i.e. 1 = 1 , (15) achieves the complex form u2 2 −i i v2  √  √   √   √          6 2i  2 i  2  2 i  2 2 3i  2  v˙1 = v1 +  +  v −  −  v1v2 −  +  v  25 5 2 1 5 5 5 10 2  √  √   √   √          2i 3 1 2i 2  2i 2 1 2i 3  + v −  −  v v2 − 1 −  v1v −  +  v  8 1 2 8 1 8 2 2 8 2 √  √   √   √  (16)         6 2i 2 2 3i  2  2 i   2 i  2  v˙2 = − v2 −  −  v −  +  v1v2 +  −  v  25 5 10 1 5 5 5 2 2   √   √   √  √         1 2i 3  2i 2 1 2i 2 2i 3  −  −  v − 1 +  v v2 −  +  v1v − v ,  2 8 1 8 1 2 8 2 8 2 involving a diagonal matrix of the linear terms. In order to reduce the second order resonant terms in (16) we determine the transformation v = n + h(n), where T T v = (v1, v2) and n = (n1, n2) , suggested by the Table 3.

~~m1 m2 Xm,1 Xm,2 Λm,1 Λm,2 hm,1 hm,2~~

~~2 0 A C λ1 3λ2 M P 1 1 −B −B λ2 λ1 N N 0 2 C A 3λ2 λ2 P M~~

~~Table 3.~~

Here Λm,1, Λm,2 are the eigenvalues√ of the associated√ Lie operator,√ Xm is a second 2 i 2 i 2 2 3i A order vector polynomial in (16) A = + , B = − , C = + , M = , 5 2 5 5 5 10 λ1 B C N = and P = . λ2 3λ2 Study of the Lyapunov stability in a biological model 91

~~We ﬁnd the transformation~~

( 2 2 v1 = n1 + Mn1 + Nn1n2 + Pn2, 2 2 v2 = n2 + Pn1 + Nn1n2 + Mn2, carrying (16) into  √  √   √        6 2i 53 55 2i 3 1 77 2i 2  n˙1 = n1 +  +  n −  +  n n2  25 36 72 1 4 72 1   √   √        23 31 2i 2 59 91 2i 3 4  +  −  n1n −  +  n + O(n ),  36 72 2 36 72 2 √  √   √  (17)       6 2i 59 91 2i 3 23 31 2i 2  n˙2 = − n2 −  −  n +  +  n n2  25 36 72 1 36 72 1   √   √        1 77 2i 2 53 55 2i 3 4  −  −  n1n +  −  n + O(n ).  4 72 2 36 72 2

Thus we eliminated the nonresonant second order terms. Now, we have to reduce the third order nonresonant terms in (17). This reduces to the determination of the T T transformation n = s + h(s), where n = (n1, n2) and s = (s1, s2) , suggested by the Table 4.

~~m1 m2 Xm,1 Xm,2 Λm,1 Λm,2 hm,1 hm,2~~

~~3 0 D −G 2λ1 4λ1 X Z 2 1 −E F 0 2λ1 - Y 1 2 F −E 2λ2 0 Y - 0 3 −G D 4λ2 2λ2 Z X~~

Table 4. √ 53 55 2i Here Xm is a third order vector polynomial in (16) D = + , √ √ √ 36 72 1 77 2i 23 31 2i 59 91 2i D F E = + , F = − , G = + , X = , Y = , 4 72 36 72 36 72 2λ1 2λ2 G and Z = − . 4λ2 We ﬁnd the transformation

( 3 2 3 n1 = s1 + Xs1 + Ys1 s2 + Zs2, 3 2 3 n2 = s2 + Zs1 + Ys1s2 + Xs2, 92 Raluca-Mihaela Georgescu carrying (17) into  √  √      6 2i 1 77 2i 2 5  s˙1 = s1 −  +  s s2 + O(s ),  25 4 72 1 √  √  (18)     6 2i 1 77 2i 2 5  s˙2 = − s2 −  −  s1 s + O(s ).  25 4 72 2

Let us come back to the real state functions by denoting s1 = w1 + iw2, s2 = w1 − iw2 [3]. In this way we obtain  √  √      6 2i 2 2  1 77 2i  5  w˙ 1 = − w2 + (w + w ) − w1 + w2 + O(w ),  25 1 2 4 72 √  √  (19)     6 2i 2 2  1 77 2i  5  w˙ 2 = w1 + (w + w ) − w1 − w2 + O(w ),  25 1 2 4 72 or, equivalently, (14), which is the normal form of (13). By [1], the equilibrium point E2(2/5, 1/5) corresponding the dynamical system generated by a s.o.d.e. of the form (14) is a nondegenerated Hopf singularity.

5. THE GLOBAL DYNAMIC BIFURCATION DIAGRAM The discussion form Section 3 shows that the parametric portrait is represented by the three curves in the space (S - corresponds to the saddle-nodes and H-corresponds to the Hopf singularities). In fig. 1 is presented the parametric portrait in three- dimensional space, and in fig. 2 are presented two sections in the parametric portrait from fig. 1 for γ = 1 (fig.2 a)) and γ = 2 (fig.2 b)). Here, the curve D corresponds to ∆ = 0 and separates the domain where E2 is a focus by the domain where E2 is a node. On D the point E2 is a sink. The portraits from zones 2, 20 and D are topological equivalent. In fig. 3 are represented the phase portraits corresponding to each stratum of the parametric portrait. This shows that, in spite of their unrealistic significance for the population dynamics, the equilibria O, E1 and E2 for m = 1 heavily contribute to the changes in the phase portraits and, so, to the dynamic bifurcation diagram (which consists of figs 1(2) and 3). The equilibria in it can be described at follows: the equilibrium O is always a saddle-node and it exists for every value of the parameters γ, m and n; the equilibrium E1 is a saddle for any m < 1, a saddle-node for m = 1 and an attractive node for m > 1; the equilibrium E2 is a repulsive focus, a Hopf singularity, an attractive focus, a sink, an attractive node, respectively, for different values of the parameter 0 < m ≤ 1. When m = 1, the equilibrium E2 collides with E1 and it becomes a saddle-node and, then, for m > 1 it has a negative component, so, from the biological point of view, we can say that it disappears. Study of the Lyapunov stability in a biological model 93

~~Fig. 1. The parametric portrait in the three-dimensional space~~

~~n n~~

~~2.0 2.0 H H 1.5 1.5 S D S 1 1 3 1.0 2 1.0 2 D 3 0.5 0.5 2’ 2’ 0.0 0.0 0.0 1.0 0.0 1.0 O 0.25 0.5 0.75 m O 0.25 0.5 0.75 m~~

~~a) b)~~

~~Fig. 2. Two sections in the parametric portrait: a) λ = 1; b) λ = 2. 94 Raluca-Mihaela Georgescu~~

~~1 H 2~~

~~2’ S 3~~

~~Fig. 3. Phase portrait for various strata in ﬁg.2.~~

~~References~~

[1] Arrowsmith, D.K., Place, C.M., Ordinary diﬀerential equations, Chapman and Hall, London, 1982. [2] Bazykin, A., Khibnik, A., On sharp excitation of self-oscillation in a Volterra-type model, Bio- physika, 26 (1981), 851-853. (in Russian) [3] Georgescu A., Georgescu R.M., Normal forms for nondegenerated Hopf singularities and bifur- cations, Proceedings of the 11-th Confenence on Applied and Industrial Mathematics, Oradea, 2003. [4] Georgescu, A., Moroianu, M., Oprea, I., Bifurcation theory. Principles and applications, Ed. Univ. Pites¸ti, Pites¸ti, 1999. (in Romanian) ROMAI J., 6, 1(2010), 95–105

A SEVEN EQUATION MODEL FOR RELATIVISTIC TWO FLUID FLOWS-II Sebastiano Giambò,Serena Giambò Department of Mathematics, University of Messina, Italy [email protected], [email protected] Abstract An interface-capturing model for relativistic two-fluid flow is presented. The flow equations are the bulk-fluid equations, combined with particle number and energy-momentum tensor equations for one of the two fluids. The latter equations, i.e. the ones describing one component of the mixture, contain source terms, accounting for the energy and momentum exchanges between the species. The fluid model enables the derivation of an exact expression for these source terms. Weak discontinuities propagating into this relativistic two-fluid system are examined and, moreover, expressions for their speeds of propagation are obtained.

~~Keywords: general relativity, relativistic ﬂuid dynamics, two-ﬂuid mixtures, nonlinear waves. 2000 MSC: 83C99, 80A10, 80A17, 76T99, 74J30.~~

1. INTRODUCTION Two-fluid flow problems appear in many topics in general relativity [1]-[27], [39]. In these problems, the medium consists of two (or more) fluids, which do not mix. In fact, a sharp interface separates the pure fluids. One way of treating two-fluid models is that of interface-capturing techniques. These methods do not use an explicit interface model. Instead, the fluid is modeled as a mixture of the pure fluids everywhere. In this paper, a capturing method is presented, which is a relativistic extension of the method introduced by Wackers and Koren [32] for classical compressible two- fluid flow. On the physical ground, the model describing the relativistic two-fluid flow is based on a relativistic two-phase flow model, in which the entire flow domain is filled with a mixture of two fluids. However, in this underlying two-phase model, the fluids are supposed not to be mixed on the molecular level. So, the fluid is a mixture in the macroscopic sense. Then, each fluid still has its own particle number density and specific internal energy, but a single pressure and a single four-velocity are defined for the the whole mixture. Moreover, heat conduction is not allowed between the fluid components. Concerning the unknown variables in our relativistic two-fluid model, seven unknowns appear in our description the full two-phase flow: the four-velocity, the par-

95 96 Sebastiano Giambò,Serena Giambò ticle number density and the pressure of the whole mixture, and two more quantities accounting for the relative concentrations of the components. This paper is organized as follows. Section 2 is devoted to a description of the relativistic mixture model and to the derivation of the flow equations. In Section 3, the source terms appearing in the flow equations are analyzed and an exact expression for this source terms is derived using a generalized Gibbs’ equation. Then, in Section 4, the complete system of governing differential equations is derived. In Section 5, the weak discontinuities propagating in the mixture are examined and the expressions for their speeds of propagation are obtained. Section 6 refers to a special case in which each fluid is supposed to satisfy the equation of state of perfect gases and the results represent the relativistic extension of the ones obtained by Wackers and Koren in the classical framework.

2. DIFFERENTIAL EQUATIONS The standard equations for a single-relativistic-fluid flow are valid for the two-fluid mixture as a whole. The particle number, r, and the energy-momentum tensor, T µν, satisfy the conservation equations: ν ∇ν ru = 0, (1) νµ ∇νT = 0 , (2) where uν is the four-velocity and the stress-energy tensor is given by T νµ = r f uνuµ − pgνµ ; (3) here f is the relativistic specific enthalpy p ρ + p f = 1 + h = 1 + + = , r r being h the “classical” specific enthalpy, the specific internal energy, p the pressure and ρ, ρ = r(1 + ) , (4) the energy density of the whole mixture. The spatial projection and the projection along uν of equation (2) give, respectively, ν µ νµ r f u ∇νu − γ ∂ν p = 0 , (5) ν ν u ∂νρ + (ρ + p) ∇νu = 0 . (6) However, suitable expressions for the bulk quantities, r, , ρ and f , have to be found. The volume fraction Xk and the mass fraction Yk of fluid k (k = 1, 2) are introduced according to the following relations X r Y = k k , (7) k r A seven equation model for relativistic two fluid flows-II 97 and in such a way that   X + X = 1 ,  1 2  (8)  Y1 + Y2 = 1 .

The volume fraction X = X1 and the mass fraction Y = Y1 of fluid 1 are chosen as field variables. The quantities X and Y allow to define any bulk quantity; the particle number density r, the specific internal energy , the energy density ρ, the relativistic specific enthalpy f and the energy-momentum tensor T µν, in terms of X and Y, are:   r = X1r1 + X2r2 ,     = Y11 + Y22 ,    f = Y f + Y f ,  1 1 2 2  (9)   ρ = X1ρ + X2ρ ,  1 2    r f = X1r1 f1 + X2r2 f2 ,    µν µν µν T = X1T1 + X2T2 , µν where rk, k, ρk, fk, Tk represents, respectively, the particle number density, the specific internal energy, the energy density, the relativistic specific enthalpy and the energy momentum tensor of each fluid (k = 1, 2) and X2 = 1 − X, Y2 = 1 − Y. Thus, for regular solutions, the mathematical study of the model can be performed with the following set of seven independent field variables uν, r, p, X, and Y. Since only five equations have already been introduced, i.e. (1), (5) and (6), two more equations are needed in order to close the governing system. The first one must, of course, be the conservation equation of the particle number density for one of the two fluids, since the fluids are not supposed to change into each other. The conservation equation corresponding to the particle number density X1r1 of fluid 1 is ν ∇ν X1r1u = 0 . (10) This equation, together with equation (1), implies that also the particle number density of fluid 2 is conserved: ν ∇ν X2r2u = 0 . (11) Consequently, as last equation, only one option remains: an equation for the energy- momentum tensor of fluid 1 which, clearly, have to be a balance equation, νµ µ ∇ν X1T1 = F , (12) with νµ ν µ νµ T1 = ρ1 + p u u − pg , (13) 98 Sebastiano Giambò,Serena Giambò and where Fµ represents the loss and source term in the separate balance.

3. DERIVATION OF THE SOURCE TERMS This section is devoted to handling the source term Fµ in equation (12). The projection along uν and the spatial projection of equation (12) are, respectively, ν ν µ Xu ∂νρ1 + ρ1u ∂νX + X ρ1 + p θ = uµF , (14) where ν θ = ∇νu , and ν µ νµ µν µ ν X ρ1 + p u ∇νu − γ ∂ν p − γ ∂νX = γν F . (15) Next, using equations (5) and (15), the following relation is obtained µν µν µν γ Fν = (χ − X) γ ∂ν p − γ X , (16) where χ = Y f1/ f denotes the relativistic enthalpy concentration. Moreover, from equation (14), being

ρ1 = r1(1 + 1) , we obtain ! ν 1 p ν Xr1u ∂ν1 + p∂ν − ∂νX = u Fν . (17) r1 Xr1 Let us turn our attention to the thermodynamic features of the model. We will assume the following axiom: the entropy S k of each fluid component is a function of the specific internal energy k, the particle number density rk of fluid k and the volume fraction X of fluid 1 S k = S k (k, rk, X) . (18) The laws of extended thermodynamics allow to express the derivatives of entropy in terms other observable variables. Thus,  ∂S 1  k = ,  ∂ T  k k   ∂S p  = − ,  2 (19)  ∂rk r Tk  k   ∂S p  = − , ∂X XrkTk where Tk is the temperature of fluid component k. It follows from equations (19) that 1 p TkdS k = dk + p d − dX (20) rk Xrk A seven equation model for relativistic two fluid flows-II 99 and then ! ν ν 1 p Tku ∂νS k = u ∂νk + p∂ν − ∂νX . (21) rk Xrk

We now also assume that the entropy S k is conserved along the ﬂow lines ν u ∂νS k = 0 (k = 1, 2) . (22) Thus, from (21), we can deduce that ! ν 1 p u ∂νk + p ∂ν − ∂νX = 0 . (23) rk Xrk

~~At this point, (17) allows to write the following relation involving Fµ ν u Fν = 0 . (24)~~

Therefore, using equations (16) and (24), the source term Fµ can now be computed as µν νµ Fν = −γ ∂νX + (χ − X) γ ∂µ p . (25) 4. PRESSURE AND VOLUME FRACTION EQUATIONS The derivation of equations for pressure and volume fraction is rather involved, as it requires the two energy equations (6) and (14). From equation (6) for total energy-momentum tensor, using (3), we deduce

2 ν ν 2 r u ∂ν + ρu ∂νr + r f θ = 0 . (26) The total speciﬁc internal energy can be expressed in terms of the variables r, p, X, Y by an equation of state of the form = (r, p, X, Y) . (27) Now, using this equation, the bulk energy equation (26) becomes ! ∂ ∂ ∂ r uν∂ p + r uν∂ X + p − r2 θ = 0 . (28) ∂p ν ∂X ν ∂r The energy balance equation for ﬂuid 1, (17), with equations (24) and (10), becomes ! ∂ ∂ ∂ r 1 uν∂ p + r 1 uν∂ X + p − rr 1 θ = 0 . (29) 1 ∂p ν 1 ∂X ν 1 ∂r From equation (28) and (29) we can deduce the following evolution equations for the pressure and the volume fraction, respectively,

~~ν u ∂ν p + ωθ = 0 , (30) 100 Sebastiano Giamb`o,Serena Giamb`o~~

ν u ∂νX + ξθ = 0 , (31) where ω and ξ are deﬁned by ∂1 2 ∂ ∂ ∂1 r1 p − r − r p − r1r ω = ∂X ∂r ∂X ∂r , (32) ∂ ∂1 ∂ ∂1 rr1 ∂p ∂X − ∂X ∂p

∂ ∂1 ∂1 2 ∂ r ∂p p − rr1 ∂r − r1 ∂p p − r ∂r ξ = , (33) ∂ ∂1 ∂ ∂1 rr1 ∂p ∂X − ∂X ∂p generalizing the classical results due to Wackers and Koren. To end this section, we recall that the complete system of governing diﬀerential equations may be written in term of variables uν, r, p, X and Y as

 ν  u ∂νr + rθ = 0 ,    ν µ µν  r f u ∇νu − γ ∂ν p = 0 ,    ν  u ∂ν p + ωθ = 0 , (34)    ν  u ∂νX + ξθ = 0 ,    ν u ∂νY = 0 . 5. WEAK DISCONTINUITIES We assume that field variables uν, r, p, X and Y are C0 and piecewise C1; that the discontinuities of their first order derivatives can occur across an hypersurface Σ of local equation ϕ (xµ) = 0, ϕ being a C2 function; that such discontinuities are well defined in each point of Σ as the difference of the limiting values of the derivatives of uν, r, p, X and Y obtained approaching each point of Σ by the two sides in which Σ divides the manifold V4; and that the above derivatives are uniformly convergent to the limiting values, which are tensor functions defined on Σ, obtained on each of the two sides. Under such hypothesis [33]-[37], introducing the infinitesimal disconti- nuity operator, δ, we investigate the conditions under which the tensor distributions, δuν, δr, δp, δX and δY, supported with regularity by Σ, are not simultaneously zero. At the same time, we obtain the differential equation (i.e. the characteristic equation) which must be satisfied by the function ϕ. To this end, it is sufficient to apply the first order compatibility conditions to the above differential equations. Then, from system µ (34), we have the following linear homogeneous system in the distribution Nµδu , δr, δp, δX and δY: A seven equation model for relativistic two fluid flows-II 101

 µ  Lδr + rNµδu = 0 ,    ν νµ  r f Lδu − γ Nµδp = 0 ,    µ  Lδp + ωNµδu = 0 , (35)    µ  LδX + ξNµδu = 0 ,    LδY = 0 , µ where Nµ is the normalized vector, (N Nµ = −1), deﬁned as

~~Lµ Nµ = √ , Lµ = ∂µϕ , (36) ν −L Lν µ and L = u Nµ. Moreover, from the unitary character of uν we have~~

~~ν uνδu = 0. (37)~~

~~Let us turn now our attention to the normal speeds of propagation, λΣ, of the various waves with respect to an observer moving with the mixture velocity uν, given by~~

~~L2 λ2 = , l2 = 1 + L2 . (38) Σ l2 The local causality condition, i.e. the requirement that the characteristic hypersurface Σ is time-like or null, is equivalent to the condition~~

2 0 ≤ λΣ ≤ 1 . (39) As first, from the above system (35), the solution L = 0 can be obtained, representing a wave moving with the mixture. The corresponding discontinuities satisfy the equations   N δuµ = 0 ,  µ (40)   δp = 0 . It can be easily seen that five degrees of freedom are still left in the coefficients characterizing the discontinuities, so we have 5 independent eigenvectors corresponding to the eigenvalue L = 0 in the space of the field variables. In what follows, we suppose L , 0. The equation (35)2, multiplied by Nµ, gives

~~ν 2 r f LNνδu − l δp = 0 . (41) 102 Sebastiano Giamb`o,Serena Giamb`o~~

Consequently, equations (35)3 and (41) represent a linear homogeneous system in the ν two scalar distributions Nνδu and δp, which may admit non zero solutions only if the determinant of the coefficients vanishes. Therefore, the following equation must hold H ≡ r f L2 − ωl2 = 0 . (42) This equation yields the hydrodynamical waves propagating in such a two-fluid system. Their speeds of propagation are given by ω λ2 = , (43) Σ r f ω where ω is given by (32), and the condition 0 < r f ≤ 1 ensures their spatial orientation. The associated discontinuities can be written in terms of µ ψ = Nµδu (44) as follows   1  δuν = − nνψ ,  l    r  δr = − ψ ,  L   ω  δp = − ψ , (45)  L    ξ  δX = − ψ ,  L    δY = 0 , where nν is the unitary space-like four-vector defined by 1 n = N − Lu . (46) µ l µ µ If the condition given above, in order to have surfaces with space-like orientation, is verified, then the governing equations represents a (not strictly) hyperbolic system. In fact, all velocities (eigenvalues) are real, and there is a complete set of eigenvectors in the space of field variables: seven independent eigenvectors (five from L = 0 and two from H = 0) for the seven independent field variables uν, r, p, X and Y.

6. APPLICATION Now, we examine the application of the preceding solution procedure to a relativistic mixture of two fluids in which each fluid k (k = 1, 2) satisfy the equation of state of perfect gases 1 Xk p k = , k = 1, 2, (47) γk − 1 Yk r A seven equation model for relativistic two fluid flows-II 103 where γk is the ratio of the specific heat capacities at constant pressure and volume, respectively, of fluid k. Using (47), (9)2 writes as ! X 1 − X p = + . (48) γ1 − 1 γ2 − 1 r Using again (47), the expressions of ω and ξ, (32) and (33) modify, respectively, as ω = Xγ1 + (1 − X) γ2 p , (49) ξ = X (1 − X) γ1 − γ2 . (50) Replacing the expression (49) of ω into equation (43) yields p λ2 = X γ + X γ . (51) Σ r f 1 1 2 2

~~Recalling that the hydrodynamical waves λk in each ﬂuid k is given by~~

~~2 p λk = γk , k = 1, 2, (52) rk fk we can rewrite equation (51) under the form~~

2 2 2 r f λΣ = X1r1 f1λ1 + X2r2 f2λ2 . (53) Equation (51) represent the relativistic generalization of the expressions of the normal velocity in a two-ﬂuid system found by Wackers and Koren [32], and allows to express the acoustic modes speeds in such a two-ﬂuid system as combination of the speeds of the individual modes.

~~References~~

[1] P. C. Vaidya, K. B. Shah, A relativistic model for a shell of flowing radiation in a homogeneous universe, Progr. Theor. Phys., 24 (1960), 111-125. [2] C. B. G. McIntosh, A cosmological model with both radiation and matter, Nature, 215 (1967), 36-37 . [3] C. B. G. McIntosh, Cosmological models containing both radiation and matter, Nature, 216 (1967), 1297-1298. [4] K. C. Jacobs, Friedmann cosmological model with both radiation and matter, Nature, 215 (1967), 1156-1157. [5] A. A. Coley, B. O. J. Tupper, Viscous fluid collapse, Phys. Rev., D 29 (1984), 2701-2704. [6] A. A. Coley, D. J. McManus, New slant on tilted cosmology, Phys. Rev., D 54 (1996), 6095-6100. [7] K. Dunn, Two-fluid cosmological models in Go¨del-type spacetimes, Gen. Rel. Grav., 21 (1989), 137-147 . [8] J. J. Ferrando, J. A. Morales, M. Portilla, Two-perfect fluid interpretation of an energy tensor, Gen. Rel. Grav., 22 (1990), 1021-1032. 104 Sebastiano Giambò,Serena Giambò

[9] V. Husain, Exact solutions for null fluid collapse, Phys. Rev., D 53 (1996), 1759-1762. [10] J. P. S.Lemos, Collapsing shells of radiation in anti-de Sitter spacetimes and the hoop and cosmic censorship conjectures, Phys. Rev., D 59 (1999), 044020 [gr-qc/9812078]. [11] A. Wang, Y. Wu, Generalized Vaidya solutions, Gen. Rel. Grav., 31 (1999), 107-114. [12] T. Harko, K. S. Cheng, Collapsing strange quark matter in Vaidya geometry, Phys. Lett., A 266 (2000), 249-253. [13] E. N. Glass, J. P. Krisch, Radiation and string atmosphere for relativistic stars, Phys. Rev., D 57 (1998), 5945-5947. [14] E. N. Glass, J. P. Krisch, Two-fluid atmosphere for relativistic stars, Class. Quantum Grav., 16 (1999) 1175-1184. [15] E. N. Glass, J. P. Krisch, Scale symmetries of spherical string fluids, J. Math. Phys., 40 (1999), 4056-4063. [16] S. Kar, Stringy black holes and energy conditions, Phys. Rev., D 55 (1997), 4872-4879. [17] F. Larsen, A string model of black hole microstates, Phys. Rev., D 56 (1997), 1005-1008. [18] P. S. Letelier, Anisotropic fluids with two-perfect fluid components, Phys. Rev., D 22 (1980), 807-813. [19] P. S. Letelier, Solitary waves of matter in general relativity, Phys. Rev., D 26 (1982), 2623-2631 . [20] S. S. Bayin, Anisotropic fluid spheres in general relativity, Phys. Rev., D 26 (1982), 1262-1274 . [21] T. Harko, M. K. Mak, Anisotropic relativistic stellar models, arXiv:gr-qc/0302104 v1 26 Feb 2003. [22] P. S. Letelier, P. S. C. Alencar, Anisotropic fluids with multifluid components, Phys. Rev., D 34 (1986), 343-351. [23] W. Zimdahl, Reacting fluids in the expanding universe: a new mechanism for entropy production, Mon. Not. R. Astron. Soc., 288 (1997), 665-673. [24] W. Zimdahl, D. Pavon, R. Maartens, Reheating and causal thermodynamics, Phys. Rev., D 55 (1997), 4681-4688. [25] J. P. Krisch, L. L. Smalley, Two fluid acoustic modes and inhomogeneous cosmologies, Class. Quantum Grav., 10 (1993), 2615-2623. [26] M. Cissoko, Wavefronts in a relativistic cosmic two-component fluid, Gen. Rel. Grav., 30 (1998), 521-534 [27] M. Cissoko, Wave fronts in a mixture of two relativistic perfect fluids flowing with two distinct four-velocities, Phys. Rev., D 63 (2001), 083516. [28] A. Murrone, Mode`les bi-fluides a` six et sept e´quations pour les ećoulement diphasiques a` faible nombre de Mach, PhD Thesis, Université de Provence, Aix-Marseille I, 2004. [29] A. Murrone, H. Guillard, A five equation reduced model for compressible two-phase flow computations, J. Comput. Phys., 202 (2005), 664-698. [30] A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problem, INRIA, Rapport de recherche 4778, 2003. [31] E. H. van Brummelen, B. Koren, A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows, J. Comput. Phys., 185 (2003), 289-308. [32] J. Wackers, B. Koren, Five-equation model for compressible two-fluid flow, Report MAS-E0414, (2004). [33] A. H. Taub, Relativistic Rankine-Hugoniot equations, Phys. Rev., 74 (1948), 328-334. A seven equation model for relativistic two fluid flows-II 105

[34] A. Lichnerowicz, Relativistic fluid Dynamics, Cremonese, Roma, 1971. [35] G. Boillat, La propagation des ondes, Gauthier-Villas, Paris, 1965. [36] G. A. Maugin, Conditions de compatibilite´ pour une hypersurface singulie`re en mećanique rela- tiviste des milieux continus, Ann. Inst. Henri Poincar, 24 (1976), 213-241. [37] A. M. Anile, Relativistic fluids and magneto-fluids, Cambridge University Press, Cambridge, 1989. [38] Y. Choquet-Bruhat, Fluides relativistes de conductibilite´ infinie, Astronautica Acta, VI (1960), 354-365. [39] S.Giambo,` S.Giambo,` A seven equation Model for relativistic two-fluid flows-I, Romai Journal, 5, 2(2009), 59-70.

~~ROMAI J., 6, 1(2010), 107–118~~

STABILITY OF BIFURCATING PERIODICAL SOLUTIONS AT POINCARE-ANDRONOV-HOPF´ BIFURCATION FOR SINGULAR DIFFERENTIAL EQUATIONS IN BANACH SPACES Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak Moscow Institute of Steel and Alloys, Russia, Ulyanovsk State Technical University, Russia, Canberra University, Australia [email protected], [email protected], [email protected] Abstract The technique of branching equations in the root subspaces and linear resolving systems [3]-[5] is applied to the problem of periodical bifurcating solutions stability at nonstationary bifurcation for singular diﬀerential equations in Banach spaces.

~~Keywords: nonstationary bifurcation, Lyapounov-Schmidt method, stability. 2000 MSC: 37G15,58E07,37J25.~~

1. INTRODUCTION In our previous articles [1]-[4] for the stability investigation of stationary and periodic bifurcating solutions for differential equations with a singular (Fredholm, Noetherian or operator of finite index), some operators multiplying the derivative branching equations in the root subspaces (BEqR) (or according to [5] linear resolving systems (LRS)) were introduced. Subsequently, Newton’s diagram method was applied to their determinants as branching equations (BEq) for the perturbed eigenvalue problems. In our article [6], a modified classic Floquet theory was developed in order to investigate the stability of periodic solutions at Poincare-Andronov-Hopf´ bifurcation, for DEqs of s ≥ 1 order with a degenerate Fredholm operator at the highest derivative. However the method of BEqRs for stability of bifurcating solutions in the indicated articles was applied only to the simple case of a pair of pure imaginary eigenvalues without or with Jordan chains presence. The authors are thankful to prof. V.A. Trenogin for paying attention on incompleteness of their results in this part. In this article we study the general case of Poincare-Andronov-Hopf´ bifurcation for differential equation dx A = Bx − R(x, ε), R(0, ε) ≡ 0, R (0, 0) = 0 (1) dt x with closed Noetherian operators A and B; the operator R(x, ε) is supposed to be continuously differentiable with respect to x and sufficiently smooth with respect to ε ∈ R. By using the results [4] the result in [6] can be generalized to the case of

107 108 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak operators of finite index. Everywhere below the terminology and notations of [1]- [8] are used, where also many other references can be found. We do not stipulate in every case here the subordination conditions of densely defined Noetherian operators [5], [6], allowing to reduce the discussion to bounded operators: for a pair of densely defined Noether operators A: DA → E2, B: DB → E2, if DB ⊂ DA, then A is subordinated to B, i.e. kAxk ≤ kBxk + kxk on DB; or if DA ⊂ DB, then B is subordinated to A, i.e. kBxk ≤ kAxk + kxk on DA. It is supposed also that A and B haven’t common zeros with the aim to avoid the complicated technique of nonfinished generalized Jordan chains (GJChs). Acknowledgement. The obtained results are supported by grant RFBR - Acad. Sci. of Romania No.07-01-91680a and by the project N 2.1.1/6194 of the program ”Development of scientific potential of Higher School” of Education Ministry of RF.

2. AUXILIARY CONSTRUCTIONS (1) n ? ˆ (1) m Let N(A) = span{φi }1 be the zero-subspace of the operator A, N (A) = span{ψi }1 (1) m ? (1) (1) (1) m be the subspace of defect functionals, {ϑi }1 ∈ E1 , hφi , ϑ j i = δi j and {ζ j }1 ∈ E2, (1) ˆ (1) (s) hζi , ψ j i = δi j are the relevant biorthogonal systems. Elements {φi }, s = 1, qi, ˆ (k) (s) (s−1) i = 1, n and {ψ j }, k = 1, q j, j = 1, m such that Aφi = Bφi , s = 2, qi, i = 1, n, (s) (1) ? ˆ (s) ? ˆ (s−1) (1) ˆ (s) hφi , ϑ j i = 0, s > 1 and A ψ j = B ψ j , s = 2, qi, j = 1, m, hζk , ψ j i = 0, s > 1 form the complete canonical generalized B-Jordan (B?-Jordan) set of the operator A(A?), if h i h i (qk) ˆ (1) ? ˆ (q j) rank hBφk , ψ j i j=1,m = rank hφk, B ψ j i j=1,m = l = min(n, m) k=1,n k=1,n

~~Lemma 2.1. [1]-[7] The elements of the complete B-Jordan (B?-Jordan) sets (JS) of the operators A and A? can be chosen so as to satisfy the biorthogonality relations~~

~~(qi+1−s) ? ˆ (k) hφi , B ψ j i = δ jkδi j, s = 1, qi, k = 1, qi, i, j = 1, l. (2) Lemma 2.2. [4] In the case n ≤ m the projections deﬁned according to (2)~~

Xn Xqi Xn Xqi (q +1−s) (s) (s) (s) (s) ? ˆ i ˆ (s) (s) (qi+1−s) p = h·, ϑi iφi , ϑi = B ψi ; q = h·, ψi iζi , ζi = Bφ i=1 s=1 i=1 s=1 (3) kA ∞−kA give rise to direct sum decompositions E1 = E1 uE1 , E2 = E2,kA uE2,∞−kA , where kA (1) (2) (qi) n E1 = k(A, B) = span{φi , φi , . . . , φi }1 is the B-root-subspace of the operator A (qi+1−k) (k) n and E2,kA = span{Bφi = ζi }1. (s) ˆ (1) (q j+1−s) ˆ (1) If hζ j , ψk i = hBφ j , ψk i = 0, j = 1, n, s = 1, q j, i = n + 1, m then the projectors p and q intertwine the operators A and B, i.e. Stability of bifurcating periodical solutions at Poincar´e-Androov-Hopf bifurcation... 109

~~Ap = qA, Bp = qB, where kA kA T ∞−kA N(A) ∈ E1 , AE1 ∈ E2,kA , A(DA E1 ) ⊂ E2,∞−kA ; (4) ∞−kA kA T ∞−kA N(B) ∈ E1 , BE1 ∈ E2,kA , B(DB E1 ) ⊂ E2,∞−kA~~

~~T ∞−kA kA and A :(DA E1 ) → E2,∞−kA , B: E1 → E2,kA are isomorphisms.~~

Deﬁnition 2.1. A solution x0(t) of the equation dx A = Bx + f (x, t), k f (x, t)k = o(kxk), kxk → 0, (5) dt is said to be Lyapunov stable if for each ε > 0 there exists a δ > 0 such that for each 0 0 0 x , kx − x0(0)k < δ, for which (5) has a solution x(t) with x(0) = x the relation kx(t) − x0(t)k < ε, t > 0 holds, and it is said to be asymptotically stable if, moreover, kx(t) − x0(t)k → 0 as t → ∞. Remark 2.1. In [4] it is proved that, if for the operator A at n > m there exists dx a complete generalized B-JS, then the zero solution of the equation A = Bx is dt unstable.

The linearized stability principle is established in [4]: the stationary solution x0(ε) of the equation (1) that branches off from x = 0 is asymptotically stable if the spectrum σA(B − Rx(x0(ε), ε)) of the Frechet´ derivative B − Rx(x0(ε), ε) on the solution x0(ε) lies in the left half of the complex plane, and unstable if there exists at least one point µ(ε) ∈ σA(B − Rx(x0(ε), ε)) in the right half of the complex plane. Everywhere in the sequel it is supposed that σA(B) \{0} lies strictly in the left half of the complex plane. (1) N ? (1) M For the Noetherian operator B, let N(B) = span{ϕi }1 , N (B) = span{ψi }1 and (1) N (1) M {γ j }1 , {z j }1 be the relevant biorthogonal systems. Then the projectors PN (1) (1) PM (1) (1) P = i=1h·, γi iϕi and Q = j=1h·, ψ j iz j generate the direct sum expansions E = EN u E∞−N, E = E u E . For the operator Bˆ which is the restriction 1 1 1 2 T2,M 2,∞−M ∞−N ˆ ∞−N ˆ−1 of B to E1 , i.e. B: DB E1 → E2,∞−M, the inverse operator B exists and ˆ−1 ∞−N according to Banach theorem about closed graph B ∈ L(E2,∞−M, E1 ). Following + −1 + to [9] introduce the pseudoinverse operator B = Bˆ (I−Q) for which N(B ) = E2,M, BB+ = I − Q, B+B = I − P, and assume that the operator B+A is bounded. If the operator B is Fredholm then instead of the pseudoinverse operator B+ we can use a −1 PN (1) (1) −1 the E. Schmidt’s regularizator [7] Γ = B = (B + i=1h·, γi izi ) ∈ L(E2, E1) + + PN (1) (1) connected with B by the equality Γ = B + i=1h·, ψi iϕi . (s) Analogously to previous definitions (2)-(4) assume that there exist elements {ϕi }, (k) (s) (s−1) (s) (1) i = 1, N, s = 1, pi, {ψ j }, j = 1, M, k = 1, p j, Bϕi = Aϕi , hϕi , γ j i = 0, s > 1, ? (k) (k−1) (s) B ψ j = Aψ j , hzk, ψ j i = 0, s > 1 which compose a complete A-Jordan set of the 110 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak operator B, i.e. h i h i rank hAϕ(pk), ψ(1)i = rank hϕ(1), A?ψ(pi)i ≡ L = min(M, N). k i i=1,M,k=1,N k i i=1,M,k=1,N If N ≤ M then by lemma 2.2 it is possible to define the projections P = PN Ppi (s) (s) (s) ? (pi+1−s) PN Ppi (s) (s) (s) (pi+1−s) i=1 s=1h·, γi iϕi , γi = A ψi ; Q = i=1 s=1h·, ψi izi , zi = Bϕi kB ∞−kB generating the direct sum decompositions E1 = E1 u E1 , E2 = E2,kB u E2,∞−kB , kB (s) where E1 = k(B, A) is the A-root-subspace of the operator B, E2,kB = span{Aϕi , i = t 1, N, s = 1, pi}. Again the operators A and B act in invariant pairs of subspaces: A, t u u u kB T ∞−kB T ∞−kB B: E1 → E2,kB , A: DA E1 → E2,∞−kB , B: DB E1 → E2,∞−kB where B t and A are isomorphisms. (s) (1) Under the assumption that < Aϕ j , ψi >= 0 for j = 1, N, i = N + 1, M in [4] the κ BEqR describing the branching of A-eigenvalues µ() of the operator B− Rx(x(), ) is constructed (where the bifurcating from x0 solution x(ε) is represented by the series on a fractional degree = ε1/κ): " u κ κ + L(µ, ε) = det h{B − µA − Rx(x0(), ) − Rx(x0(), )[I − µB A− # u u (6) + κ −1 + κ (s) (p j+1−s) −B (I − Q)Rx(x0(), )] B (I − Q)Rx(x0(), )}ϕi , ψ j i = 0

(s) (p j+1−s) p1+...+pN L(µ, 0) = det[h(B − µA)ϕi , ψ j i] = (−µ) Consequently the Newton diagram method [8] for the µ-BEqR (6) enables us to PN compute the principal terms p = 1 pi of asymptotics of the A-eigenvalues of B − κ Rx(x0(), ) that branch oﬀ from the eigenvalue µ = 0, and thereby to determine the stability of stationary solution x0() to (1). Remark 1. In the case N > M the bifurcating solutions of (6) are unstable. Now we consider bifurcation and stability of stationary solutions to (1) with Fred- holm operator B. Here Lyapunov resolving system [2]-[5] gives the Lyapunov’s BEqR (6) or (that is the same) according [2],[3] "*( ) +# u u −1 (s) (σ) f (µ, ) ≡ det B − µA − Rx − Rx[B − µA − (I − Q)Rx] (I − Q)Rx ϕi , ψ j = 0 (7)    0, 0,... 0, −µ     0, 0,... −µ, 1  f (µ, 0) = det diag   = µkB (−1)p1+...+pN . (8)  ......    −µ 1,... 0, 0 Consequently the same conclusion about Newton diagram method for the stability of stationary solution x0() determination is true. Stability of bifurcating periodical solutions at Poincar´e-Androov-Hopf bifurcation... 111

a a Analogously the introduction of E.Schmidt regularizator [7] Γ = B−1, B = B + n P (1) (1) κ h·, γi izi allows to rewrite the equation By − Rx(x0(), )y = µAy in the form of i=1 a N κ P (1) (σ) (σ) equivalent system By = µAy + Rx(x0(), )y + ξi1zi , ξsσ = hy, γs i. Here γs = i=1 ? (ps+1−σ) (σ) (ps+1−σ) ( j) (l) ( j) (l) A ψs , zs = Aϕs , hϕi , γk i = δikδ jl, hzi , ψk i = δikδ jl according to A − JS biorthogonality lemma [1]-[4] and x() = x(ξ(), ) for brevity. Writing the N p P Pk (l) solution of the ﬁrst equation in the form y = w + ξklϕk gives  k=1 l=1 h i  XN h −1 −1  −1 (1) (2) w = I − (I − µΓA) ΓRx (I − µΓA) µξ ϕ + (µξ − ξ )ϕ + ... +  kpk k k1 k2 k k=1  i XN Xpk  (pk) −1 (l) +(µξ − ξ )ϕ + (I − µΓA) ΓRx ξ ϕ  = kpk−1 kpk kl k  k=1 l=1

N pk N X X h i−1 X h i (l) −1 (1) (2) pk−2 (pk−1) pk−1 (pk) = − ξklϕk + I − (I − µΓA) ΓRx ξk1 ϕk + µϕk + ... + µ ϕk + µ ϕk . k=1 l=1 k=1 Substitution of w in the second equation leads to Schmidt’s (linear with respect to ξ) resolving system

~~N * p + ps X h i Xk µ ξk1 −1 −1 −1 r−1 (r) (1) − ξs1 − (I − µAΓ) Rx I − Γ(I − µAΓ) Rx µ ϕ , ψs = 0, (9) 1 − µps 1 − µpk k k=1 r=1~~

~~N * pk + σ−1 X ξ h i−1 X µ k1 −1 −1 r−1 (r) (ps+2−σ) ξsσ − ξs1 − (I − µAΓ) Rx I − Γ(I − µAΓ) Rx µ ϕ , ψs = 0, 1 − µps 1 − µpk k k=1 r=1~~

~~σ = 2,..., ps,, s = 1,..., N.~~

By equating to zero the determinant of the resolving system (9) we obtain the µ-BEqR and the exact answer about stability of bifurcating solution x0(). Theorem 2.1 is similar to Theorem 4 [2,3]. By using the results of [19] we give here a more detailed proof of this theorem. ( j) According to Schmidt’s approach to BEq construction introduce GJS Φi , j = ∞ κ P k (1) 1, rs, s = 1, N of the operator-function B − Rx(x(), ) = B − Rk, Bϕi = k=1 k (1) (k+1) P (k+1− j) ( j) (1) BΦi = 0; BΦi = R jΦi , k = 2, rs − 1, hΦi , γk i = 0, j = 2, rs, k, s = * j=1 + Prs 1, N ∃ ψ R Φ(ri+1− j), ψ , 0. This GJS is complete canonical [7] if ∆ = k0 j i k0 r "* j=1 +# ri P (rs+1− j) det R jΦi , ψk , 0, r = (r1,..., rN). j=1 112 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak

a a By using the Schmidt regularizator B, Γ = B−1 we rewrite the equation Bx = a PN R(x, ε) in the form of the system Bx = R(x, ε) + ξ jz j, zk = hx, γki. The ﬁrst 1 equation has the unique suﬃciently smooth solution x = x(ξ, ε). Its substitution into the second equation gives Schmidt’s BEq

tk(ξ, ε) ≡ ξk − hx(ξ, ε), γki = −hR(x(ξ, ε), ε), ψki = 0. The Schmidt’s BEq and the original equation have the same quantity of small solutions ξ() and x(ξ(), ), decomposing on the fractional degrees of ε, i.e. ε = κ,

κ κ tk(ξ(), ) = −hR(x(ξ(), ), ), ψki = 0, k = 1, n. (10) Pn Theorem 2.1. [2], [3] Let pi = 1, i = 1, N and let the root number k(A; B) = q j be j=1 N ﬁnite. Suppose also that to the elements {ϕi}1 ∈ N(B) there corresponds a complete κ canonical GJS relative to the operator-function B − Rx(x(ξ(), ), ) with GJChs of the lengths r1 ≤ r2 ≤ ... ≤ rN. Then the stationary solution x(ξ(), ) to (1) is stable if for the corresponding solution ξ() of BEq (10) the nonvanishing main terms of asymptotics of the eigenvalues" # ν j(), j = 1, N deﬁned by the main part of the ∂tk Jacobian matrix J = tξ = ∂ξ j h i κ det tξ(ξ(), ) − νI = hD E i N κ κ −1 (1) (1) = (−1) det Rx(x(ξ(), ), ) I − ΓRx(x(ξ(), ), ) Φk , ψ j + νδk j = 0, (11) have negative real parts, and unstable if one of them is positive. κ D E ∂tk(ξ(), ) κ κ −1 (1) (1) In fact, = − Rx(x(ξ(), ), ) I − ΓRx(x(ξ(), ), ) Φ j , ψk ∂ξ j whence it follows (11). According to Schmidt lemma, we reduce the problem about bifurcation of A-eigenvalues and A-eigenvectors of the Fredholm operator B at the perturbation Rx, to the equivalent µ-BEq, L(µ, ) = 0

κ [B − Rx(x(ξ(), ), ) − µA]y = 0 => N a X D E κ (1) (1) By = µAy + Rx(x(ξ(), ), )y + ξ jz j , ξk = y, γk => j=1 XN −1 (1) => y = ξ j[I − µΓA − ΓRx] Φ j => j=1 Stability of bifurcating periodical solutions at Poincar´e-Androov-Hopf bifurcation... 113

XN D E −1 (1) (1) L j(µ, ) ≡ ξk − ξ j I − ΓRx − µΓA Φ j , γk = j=1 XN h i −1 −1 −1 (1) (1) = ξk − ξ j I − µ(I − ΓRx) ΓA (I − ΓRx) Φ j , γk = 0 => j=1 hD E N −1 (1) (1) L(µ, ) ≡ (−1) det Rx(I − ΓRx) Φ j , ψk +  ∞ X h il  l −1 −1 (1) (1)  + µ (I − ΓRx) ΓA (I − ΓRx) Φ j , γk  = 0 (12) l=1 h i ∞ n o n o −1 l −1 P s l s l here (I − ΓRx) ΓA (I − ΓRx) = (ΓRx) , (ΓA) , (ΓRx) , (ΓA) denote the s=0 sum of all possible products of s operators ΓRx and l operators ΓA. Now from ∞ P k the completeness of canonical GJS of the operator-function B − Rk ≡ B − k=1 κ Rx(x(ξ(), ), ) it follows that

D E σX−1 −1 (1) (1) r j (r j) (σ) (σ−s) a jk() ≡ Rx(I − ΓRx) Φ j , ψk = (a jk + o(||)), Φ j = Γ RsΦ j (13) s=1 D E (1) −1 −1 (1) (1) and b jk = (I − ΓRx) ΓA(I − ΓRx) Φ j , γk = δ jk + o(||). On the base of Newton diagram method, µ-BEq (12) has n small solutions µ = µ() → 0 at → 0, that are found in the form µ() = ρ(C + O()), (14) where ρ ∈ (r1,..., rN), C , 0. Let us denote ρ = rs and

~~r1 ≤ . . . < rs = ... = rs+σ < . . . ≤ rN, σ ≥ 0 (15)~~

Here (σ + 1) of Jordan chains has the length rs. Substitute µ() in the determinant (12). Take out the common factor ρ from every j-th line, j = s, N. After crossing the factor ρ(N+1−s) tend to zero. As result it turns out that the main term of the solution N−s−σ (p j) s+σ (14) must be satisfied the equation C det[a jk + Cδ jk] j,k=s = 0 and the main (p j) terms of the solutions (14) are nonzero roots of the polynomial Pσ+1(C) ≡ det[a jk + s+σ (ps) Cδ jk] j,k=s. For σ = 0, C = −ass . For the case r1 = r2 = ... = rN = ρ the magnitude (ρ) N C is determined by the polynomial of N-th degree PN(C) ≡ det[a jk + Cδ jk] j,k=1 = 0, (ρ) N where det[a jk ] j,k=1 , 0. In the last case all N small eigenvalues µ() of the operator κ B − Rx(x(ξ(), ), ) have the same order ρ. Thus from Newton diagram method it follows that the main terms of the asymptotics of µ() defined from (12) and ν j() defined from (11) coincide. 114 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak 3. JORDAN STRUCTURE OF SOME OPERATOR-FUNCTIONS OF SPECTRAL PARAMETERS AT POINCARE-ANDRONOV-HOPF´ BIFURCATION

Let in the Fredholm case (for simplicity of presentation) the A-spectrum σA(B) of − 0 the operator B consists of two parts: σA(B) lying in the left half-plane and σA(B) on the imaginary axis consisting of a ﬁnite number of the points ±iβ, that are nonzero eigenvalues of ﬁnite multiplicity. The set of pure imaginary eigenvalues is split into disjoined subclasses ±iαr, αr = krα, r = 1, ν (kr are natural numbers without nontrivial common divisors). Let iαr be eigenvalues of multiplicity Nr with eigenelements (1) (1) (1) (1) (1) (1) (1) (1) ur j = u1r j + iu2r j, i.e. Bur j = iαrAur j , Bur j = −iαrAur j and vr j = u1r j + iv2r j ? (1) ? (1) ? (1) ? (1) be eigenvalues of adjoint operators, i.e. B v2r j = −iαrA vr j , B vr j = iαrA vr j , j = 1, Nr with A-Jordan chains of lengths pr j. The last means the existence of ele- (k) (k) (k) (k) ments ur j , ur j and vr j , vr j such that (s) (s−1) (s) (s−1) (B − ikrαA)ur j = A ur j , (B + ikrαA)ur j = Aur j , ? ? (s) ? (s−1) ? ? (s) ? (s−1) (16) (B + ikrαA )vr j = A vr j , (B − ikrαA )vr j = A vr j and moreover by virtue of Lemma 1 about the biorthogonality at the complete GJS these can be chosen so that

(s) (pρl+1−σ) (s) (pρl+1−σ) hAu , v i + hAu , v i = δrρδ jlδρσ hAu(s), v(pσρ+1−l)i δ δ δ ⇒ 1r j 1ρl 2r j 2ρl r j σρ rσ jρ kl (s) (pρl+1−σ) (s) (pρl+1−σ) hAu1r j, v2ρl i = hAu2r j, v1ρl i (17) (1) (1) (1) (1) The deﬁnition of eigenelements ur j , ur j , vr j , vr j for the eigenvalues ±iαr in 0 0 ? σA(B) and σA? (B ) according to the equalities !  (1)  !  (1)  B α A u  B? α A v  B(α )Φ(1) ≡ r  1r j = 0, B?(α )Ψ(1) ≡ r  1r j = 0 r r j −α AB  (1)  r r j −α AB?  (1)  r u2r j r v2r j

j = 1, Nr, r = 1, ν, N = N1 + ... Nν mean that the zero-subspaces N(B(αr)) and N(B(αr)) have the form   (1)   (1)    u   u   N(B(α )) = span Φ(1) =  1r j , Φ(1) =  2r j  , j = 1, N , r = 1, ν r  1r j  (1)  2r j  (1)  r  u2r j −u1r j   (1)   (1)    −v  v   N(B?(α )) = span Ψ(1) =  2ρl , Ψ(1) =  1ρl , l = 1, N , ρ = 1, ν r  1ρl  (1)  2ρl  (1)  r  v1ρl v2ρl (1) (1) Introduction of complete GJSets of the ﬁnite lengths pr j (pρl) to the elements Φir j,Ψkρl ? ? ? for the operator functions B(αr +ε) = B(αr)−εA and B (αr +ε) = B (αr)−εA by Stability of bifurcating periodical solutions at Poincar´e-Androov-Hopf bifurcation... 115

(s) (s−1) ? (s) ? (s−1) the equalities B(αr)Φir j = AΦir j , B (αr)Ψkρl = A Ψkρl that respect (according (s) (pρl+1−σ) to lemma 2.2) the biorthogonality conditions hAΦir j, Ψkρl i = δikδrρδ jlδsσ leads to the same relations (17). For every α its own bifurcation equation is constructed [10], [8], [11]-[13]. The τ H. Poincare´ change of variables [14] t = , x(t) = y(τ) reduces the problem α + µ 2π of the construction of -periodic solutions for (1) to the problem of 2π-periodic α + µ solutions determination for the equation dy dy By = µCy + R(y, ε) ≡ R(y, µ, ε), (By)(τ) ≡ By(τ) − αA , (Cy) ≡ A (18) dτ dτ with two small parameters in the complexiﬁed Banach spaces E = Ek u iEk, k = 1, 2 1 R 2π with the functionals hhy, f ii = hy(τ), f (τ)idτ, y ∈ Y, f ∈ Y? or y ∈ Z, f ∈ Z?. 2π 0 d Here the zero-subspaces N(B) and N(B?)(B? = B? + αA? ) are 2N-dimensional, n dτ o (1) (1) ikrτ (1) (1) −ikrτ N(B) = span ϕr j = ur j e , ϕr j = ur j e , j = 1, Nr, r = 1, ν and ? (1) (1) ikrτ (1) −ikrτ ? N(B ) = span{ψr j = vr j e , ψr j = vr j e , j = 1, Nr, r = 1, ν} with A-Jordan (A - (s) (s) ikrτ (s) (s) ikrτ Jordan) chains ϕr j = ur j e , ψr j = vr j e , s = 1, pr j of lengths pr j that satisfy, according to lemma 2.2, the biorthogonality relations

(k) (l) (l) ? (pσρ+1−l) (k) (l) hhϕr j , γσρii = δrσδ jρδkl, γσρ = A ψσρ ; hhzr j , ψσρii = δrσδ jρδkl (l) ? (pσρ+1−l) (k) (pr j+1−k) γσρ = A ψσρ , zr j = Aϕr j , k(l) = 1, pr j(pσρ), ρ( j) = 1, Nσ(Nr), σ, r = 1, v. (19) Conditions (19) lead again to relations (17). Thus the three considered operator-functions of spectral parameter lead to the equivalent biorthogonality conditions.

4. BIFURCATING SOLUTIONS CONSTRUCTION AND THEIR STABILITY INVESTIGATION Branching equation for nonstationary bifurcation is the system on n complex- valued equations f (ξ, ξ, µ, ε) = 0, with respect to unknowns ξ ∈ Cn and unknown small addition µ to the frequency of oscillations . Therefore at n > 2 for the construction of small solutions asymptotics on small parameter ε these solutions should be found in the subspaces, which are invariant with respect to the left-hand side f of the BEq. Under group invariance conditions after solutions determination in some n invariant subspace Ξ0 ⊂ C by the action of the group G we get the relevant orbit (trajectory) of solutions, i.e. the solution in the orbit eΞ = Ug∈GImg(Ξ0) of the subspace Ξ0. 116 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak

The following assertion allows to construct the complete system of invariant subspaces, generating the others. Lemma 1. [15] Let f, L: Ξn → Ξn and f L = L f . If a subspace M ⊂ Ξn is invariant with respect to f , then the subspaces L(M) ⊂ Ξn is also invariant with respect to f . In the articles [16],[17] a computer program is suggested for the construction of a complete system of invariant subspaces at nonstationary bifurcation. To every invariant subspace in their complete system there corresponds its stationary subgroup of allowed symmetry. Consequently for every Poincare-Andronov-Hopf´ bifurcation the complete system of invariant subspaces (only one from every orbit) must be indicated with indication of relevant stationary subgroup.

Conclusion 2. Let y = y(τ, ) be a periodic solution of the equation (18) constructed for concrete Poincar´e-Andronov-Hopf bifurcation problem, which corresponds to some invariant subspace Ξ0 of their complete system for the relevant BEq. The question about its stability may be solved by applying the Newton’s diagram method [8] to the µ-BEq (as determinant of the corresponding linear resolving system analog of BEqR) of the form (6) for the problem about branching of A-eigenvalues and elements of the Frech´etderivative of (18) in the root subspace K(B; Ad/dτ). Here the results of [18] may be used keeping in mind the stationary subgroup of the solution y. Tak- ing into account the stationary subgroup of the solution y here may be considered also the question about its orbital stability. Remark 2. Stability of bifurcating periodic solutions can be investigated on the base of [6].

5. REMARKS ABOUT STABILITY OF SOLUTIONS IN CONDITIONS OF GROUP SYMMETRY OR INTERTWINING The articles [3], [4] contain the program of stability investigation of bifurcating stationary and periodic solutions under group symmetry conditions on the base of group symmetry inheritance theorem by the relevant bifurcation equations. In the articles [5], [20] the equation (1) is considered at intertwining of operators A, B, R(·, ε) by linear operators L ∈ L(E1) and K ∈ L(E2), i.e. AL = KA on DA, BL = KB on DB, R(Lx, ε) = KR(x, ε) on a neighborhood Ω of x = 0, ε = 0. In this articles the relevant inheritance theorems are proved, i.e. the branching equations inherit the intertwining property by the matrices XN XM N M ∗ A = [ai j]i, j=1, Lϕi = a jiϕ j ;B = [bi j]i, j=1, K ψi = bi jψ j, j=1 j=1 If det B , 0, the Lyapunov BEq and Schmidt BEq (Lyapunov BEqR and Schmidt BEqR) are equivalent. Stability of bifurcating periodical solutions at Poincar´e-Androov-Hopf bifurcation... 117

If det A , 0, then all small solutions (stationary or periodic) can be represented in the special form with Schmidt’s BEq (BEqR). By using the additional structure of intertwining analogues of the results from Sections 2-4 herein on bifurcating solutions, stability can be proved. In the article [20] parametric families which intertwine the operators of the equation (1) are considered. Here also new results about bifurcating solutions stability can be obtained.

References [1] B.V. Loginov, On the stability of solutions of differential equations with degenerate operator at the derivative, Izvestiya Acad. Nauk Uzbek SSR 1 (1988), 28-31 (in Russian); Letter to the Editor 2(1988), 78. [2] B.V. Loginov, Yu.B. Rousak, On the role of generalized Jordan structure in the problem of the stability of bifurcating solutions, III Colloq. on Qualitative Theory of Diff. Eqs., Szeged, Hun- gary, 22-26.08.1988, Book of Abstracts. [3] B.V. Loginov, Yu.B. Rousak, Generalized Jordan structure in the problem of the stability of bifurcating solutions, Nonlinear Analysis. TMA, 17, 3(1991), 219-232. [4] B.V. Loginov, L.R. Kim-Tyan, Yu.B. Rousak, Modification of the Lyapunov-Schmidt method and the stability of solutions of differential equations with a singular operator of finite index multiplying the derivative, Russian Acad. Sci. Dokl. Math., 47, 3(1993), 599-603. [5] B. Karasozen,¨ I. Konopleva, B. Loginov, Hereditary symmetry of resolving systems in nonlinear equations with Fredholm operator, Nonl. Anal. and Appl.: To Prof. V.Lakshmikantham on his 80-th Birthday, Ravi P. Agarwal, Donal O’Regan (Eds), Kluwer Acad. Publ. Dordrecht, 2(2003), 617-644. [6] L.R. Kim-Tyan, B.V. Loginov, Yu.B. Rousak, On the stability of periodic solutions for differential equations with a Fredholm operator at the highest derivative, Nonlinear Analysis, TMA, 67(2007), 1570-1585. (Russian). [7] M.M. Vainberg, V.A. Trenogin, Branching theory of solutions of nonlinear equations, Moscow, Nauka, 1969; Engl. transl. Volters-Noordorf Int. Publ., Leyden 1974. [8] V.A. Trenogin, Periodic solutions and solutions of transition type in abstract reaction-diffusion equations, in: Questions of Qualitative Theory of Diff. Eqs., Siberian Branch Acad. Sci. USSR, Novosibirsk, Nauka, 1988, 134-140. [9] M.Zuhair Nashed (ed), Generalized Inverses and Applications, (Proc. Adv. Sem., Madison, WI, 1973), Acad. Press, NY, 1976. [10] V.I. Yudovich, Autooscillations arising in a fluid, Prikl. Mat. Mekh.35(4), (1971), 638-655; Engl. transl. J. Appl. Math. Mech., 35(1971). [11] B.V. Loginov, On the determination of branching equation in nonstationary bifurcation by its group symmetry, in: Modern Group Analysis and Problems of Math. Modeling (Proc. XI Russian Colloq. on Group Anal., 7-11 June, Samara Univ.), 1993, 112-124. [12] B.V. Loginov, Determination of the branching equation by its group symmetry – Andronov-Hopf bifurcation, Nonlinear Analysis, TMA 28, 12(1997), 2033-2047. [13] B.V. Loginov, V.A. Trenogin, Branching equation of Andronov-Hopf bifurcation under group symmetry conditions, CHAOS, Amer. Inst. Phys., 7, 2(1997), 229-238. [14] N.N. Moiseev, Asymptotic Methods of Nonlinear Mechanics, M., Nauka, 1969 (Russian); Engl. Transl. 118 Luiza R. Kim-Tyan, Boris V. Loginov, Youri B. Rousak

[15] I.V. Morshneva, V.I. Yudovich, On cycles bifurcation from equilibria of systems with inversion and rotation symmetry, Siberian Math. J., 26, 1(1985), 124-133. [16] O.V.Makeev, Andronov-Hopf bifurcation with symmetries of square and hexagonal lattices, Proc. Middle-Volga Math. Soc. 7, 1(2005), 215-223. (in Russian). [17] O.V. Makeev, An Review of Computer Realization of the branching equation construction on allowed group symmetry and subgroup invariant solutions, ROMAI Journal 2, 1(2006), 135-148. [18] I.V. Konopleva, B.V. Loginov, Yu.B. Rousak, Branching equation in the root-subspace for differential equations nonresolved with respect to derivative and stability of bifurcating solutions, ROMAI Journal 5, 2(2009), 97-107. [19] N.A.Sidorov, D.A.Tolstonogov.Morse lemma application in the investigation of bifurcation points and stability of diﬀerential equations , Preprint N 104, Irkutsk Comput. Center of Siberian Branch Acad.Sci.USSR, 1989. [20] N.A.Sidorov, V.R.Abdullin.Intertwined branching equations in the theory of nonlinear equations, Matem. Sbornik 192, 7(2001)107-124. ROMAI J., 6, 1(2010), 119–124

STRICTLY INCREASING MARKOV CHAINS AS WEAR PROCESSES Mario Lefebvre Department of Mathematics and Industrial Engineering, Ecole´ Polytechnique, Montr´eal,Canada [email protected] Abstract To model the lifetime of a device, increasing Markov chains are used. The transition probabilities of the chain are as follows: pi, j = p if j = i+δ, and pi, j = 1− p if j = i+2δ. The mean time to failure of the device, namely the mean number of transitions required for the process, starting from x0, to take on a value greater than or equal to x0 + kδ is computed explicitly. A second version of this Markov chain, based on a standard Brownian motion that is discretized and conditioned to always move from its current state x to either x + δ or x + 2δ after time units, is also considered. Again the expected value of the time it takes the process to cross the boundary at x0 + kδ is computed explicitly.

~~Keywords: reliability, hitting time, Brownian motion, diﬀerence equations. 2000 MSC: 60J20, 60J65, 60K10.~~

1. INTRODUCTION Let X(t) denote the wear of a machine at time t. Although wear should obviously increase with time, some authors have used a one-dimensional diﬀusion process as a model for X(t). For example, in a related problem, Tseng and Peng [4] (see also Tseng and Peng [3]) considered the following model for the lifetime of a device: they assumed that the device possesses a certain quality characteristic that is closely corre- lated with its lifetime, and they denoted by D(t) the value of this quality characteristic at time t. Next, they assumed that D(t) is a decreasing function of t and that it can be represented as follows:

Z t D(t) = M(t) + s(u)dB(u), (1) 0 in which M(t) designates the mean value of the random variable D(t), and {B(t), t ≥ 0} is a standard Brownian motion. The function s(u) could be a constant (as in [3]). Thus, Tseng and Peng wanted to use a stochastic integral as a noise term. Finally, in their model, the device is considered to be worn out the ﬁrst time D(t) takes on a value smaller than or equal to the critical level c (a constant). Hence, the lifetime Lc of the device is deﬁned by

~~Lc = inf{t > 0 : D(t) ≤ c}.~~

~~119 120 Mario Lefebvre~~

Now, if the function s(·) is very small, the model proposed by Tseng and Peng is probably a good approximation of reality. However, even if we suppose that M(t) is a decreasing function, we cannot claim that D(t) is decreasing as well. Indeed, a stochastic process that satisfies Eq. (1) can both increase and decrease on any interval. Next, to obtain a function X(t) that is strictly increasing with time, as should be, Rishel [2] proposed to use a degenerate two-dimensional diffusion process defined by the following system of stochastic differential equations: dX(t) = ρ[X(t), Y(t)]dt, dY(t) = f [X(t), Y(t)]dt + σ[X(t), Y(t)]dB(t). In this model, ρ(·, ·) and σ(·, ·) are positive functions in the domain of interest, and Y(t) is a variable (it could actually be a vector) that directly influences the wear. No- tice that we could use this model for the remaining lifetime of a device by assuming instead that ρ(·, ·) is a negative function. Recently, the author (see [1]) used the above model in the particular case when {Y(t), t ≥ 0} is a geometric Brownian motion and the function ρ(·, ·) is given by ρ Y(t) ρ[X(t), Y(t)] = − 0 , [X(t) − c]κ where ρ0 > 0 and κ ≥ 0 are constants. He computed the expected value of the first passage time

~~Tc(x, y) = inf{t > 0 : X(t) = c | X(0) = x (> c), Y(0) = y}~~

(that corresponds to the random variable Lc defined in [4]). This expected value yields the Mean Time To Failure (MTTF) for the device considered. In discrete time, we don’t have to define a two-dimensional stochastic process. Indeed, we can consider a strictly increasing one-dimensional Markov chain {Xn, n = 0, 1,... }. In this paper, we suppose that after the nth time unit, for n = 1, 2,..., the process {Xn, n = 0, 1,...} takes on the value Xn−1 + δ with probability p ∈ (0, 1) or Xn−1 + 2δ with probability 1 − p. In Section 2, we compute the mean number of transitions required for Xn to become greater than or equal to x0+kδ, with k = 1, 2,..., a value for which the device is considered to be worn out. In Section 3, we consider a version of the Markov chain defined above that is based on a continuous-time stochastic process: we condition a discretized Brownian motion process to always move from its current state x to either x + δ or x + 2δ after time units. Again the expected value of the time it takes the process to cross the boundary at x0 + kδ is computed explicitly. 2. A STRICTLY INCREASING MARKOV CHAIN

Let Xn, for n ∈ {0, 1,...}, represent the wear of a certain device after n time units. We assume that the initial wear is X0 = x0 (≥ 0) and that {Xn, n = 0, 1,...} is a Markov Strictly increasing Markov chains as wear processes 121 chain with state space {x0, x0 + δ, . . . , x0 + (k + 1)δ}, where δ > 0 and k ∈ {1, 2,...}, and having transition probabilities ( p if j = i + δ, p = i, j 1 − p if j = i + 2δ for i = x0, x0 + δ, . . . , x0 + (k − 1)δ, where 0 < p < 1. The Markov chain is thus strictly increasing: after each time unit, the process increases by δ or 2δ units. The critical level for the device is the value x0 + kδ. Let m(i), for i = 0, 1,..., k, denote the expected value of the random variable

~~T(i) = inf{n > 0 : Xn ≥ x0 + kδ | X0 = x0 + iδ}.~~

Notice that T(i) is the number of transitions required for the Markov chain to increase by at least (k − i)δ units, if it starts from x0 + iδ. To obtain a difference equation satisfied by the function m(i), we can condition on the first transition. We obtain that

~~m(i) = m(i + 1)p + m(i + 2)(1 − p) + 1. (2)~~

Indeed, if X1 = x0 + δ (respectively X1 = x0 + 2δ), then, by the Markov property, the process starts anew from x0 + δ (respectively x0 + 2δ). Furthermore, we must take the first transition into account. Hence we add one time unit to the expected value from the new starting point. To obtain the general solution of (2), we must first find two linearly independent solutions of the corresponding homogeneous equation, that is,

~~m(i) = m(i + 1)p + m(i + 2)(1 − p).~~

~~One such solution is trivially m(i) ≡ c0, a constant. A second solution is given by~~

~~m(i) = (p − 1)k−i.~~

~~It follows that the general solution of the homogeneous equation can be expressed as~~

k−i m(i) = c0 + c1(p − 1) , in which c0 and c1 are arbitrary constants. Next, we can check that m(i) = c(k − i), where c is a constant, is a solution of the non-homogeneous equation (2) if we take c = (2 − p)−1. Hence, we can write that k − i m(i) = c + c (p − 1)k−i + . 0 1 2 − p 122 Mario Lefebvre

Finally, to obtain the particular solution to our problem, we simply have to make use of the boundary conditions m(k) = 0 and m(k − 1) = 1. Indeed, if the initial value of the Markov chain is X0 = x0 + kδ then the device is already worn out, whereas if X0 = x0 + (k − 1)δ then X1 will be equal to either x0 + kδ or x0 + (k + 1)δ. We can now state the following proposition.

~~Propozit¸ia2.1. The expected lifetime of the device, when X0 = x0 + iδ, is given by (1 − p) h i k − i m(i) = 1 − (p − 1)k−i + (2 − p)2 2 − p for i = 0, 1,..., k.~~

Thus, when X0 = x0 the expected lifetime of the device is (1 − p) h i k m(0) = 1 − (p − 1)k + . (2 − p)2 2 − p Notice that if p decreases to zero, then m(0) tends to k/2 if k is an even integer, and to (k + 1)/2 if k is an odd integer, as should be. Similarly, if p increases to 1, then m(0) tends to k, which again is obviously correct.

3. A CONDITIONED BROWNIAN MOTION If we want to use a Brownian motion (or Wiener process) to model the wear of a device, we can condition it to increase by a small quantity in a short interval. By doing so, we take into account the fact that there are always small errors when we measure wear, so that it can slightly decrease in a short interval. However, the probability that wear will decrease (or increase) greatly in a short interval is negligible. Assume that {B(t), t ≥ 0} is a standard Brownian motion starting at B(0) = x0. Let Xδ,(0) = B(0) = x0 and

~~Xδ,(t) = B(t) | {{B(t) = B(t − ) + δ} ∪ {B(t) = B(t − ) + 2δ}} for t = , 2, . . . Next, consider the random variable~~

τδ,(x0) = inf{t > 0 : Xδ,(t) = x0 + kδ | Xδ,(0) = x0}. We obtain the following diﬀerence equation for the expected value of the random variable τδ,(x0), which we will denote simply by µ(k) to emphasize that the process must increase by k times δ from its current position:

~~µ(k) = + µ(k − 1)p + µ(k − 2)(1 − p), (3) where~~

~~p = P[B(t) = B(t − ) + δ | {B(t) = B(t − ) + δ} ∪ {B(t) = B(t − ) + 2δ}]. Strictly increasing Markov chains as wear processes 123~~

~~Remark 3.1. Because, by deﬁnition, the Wiener process has stationary increments, only the distance that it must cover, in time units, is important.~~

~~The probability p is given by~~

fB()(δ) p = , fB()(δ) + fB()(2δ) in which fB() is the probability density function of a Gaussian distribution with mean 0 and variance . That is, ( ) 1 x2 fB()(x) = √ exp − 2π 2 for x ∈ R. It follows that

~~δ2 exp{− 2 } p = . δ2 2δ2 exp{− 2 } + exp{− }~~

2 2 If we write that δ = σ0, where σ0 > 0, we have: 1 p = 2 . 3σ0 1 + exp{− 2 } Hence, contrary to the case of the Markov chain in the preceding section, the probability p cannot take on any value in the interval (0, 1). Actually, we ﬁnd that p ∈ (1/2, 1). Now, (3) is of the same form as (2). Proceeding as in Section 2, we obtain the following result. Propozit¸ia3.1. Under the assumptions made in this section, the expected lifetime of the device, when Xδ,(0) = x0, is (1 − p) h i k µ(k) = 1 − (p − 1)k + , (2 − p)2 2 − p where p ∈ (1/2, 1). Proof. The general solution of Eq. (3) can be written as k µ(k) = k + k (p − 1)k + . 1 2 2 − p

The constants k1 and k2 are obtained from the boundary conditions µ(0) = 0 and µ(1) = . 124 Mario Lefebvre 4. CONCLUSION To model the wear of a device, in Section 2 we proposed a strictly increasing Markov chain. In Section 3, we used a standard Brownian motion that we discretized and conditioned to increase by either δ or 2δ units every time units. In both cases we obtained a second-order non-homogeneous diﬀerence equation with constant coeﬃ- cients for the expected value of the random variable representing the time to failure of the device. We could try to compute the variance of the lifetime. Ideally, we would like to obtain the probability mass function of the lifetime. Finally, in Section 2 we could consider the case when the process spends a random amount of time, say S , in a given state before making a transition. If the random variable S has an exponential distribution for each state, then the stochastic process {X(t), t ≥ 0}, where X(t) denotes the state of the process at time t, would be a continuous-time Markov chain. Otherwise, it would be a semi-Markov process.

~~Acknowledgments~~

~~This work was supported by the Natural Sciences and Engineering Research Coun- cil of Canada.~~

~~References~~

[1] M. Lefebvre, Mean ﬁrst-passage time to zero for wear processes, Stoch. Models. (To appear) [2] R. Rishel, Controlled wear processes: modeling optimal control, IEEE Trans. Automat. Contr., 36(1991), 1100-1102. [3] S. T. Tseng, C. Y. Peng, Optimal burn-in policy by using an integrated Wiener process, IIE Trans., 36(2004), 1161-1170. [4] S. T. Tseng, C. Y. Peng, Stochastic diﬀusion modeling of degradation data, J. Data Sci., 5(2007), 315-333. ROMAI J., 6, 1(2010), 125–134

GENERALIZED PRIORITY MODELS WITH “LOOK AHEAD” STRATEGY: NUMERICAL ALGORITHMS FOR BUSY PERIODS Gheorghe Mishkoy, Olga Benderschi Free International University of Moldova, Chi¸sin˘au,Republic of Moldova State University of Moldova, Chi¸sin˘au,Republic of Moldova [email protected], [email protected]

Abstract Some analytical results concerning busy period distributions for M2|G2|1 generalized queueing systems with semi-Markov switching and so called “look ahead” strategy are discussed in this paper. We show that the presented analytical results can be viewed as a 2- dimensional analog of the well-known in queueing theory Kendall-Takacs functional equation. Numerical algorithms and modelling examples for busy periods are also presented. Acknowledgement. This work is partially supported by Russian Foundation for Basic Research (RFFI) grant 08.820.08.09RF and grant 09.820.08.01GF of the Federal Min- istry of Education and Research (BMBF) of Germany.

~~Keywords: generalized queueing systems, busy period, numerical algorithms. 2000 MSC: 60K25, 90B22.~~

1. INTRODUCTION By the Generalized Priority Models (GPM) we understand mathematical models of queueing systems, in which the switching of the service process from a class of requests (messages) to another is non-zero. Such switching between the priority classes is considered to be a random variable with arbitrary distribution function. The GPM can be defined by setting four identifiers: “priority type”, “strategy in free state”, “discipline of service” and “discipline of switching”. As shown in [1] and in some recent publications (see, for example [2] and [3]), GPM have a number of important distinguished features, compared with classical priority models. One of these distinguished features consists in the fact that mathematical formalization of switchover times leads to various new priority laws enabling one to consider more flexible real time processes, such as, absolute, semi-absolute, relative, etc., priority disciplines. Another import feature of GPM consists in the fact that they enable one to consider the strategy of server in the free states. There are several models of behaviour of the server when the system becomes empty. One of the most studied models is “set to zero” – upon completion of service

125 126 Gheorghe Mishkoy, Olga Benderschi of the last request in the system the server immediately switches to the neutral state. In the following we analyze a model which is less investigated, namely “look ahead” – the server switches itself to the 1-requests line at the moment when the system becomes empty.

2. DEFINITIONS AND NOTATIONS Consider the case of “Look Ahead” strategy for priority queueing systems with two priority classes M2|G2|1. Consider a queueing system with a single station and 2 classes of incoming requests, each having its own flow of arrival and waiting line. We call the requests th from the i queueing line Li i-requests. i-Requests have a higher priority than j- requests if 1 ≤ i < j ≤ r. The station gives preference in service to the requests of the highest priority among those presented in the system. Adopting and slightly extending the standard Kendall notation we write M2|G2|1|∞ to denote a priority queueing system with two Poisson incoming flows of requests and random switchover times. Suppose that the time periods between two consecutive arrivals of the requests of the class i are independent and identically distributed with some common cumulative distribution function (cdf) Ai(t) with mean E[Ai], i = 1, 2. Similarly, suppose that the service time of a customer of the class i is a random variable Bi with a cumulative distribution function Bi(t) with mean service time E[Bi], i = 1, 2. However, some time is needed for server to proceed with the switching from one line of requests to another. This time is considered to be a random variable, and we say that Ci j is the time of switching from the service of i- requests to the service of j-requests, if 1 ≤ i, j ≤ r, i , j. We adopt classification and terminology introduced in [1]. We also explain some additional notions and notations. Definition 2.1. By a k-busy period we understand the period of time which starts when an i-request enters the empty system, i ≤ k, and finishes when there are no longer k-requests in the system. Denote the k-busy period by Πk.

~~Note, that a 2-busy period is nothing but the system’s busy period Π, i.e. Π ≡ Π2.~~

Definition 2.2. By a Πk – period we understand the period of time which starts from the moment of arrival of a k-request when there are no i-requests (i < k) in the system and ending when the system is free of k-requests. Definition 2.3. By a k-cycle of service we understand the period of time which starts when server begins the servicing of a k-request, and finishes when this request leaves the system. Denote the k-cycle of service by Hk. Definition 2.4. By a k-cycle of switching we understand the period of time which starts when server begins to switch to the line of k-requests, and finishes when server is ready to provide service to these requests. Denote the k-cycle of switching by Nk. Generalized priority models with “Look Ahead” strategy... 127

Let Πk(t), Πk(t), Hk(t) and Nk(t) be the cumulative distribution functions of Πk- busy periods, k-busy periods, k-cycle of service and k-cycle of switching, correspond- ingly. Let also πk(s), πk(s), hk(s) and νk(s) be their Laplace–Stieltjes transform, i.e. Z∞ Z∞ −st −st πk(s) = e dΠk(t), . . . , νk(s) = e dNk(t). 0 0

~~Finally, let βk(s) be the Laplace–Stieltjes transform of Bk(t), i.e. Z∞ −st βk(s) = e dBk(t). 0~~

~~Let C12(t) and C21(t) be the cumulative distribution functions of C12 and C21. Let also c12 and c21 be their Laplace–Stieltjes transform, i.e. Z∞ −st c12(s) = e dC12(t), 0~~

~~Z∞ −st c21(s) = e dC21(t). 0~~

In this paper we consider next situation for queueing system M2|M2|1: for the orientation ON – non-identical orientation again when 1-request arrives in the system the orientation to 2-request is interrupted; after the service of 1-request is ﬁnished, the interrupted orientation begins again non-identical time of orientation, but this time has the same distribution.

OR – resume interrupted orientation when 1-request arrives in the system the orientation to 2-request is interrupted; after the service of 1-request is ﬁnished, the interrupted orientation is resumed.

OC – orientation is not interrupted, it continues when 1-request arrives in the system the orientation to 2-request is not interrupted. on the service SN – non-identical service again 128 Gheorghe Mishkoy, Olga Benderschi

when 1-request arrives in the system the service of 2-request is interrupted, after the service of 1-request is ﬁnished, the interrupted service begins again non-identical time of service, but this time has the same distribution. SR – resume interrupted service when 1-request arrives in system the service to 2-request is interrupted, after the service of 1-request is ﬁnished, the interrupted service is resumed. SL – the request is lost when 1-request arrives in system the service of 2-request is interrupted and 2-request is lost. By combining possible regimes of orientation and service one can obtain 9 types of station operation. Using the above notation, for example ON SR indicate non- identical orientation again and resume interrupted service.

3. BUSY PERIOD AND ITS EVALUATION As mentioned above the busy period is the period of time which starts when request enters in the empty system, ﬁnishes when there are no requests in the system. The Laplace–Stieltjes transform of busy period can be determined from following system of functional equations [1]:

~~(a1 + a2)π2(s) = a1π21(s) + a2π22(s), n o π21(s) = π1(s + a2) + π1(s + a2 1 − π2(s)]) − π1(s + a2) ν2 s + a2[1 − π2(s)] ϕ1(s), π22(s) = ν2 s + a2[1 − π2(s)] ϕ1(s),~~

~~π2(s) = h2(s + a2[1 − π2(s)]), (1)~~

~~π1(s) = β1(s + a1[1 − π1(s)]), (2) −1 ϕ1(s) = c21(ξ(s + a2)){1 − [c21(ξ(η(s)) − c21(ξ(s + a2))]ν2(η(s))} ,~~

~~ξ(s) = s + a1 − a1π1(s), η(s) = s + a2 − a2π2(s), were for respective case: “non-identical orientation again” – ON~~

−1 a1 ν2(s) = c12(s + a1) 1 − [1 − c12(s + a1)] c21 s + a1[1 − π1(s)] π1(s) ; s + a1 “resume interrupted orientation” – OR h i ν2(s) = c12 s + a1 1 − c21(s + a1[1 − π1(s)])π1(s) ; Generalized priority models with “Look Ahead” strategy... 129

~~“orientation is not interrupted” – OC~~

~~ h i −1 ν2(s) = c12(s + a1) 1 − c12(s + a1[1 − π1(s)]) − c12(s + a1) c21(s + a1[1 − π1(s)]) ;~~

~~“nonidentical repeat again” – SN~~

~~ −1 a1 h2(s) = β2(s + a1) 1 − 1 − β2(s + a1) c21(s + a1[1 − π1(s)])π1(s)ν2(s) ; s + a1 “resume” – SR h i h2(s) = β2 s + a1 1 − c21(s + a1[1 − π1(s)])π1(s)ν2(s) ;~~

~~“loss” – SL~~

~~a1 h i h2(s) = β2(s + a1) + 1 − β2(s + a1) c21(s + a1[1 − π1(s)])π1(s)ν2(s); s + a1~~

Remark 3.1. Assume that in the above equations the functions c12(s) and c21(s) are null and the system has only one arrival ﬂow then for this case we obtain next equation: a1π1(s) = a1π1(s) = a1β1(s + a1[1 − π1(s)]).

~~In this case π(s) = π1(s) and~~

~~π1(s) = β1(s + a1[1 − π1(s)]) This equation is known as the classical Kendall–Takacs equation.~~

~~3.1. ALGORITHMS FOR EVALUATION LAPLACE-STIELTJES TRANSFORM OF BUSY PERIOD.~~

As can be seen from equations (1) and (2) the function values π2(s) and π1(s) can be determined using numerical methods only. Thus, to evaluate the Laplace–Stieltjes transform of the busy period and the Laplace–Stieltjes transforms ν2(s) and h2(s) one needs to use numerical algorithms. An efficient method for doing so elaborated in [4] is used in the following algorithms. In the following two algorithms for two different situations are presented. The other situation mentioned in the first section can be treated in a similar way.

~~Algorithm 1 (M2|G2|1 ON SN) 2 2 Input: r, {ai}i=1, {βi(s)}i=1, c12(s), c21(s), ε; ∗ ∗ ∗ Output: π2(s ), h2(s ), ν2(s ); 130 Gheorghe Mishkoy, Olga Benderschi~~

Description: σ := a1 + a2; a π (s∗) a π (s∗) π (s∗) = 1 21 + 2 22 , 2 σ σ n o ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π21(s ) = π1(s +a2)+ π1(s +a2 1−π2(s )])−π1(s +a2) ν2 s +a2[1−π2(s )] ϕ1(s ), π22(s) = ν2 s + a2[1 − π2(s)] ϕ1(s), ∗ ∗ ∗ ∗ ∗ −1 ϕ1(s ) = c21(ξ(s + a2)){1 − [c21(ξ(η(s )) − c21(ξ(s + a2))]ν2(η(s ))} , ∗ ∗ ∗ ξ(s ) = s + a1 − a1π1(s ), ∗ ∗ ∗ η(s ) = s + a2 − a2π2(s ), −1 ∗ ∗ a1 ∗ ∗ ∗ ∗ ν2(s ) = c12(s + a1) 1 − ∗ 1 − c12(s + a1) c21 s + a1[1 − π1(s )] π1(s ) ; s + a1 −1 ∗ ∗ a1 ∗ ∗ ∗ ∗ ∗ h2(s ) = β2(s +a1) 1− ∗ 1 − β2(s + a1) c21(s +a1[1−π1(s )])π1(s )ν2(s ) ; s + a1

~~(n) e(n) n := 1; π (0) := 0; π1 (0) = 1; e1 Repeat (n) (n−1) e ∗ ∗ e ∗ π1 (s ) = β1(s + a1[1 − π1 (s )); (n) ∗ ∗ (n−1) ∗ π (s ) = β1(s + a1[1 − π (s )); e1 e1 inc(n);~~

(n) e ∗ (n−1) ∗ π1 (s ) − π (s ) Until e1 < ε; 2 e(n) ∗ (n−1) ∗ π1 (s ) + π (s ) π (s∗):= e1 ; 1 2 (n) e(n) n := 1; π (0) := 0; π2 (0) = 1; e2 Repeat (n) (n−1) e ∗ ∗ e ∗ π2 (s ) = h2(s + a1[1 − π2 (s )); (n) ∗ ∗ (n−1) ∗ π (s ) = h2(s + a1[1 − π (s )); e2 e2 inc(n); (n) e ∗ (n−1) ∗ π2 (s ) − π (s ) Until e2 < ε; 2 e(n) ∗ (n−1) ∗ π2 (s ) + π (s ) π (s∗):= e2 ; 2 2 End of Algorithm 1 Generalized priority models with “Look Ahead” strategy... 131

Algorithm 2 (M2|G2|1 OC SL) 2 2 Input: r, {ai}i=1, {βi(s)}i=1, c12(s), c21(s), ε; ∗ ∗ ∗ Output: π2(s ), h2(s ), ν2(s ); Description: σ := a1 + a2; a π (s∗) a π (s∗) π (s∗) = 1 21 + 2 22 , 2 σ σ n o ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π21(s ) = π1(s +a2)+ π1(s +a2 1−π2(s )])−π1(s +a2) ν2 s +a2[1−π2(s )] ϕ1(s ), π22(s) = ν2 s + a2[1 − π2(s)] ϕ1(s),

∗ ∗ ∗ ∗ ∗ −1 ϕ1(s ) = c21(ξ(s + a2)){1 − [c21(ξ(η(s )) − c21(ξ(s + a2))]ν2(η(s ))} , ∗ ∗ ∗ ξ(s ) = s + a1 − a1π1(s ), ∗ ∗ ∗ η(s ) = s + a2 − a2π2(s ), h i −1 ∗ ∗ ∗ ∗ ∗ ∗ ν2(s) = c12(s +a1) 1− c12(s +a1[1−π1(s )])−c12(s +a1) c21(s +a1[1−π1(s )]) ; h i ∗ ∗ a1 ∗ ∗ ∗ ∗ ∗ h2(s ) = β2(s + a1) + ∗ 1 − β2(s + a1) c21(s + a1[1 − π1(s )])π1(s )ν2(s ); s + a1

e(n) ∗ (n−1) ∗ π2 (s ) + π (s ) π (s∗):= e2 ; 2 2 End of Algorithm 2 Remark 3.2. One can eﬃciently evaluate the Laplace–Stieltjes transform of busy period using the algorithms described. In order to determine the value of the busy period one needs to use inversion algorithms (for example see [4]).

~~3.2. EXAMPLES OF EVALUATIONS OF THE BUSY PERIOD~~

Example 3.1. Consider the system M2|M2|1 (ON SN) with interarrival times being distributed exponentially Exp(ak) k = 1, 2 and exponential service times Exp(bk), bk = 100, k = 1, 2. The switchover times Ck are all distributed exponentially Exp(ω), ω = ω12 = ω21 = 100. The quantity ε was taken to be 0.000001.

~~bk βk(s) = , k = 1, 2 s + bk ω12 ω21 c12(s) = , c21(s) = . ω12 + s ω21 + s~~

~~ak π2(0) ν2(0) h2(0) 10 0.999999 1.000000 1.000000 50 0.545809 0.999999 0.999999 80 0.295837 0.999999 0.999998~~

~~Table 1 Calculation of the π2(0), ν2(0) and h2(0)~~

Example 3.2. Consider the system M2|M2|1 (ON SR) with interarrival times being distributed exponentially Exp(ak), ak = 10, k = 1, 2 and service times being distributed according to Erlang law Er(3, bk), k = 1, 2. The switchover times Ck are all distributed exponentially Exp(ω), ω = ω12 = ω21 = 200. The quantity ε was taken to be 0.000001.

~~ bk 3 ω12 ω21 βk(s) = , k = 1, 2; c12(s) = , c21(s) = . s + bk ω12 + s ω21 + s~~

Example 3.3. Consider the system M2|M2|1 (OR SN) with interarrival times being distributed exponentially Exp(ak) k = 1, 2 and Erlang service times Er(2, bk), bk = Generalized priority models with “Look Ahead” strategy... 133

~~bk π2(0) ν2(0) h2(0) 35 0.385793 1.000000 0.999999 50 0.689388 1.000000 1.000000 70 0.999996 1.000000 1.000000~~

~~Table 2 Calculation of the π2(0), ν2(0) and h2(0)~~

~~200, k = 1, 2. The switchover times Ck are all distributed exponentially Exp(ω), ω = ω12 = ω21 = 100. The quantity ε was taken to be 0.000001.~~

~~ bk 2 ω12 ω21 βk(s) = , k = 1, 2; c12(s) = , c21(s) = . s + bk ω12 + s ω21 + s~~

~~ak π2(0.5) ν2(0.5) h2(0.5) 1 0.989696 0.994926 0.994857 10 0.984776 0.993935 0.993169 50 0.469865 0.985422 0.975887~~

~~Table 3 Calculation of the π2(0.5), ν2(0.5) and h2(0.5)~~

Example 3.4. Consider the system M2|M2|1 (OC SL) with interarrival times being distributed exponentially Exp(ak), a1 = 70, a1 = 1, k = 1, 2 and exponential service times Exp(bk), bk = 100, k = 1, 2. The switchover times Ck are all distributed according Gamma Ga(2.5; ω) law, ω = ω12 = ω21, ω = 200. The quantity ε was taken to be 0.000001.

~~bk βk(s) = , k = 1, 2; s + bk ω 2.5 c (s) = , 12 ω + s ω 2.5 c (s) = . 21 ω + s 134 Gheorghe Mishkoy, Olga Benderschi~~

~~s π2(s) ν2(s) h2(s) 0 0.999999 0.999999 0.999999 0.5 0.980521 0.963790 0.968226 1.0 0.963587 0.932315 0.940896 5.0 0.870813 0.768586 0.803302~~

~~Table 4 Calculation of the π2(s), ν2(s) and h2(s)~~

~~References~~

[1] Gh. Mishkoy, Generalized priority systems, Academy of Sciences of Moldova, S¸tiint¸a, Chis¸inau,˘ 2009 (in Russian, in print). [2] Gh. Mishkoy, On multidimensional analog of Kendall-Takacs equation and its numerical solution, Lecture Notes in Engineering and Computer Sciences, World Congress of Engineering, London, U.K. 2008, Vol. 2 , 928-932. [3] Gh. Mishkoy, A virtual analog of Pollaczek-Khintchin transform equation, Bul. Acad. of Sciences of Moldova, Mathem., 2(2008), 81-91. [4] A. Bejan, Switchover time modelling in priority queueing systems, Chis¸inau,˘ PhD thesis, 2007. ROMAI J., 6, 1(2010), 135–143

AN INVERSE PROBLEM IN THE PHASE-FIELD TRANSITION SYSTEM. THE 2D CASE Costic˘aMoro¸sanu Faculty of Mathematics, ”Al. I. Cuza” University of Ia¸si,Romania [email protected] Abstract The invers problem, denoted by (P), in 2D space dimension governed by the nonlinear parabolic system (the phase-field transition system, introduced by Caginalp [3]), is considered. For every ε > 0, we associate to the nonlinear system an approximating scheme of fractional steps type; corresponding, we consider for (P) the approximating boundary optimal control problem, denoted by (Pε). On the basis of the convergence of (Pε) to (P), the necessary optimality conditions are established for (Pε) and, a conceptual algorithm of gradient type is elaborated in order to compute the sub(optimal) boundary control. The advantage of such approach is that the new method simplifies the numerical computations due to its decoupling feature. The finite element method (fem) is used to deduce the discrete equations and numerical results regarding the stability and accuracy of the fractional steps method, as well as the physical aspects (separating zone of solid and liquid states, supercooling, superheating), are reported.

Keywords: boundary value problems for nonlinear parabolic PDE, optimal control, free boundary problem, fractional steps method, ﬁnite element method, computer science. 2000 MSC: 35K60, 49, 65, 68, 93.

1. INTRODUCTION The following nonlinear parabolic system in Q = [0, T] × Ω, T > 0:  `  ρVut + ϕt = k∆u in Q, 2 (1.1)  1 τϕ = ξ2∆ϕ + (ϕ − ϕ3) + 2u in Q, t 2a ( u(0, x) = u (x) x ∈ Ω, 0 (1.2) ϕ(0, x) = ϕ0(x) x ∈ Ω,  ∂u  + hu = w(t)g(x) on Σ = [0, T] × ∂Ω, ∂ν (1.3)  ∂ϕ  = 0 on Σ, ∂ν was proposed by Caginalp [3] to describe the phase transition in a domain Ω ⊂ Rn, n = 1, 2, 3. System (1.1)–(1.3) is derived from classical Fourier model via Landau- Ginzburg theory. Here u is the reduced temperature, ϕ is the phase function used

135 136 Costic˘aMoro¸sanu to distinguish between the phases of the material Ω that is involved in the transition process and u0, ϕ0 : Ω 7→ R, w : [0, T] 7→ R are given functions. The positive parameters τ, ξ, `, k, h, a have the following physical meaning: τ is the relaxation time, ξ is the length scale of the interface, ` denotes the latent heat, k the heat conductivity, h the heat transfer coeﬃcient and a is an probabilistic measure on the individual atoms, depending on ξ. The function w in (1.3) represents the temperature of the surrounding at x ∈ ∂Ω and it is manipulated by a cooling system according to the equation ( w0(t) = βw(t) + v(t) t ∈ [0, T], (1.4) w(0) = 0,. where v ∈ U, U = v(t) ∈ L∞(0, T), 0 ≤ v(t) ≤ R, a.e. t ∈ [0, T] . (1.5)

As regards the existence in (1.1)-(1.4), see [7, Proposition 2.1]. Consider that at the moment t the separating region between the phases of the material (solid and liquid, for example) is given by the surface x = σ(t) (denoted also −1 2 by t = l(x) = σ (x)) that is a function of class C (Ω¯ t) such that (see Figure 1)

~~Ωt = {x ∈ Ω, l(x) < t} is increasing in t, l(x) = 0 for all x ∈ Ω0, |∇l(x)| , 0 for all x ∈ Ω \ Ω0 and ∆l(x) > 0.~~

~~Fig. 1. A material Ω exists in two phases~~

~~Let δ and α be two positive constants. We deﬁne solid region: {(t, x); u(t, x) < −δ + α, |ϕ(t, x) + 1| ≤ α}, An inverse problem in the phase-ﬁeld transition system. The 2D case 137~~

~~liquid region: {(t, x); u(t, x) > δ − α, |ϕ(t, x) − 1| ≤ α}, separating region: {(t, x); |u(t, x)| ≤ δ, |ϕ(t, x)| ≤ α} , and we set Σ0 = {(t, x) ∈ Q, t = l(x)},~~

~~Q0 = {(t, x) ∈ Q, l(x) < t < T}. The inverse problem that we will study in this paper can be formulated as follows:~~

2 Given Σ0, ﬁnd the boundary control w ∈ L (Σ) such that Q0 is in the liquid region, Q1 = Q\Q¯ 0 is in the solid region and a neighbourhood of Σ0 is the separating region between the liquid and the solid region. This inverse problem is in general ”ill posed” and a common way to treat it is to reformulate it as an optimal control problem with an appropriate cost functional (see [5]). So we will concern in the sequel with an optimal control problem associated to the above inverse problem, namely: we look for w ∈ L2(Σ) which minimizes the functional Z Z T 1 2 − 2 2 (P) [(ϕ + 1) + γ((u + δ − α) ) ]χQ dxdt + w (t)dt, 2 Q 0 0 for all (u, ϕ) solutions of the system (1.1)-(1.4) and for all v ∈ U. γ > 0 is a given constant and χ is the characteristic function of Q0. In the statement above we have Q0 denoted by u− the negative part of u, i.e., ( 0, if u > 0; u− = − inf{u, 0} = −u, if u < 0.

~~2. Approximating optimal control problem~~

~~For every ε > 0 we associate to problem (P) the following approximating optimal control problems: (Pε) Minimize~~

Z Z T ε 1 v 2 v − 2 v 2 j (v) = [(ϕ + 1) + γ((u + δ − α) ) ]χQ dxdt + (w ) (t)dt, 2 Q 0 0 on all (uε, ϕε, w, v) subject to   `  ρVuε + ϕε − k∆uε = 0 in Qε = (iε, (i + 1)ε) × Ω,  t t i  ε 2 ∂u (2.1)  + huε = w(t) on Σε = (iε, (i + 1)ε) × ∂Ω,  ∂ν i  ε u (0, x) = u0(x) x ∈ Ω, 138 Costic˘aMoro¸sanu ( w0(t) = βw(t) + v(t), t ∈ [0, T], (2.2) w(0) = 0,  1  ε 2 ε ε ε ε  τϕt − ξ ∆ϕ = ϕ + 2u in Qi ,  2a ∂ϕ (2.3)  ε = 0 on Σε,  ∂ν i  ε ε ϕ+(iε, x) = z(ε, ϕ−(iε, x)), ε where z(·, ϕ−(iε, x)) is the solution of the Cauchy problem   1  z0(s) + z3(s) = 0, s ∈ [0, T]  2a  (2.4)   ε ε z(0) = ϕ−(iε, x)) ϕ−(0, x)) = ϕ0(x), h i T ε computed at s = ε, for i = 0, Mε − 1, with Mε = and Q = ((Mε −1)ε, T)×Ω. ε Mε−1 ε ε ε ε Here ϕ+(iε) = lim ϕ (t), ϕ−(iε) = lim ϕ (t). t↓iε t↑iε The convergence of the optimal solution of problem (Pε) to the optimal solution of problem (P) (as ε → 0) as well as the necessary optimality conditions in (Pε), that is:   ε ε ` ε 2 ε ε  pt + k∆p − τ p + τ q = 0 in Qi ,  ∂pε  + hpε = 0 on Σε, (2.5)  ∂ν i  ε ε p−((i + 1)ε, x) = 0, p−(T, x) = 0 x ∈ Ω,

  `ξ2 ` ξ2 1  τqε − ∆pε − pε + ∆qε + qε = ϕεχ in Qε,  t 2τ 4aτ τ 2aτ 0 i  `  qε = pε on Σε,  2 i (2.6)  Rε  qε ((i + 1)ε, ·) = exp 3 (z(t, ·) + 1)2dt qε ((i + 1)ε, ·), on Ω,  − 2a +  0  ε q−(T, ·) = 0, for i = Me − 2, Me − 3, ..., 1, 0, where z(t, ·) is the solution of (2.4) and ( R, if r(t) < 0, v∗(t) = (2.7) 0, if r(t) > 0,

~~Z T Z r(t) = (w(s) − kpε(s, x))dx eβ (s−t)ds, t ∂Ω are proved in [6]. An inverse problem in the phase-ﬁeld transition system. The 2D case 139~~

~~3. Numerical algorithm and results~~

The aim of this section is to give a numerical algorithm in order to compute the approximating optimal control v∗ in problem (Pε) given by (2.7). In this sense, we have proposed a gradient type method (see [2], [8]). We assume that Ω ⊂ lR2 is a polygonal domain. Let ε = T/M be the time step size ¯ (M ≡ Mε in the sequel), let Tr be the triangulation (mesh) over Ω, Ω = ∪K∈Tr K and, let N j = (xk, yl), j = 1, nn, be the nodes associated to Tr. Now, we will construct the discreet form of the problem (Pε). Using an implicit (backward) finite difference scheme in time and the finite element method in the space, the discrete equations corresponding to (2.1)-(2.3) and (2.5)-(2.6) are, respectively (i = 1, M):

( ε, i ` ε, i ε, i ε, i−1 ` ε, i−1 i−1 Cul + 2 Bϕl + εkhFRul = B(u + 2 ϕ + εkw g¯l), ε, i ε, i ε, i−1 ε (3.1) Dϕl − 2εBul = B · (τϕl + 2a ),   ε, i 2 ε, i ε, i ε, i+1  Epl + ε τ Bql − εkhFRpl = Bpl ,  2 (3.2)  Fqε, i + Rpε, i + (εh + ε ξ ` )FR pε, i = B(ϕε, iχ + pε, i+1 + qε, i+1) l l τ 2 l l 0 l l (see [9] for more details). ∗ The conceptual algorithm of gradient type for the calculation of the controller vε in (2.7) is: Algorithm CPHT-2D (Control PHase Transition-2D) P0. Set iter:=0; ε,iter ε, iter ε, iter ε, iter ε, iter Choose v = (v0 , v1 , ..., vM ), vi ∈ U, i = 0, M; ε, iter ε, iter ε, iter ε, iter P1. Compute w = (w1 , w2 , ..., wM ) from (1.4); P2. Compute the approximate matrix

~~ε, iter ε, 1 ε, M ε, iter ε, 1 ε, M u = (ul , ..., ul ), ϕ = (ϕl , ..., ϕl );~~

i.e., for i = 1, M): ε |ϕ−(ti, ·) + 1| • Compute z(ti, ·) = ± q − 1, on Ω; t ε 2 1 + a (ϕ−(ti, ·) + 1) ε • Set ϕ+(ti, ·) = z(ti, ·); ε,i ε,i • Compute the column vectors ul , ϕl , l = 1, nn, solving the linear systems (3.1); P3. Compute the approximate matrix

~~ε, iter ε,0 ε,1 ε,M−1 ε, iter ε,0 ε,1 ε,M−1 p = (pl , pl , ..., pl ), q = (ql , ql , ..., ql ); ε, i ε, i The column vectors pl , ql , i = 0, M − 1, l = 1, nn, 140 Costic˘aMoro¸sanu~~

~~are obtained solving the linear systems (3.2); ε, iter ε, iter P4. For all i ∈ {0, 1, ..., M} compute r (ti) andv ˜ (ti), by (2.7);~~

~~P5. Compute λiter ∈ [0, 1] (the steplenght of the gradient method) solution of the minimization process:~~

~~min{ jε(λvε,iter + (1 − λ)˜vε,iter), λ ∈ [0, 1]};~~

ε,iter+1 ε,iter ε,iter Set v := λiterv + (1 − λiter)˜v ; P6. (the ”Stopping Criterion”) if kvε,iter+1 − vε,iterk ≤ η then STOP (the algorithm is convergent) else iter := iter + 1 ; Go to P1. Let us brieﬂy discuss the main steps in algorithm CSR-2D. For approximating the solution of the nonlinear parabolic system (1.1)-(1.4) we have used a numerical method of fractional steps type (see [7], Section 4.2). This method (expressed in step P2) avoids the iterative process required by the classical approaches (e.g., Newton’s type method) in passing from a time level to another (see [6] for additional details). Moreover, we point out that the equation (2.4) can be solved directly by the separation of variables (the relation (4.7) in [7]). In the above algorithm the variable iter represents the number of iterations after which the algorithm CSR-2D found the optimal value of the cost functional jε(v) in (Pε).

~~Fig. 2. The domain Q0~~

Numerical experiments to compute the boundary control in (P) in 1D was made in the work [5] where an iterative Newton method is used in order to approximate the solution of nonlinear parabolic system (1.1)-(1.4). In order to test the computer program implementing the algorithm CPHT-2D, we have used the following experimental values of parameters:

~~the casting speed (V = 12.5 mm/s), An inverse problem in the phase-ﬁeld transition system. The 2D case 141~~

~~Fig. 3. The triangulation over Ω=[0,650]x[0,220]~~

physical parameters: the density (ρ = 7850 kg/m3), the latent heat (` = 65.28kcal/kg), the relaxation time (τ = 1.0e + 2 ∗ ξ2), the length of separating zone (ξ = .5), the coeﬃcients of heat transfer (h = 32.012), a = .00008, T = 44s; the boundary conditions (w(t), t ∈ [0, T]) in the primary cooling zone: dimensions of cristallizer (550 x 1300 x 220), in mm;

0 the casting temperature (u0 = 1530 C); the termal conductivity k(u): k(u) = [20 100 200 300 400 500 600 700 800 850 900 1000 1100 1200 1600;1.43e-5 1.42e-5 1.42e-5 1.42e-5 1.42e-5 9.5e-6 9.5e-6 9.5e-6 8.3e-6 8.3e-6 8.3e-6 7.8e-6 7.8e-6 7.4e-6 7.4e-6].

Figure 2 illustrates the domain Q0 we have considered in our experiments, while in Figure 3 the number of nodes associated to the mesh in the x1 and x2 – axis directions of one half of a rectangular proﬁle is represented. Only a half of the cross-section is used in the computation program. The numerical model (3.1) uses the temperatures w(t), t ∈ [0, T] measured by the termocouples; corresponding to tM, the values are ilustrated in the Figure 6 (the line ploted by ∗). ∗ Figures 4 and 5 represents the approximate solution uM for iter = 1 and iter = 5, respectively. A close examination of them tell us the dimension of the solid and liquid zone resulting by runing the Matlab computation program developed on the basis of the conceptual algorithm CPHT-2D. 142 Costic˘aMoro¸sanu

~~∗ Fig. 4. The approximate temperature uM ( iter=1)~~

~~∗ Fig. 5. The approximate temperature uM ( iter=5)~~

~~1 5 5 Fig. 6. The boundary optimal control: ∗ - wM, • - w1, ◦ - wM An inverse problem in the phase-ﬁeld transition system. The 2D case 143~~

The shape of the graphs in ﬁgures 4-5 shows the stability and accuracy of the numerical results obtained by implementing the fractional steps method, but the most interesting aspect that we can observe when analyzing Figure 5, are the presence of supercooling and superheating phenomenon. Acknowledgment. The work has been partial elaborated under the support of Con- tract CEx 05-D11-84/28.10.2005, ﬁnanced by Romanian Ministry of Education and Research.

~~References~~

[1] O. Axelson, V. Barker, Finite element solution of boundary value problems, Academic Press, 1984. [2] V. Barbu, A product formula approach to nonlinear optimal control problems, SIAM J. Control Optim., 26(1988), 496-520. [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal., 92(1986), 205-245. [4] K.-H. Hoffman, L. Jiang, Optimal control problem of a phase field model for solidification, Nu- mer. Funct. Anal., 13 (1992), 11-27. [5] C. Moros¸anu, Numerical approach of an inverse problem in the phase field equations, An. S¸t. Univ. ”Al.I.Cuza” Iai¸, T XXXIX, s. I-a, f.4, (1993). [6] C. Moros¸anu, Boundary optimal control problem for the phase-field transition system using fractional steps method, ”Control & Cybernetics”, 32, 1(2003), 5-32. [7] C. Moros¸anu, Analysis and optimal control of phase-field transition system, Nonlinear Funct. Anal. & Appl., 8, 3(2003), 433–460. [8] C. Moros¸anu, Fractional steps method for approximation the solid region via phase field transition system, 6-th International Conference APLIMAT 2007, PART II, Bratislava, Slovak Repub- lic, February 06-09, 2007, 235-240, (2007). [9] C. Moros¸anu, al., Report Stage II/2006, CEEX program no. 84/2005, (2007).

~~ROMAI J., 6, 1(2010), 145–152~~

KACZMARZ EXTENDED VERSUS AUGMENTED SYSTEM SOLUTION IN IMAGE RECONSTRUCTION Aurelian Nicola, Constantin Popa “Ovidius” University of Constant¸a, Romania [email protected], [email protected] Abstract In this paper we make a comparative analysis of two projection based iterative algorithms for systems of linear equations arising from image reconstruction in computerized tomography. The ﬁrst one - Kaczmarz’s iterations - is used for solving the (consistent) augmented system, whereas the second - Kaczmarz extended algorithm - is used for solving the original (inconsistent) system. We obtain bounds for the generalized spectral condition numbers for both augmented and original system, which give us information about the theoretical behaviour of the two iterative methods. Numerical experiments, comparing the two solvers are made on a phantom widely used in the image reconstruction literature.

~~Keywords: linear least squares problem; augmented system; Kaczmarz’s projections iteration; Kacz- marz Extended method. 2000 MSC: 65F10, 65F20.~~

~~1. INTRODUCTION Image reconstruction from projections in computerized tomography gives rise to general least squares problem (LSP, for short): ﬁnd x∗ ∈ Rn such that~~

k Ax∗ − b k= min{k Az − b k, z ∈ Rn}, (1) where h·, ·i, k · k will denote the Euclidean scalar product and norm. We shall denote by LS S (A; b) the set of its all (least squares) solutions and by xLS its (unique) minimal norm one. AT , R(A), N(A), σ(A), ρ(A) will denote the transpose, range, null space, spectrum and spectral radius of A, whereas PS will be the orthogonal projection onto a vector subspace S ∈ Rq, for some q ≥ 1. Moreover, if r = rank(A) and

T U AV = diag(σ1, . . . , σr, 0,..., 0), σ1 ≥ σ2 ≥ · · · ≥ σr > 0, (2) is a singular value decomposition (SVD) of A we deﬁne its generalized spectral condition number by σ1 gk2(A) = . (3) σr

~~145 146 Aurelian Nicola, Constantin Popa~~

Remark 1.1. If B is a symmetric n×n matrix with r = rank(B), it exists an orthogonal one U such that T U BU = diag(λ1, . . . , λn), (4) where λi ∈ R are the eigenvalues of B. If in addition λi ≥ 0, ∀i = 1,..., n (i.e. B is positive semideﬁnite), we can construct U such that

~~λ1 ≥ λ2 ≥ · · · ≥ λr > 0 = λr+1 = . . . λn, which tells us that the decomposition (4) is an a SVD of B and~~

~~λ1 gk2(B) = . (5) λr From all these considerations and (2) we get for a general m×n matrix A the equality s q T T λmax(A A) gk2(A) = gk2(A A) = ∗ T , (6) λmin(A A)~~

T ∗ T where λmax(A A), λmin(A A) are the maximal, respectively minimal-nonzero eigenvalues of the symmetric and positive semideﬁnite matrix AT A. The following results are known. Proposition 1.1. We have the following two equivalent formulations for (1):

~~∗ ∗ x ∈ LS S (A; b) ⇐⇒ Ax = PR(A)(b). (7) and the normal equation~~

~~x∗ ∈ LS S (A; b) ⇔ AT Ax∗ = AT b. (8)~~

Remark 1.2. The normal equation (8) is not only a theoretical characterization of LS S (A; b), but because it is a square, positive semideﬁnite and always consistent system, several direct and iterative solvers have been designed for its solution (see e.g. [1, 2]). Although the most of these methods do not require the computation of AT A, a big inconvenient still remains: the generalized spectral condition number for T 2 the matrix involved in the normal equation satisﬁes (see (6)) gk2(A A) = (gk2(A)) , which will determine a poor behaviour of the corresponding methods when applied to (8). As an alternative with respect to the inconvenients mentioned in the above remark, we may consider the associated (m + n) × (m + n) augmented system (see [1]) " #" # " # IA r b = . (9) AT 0 x 0 Kaczmarz Extended versus augmented system solution in image reconstruction 147

Proposition 1.2. (i) The normal equation (8) and the augmented system (9) are equivalent in the following sense: if x ∈ Rn satisfies (8) and r = b − Ax, then (r, x)T ∈ Rm+n satisfies (9) and conversely, if (r, x)T ∈ Rm+n is a solution for (9), then x satisfies (8). In particular, the augmented system is consistent. (ii) Let xLS be the minimal norm solution of (1) and

~~rLS = b − AxLS = PN(AT )(b). (10)~~

~~m+n Then, the minimal norm solution of the augmented system zLS ∈ R is given by~~

~~T zLS = (rLS , xLS ) . (11)~~

Proof. (i) If x ∈ Rn is a solution for (8) and r = b − Ax then " #" # " # IA b − Ax b = , AT 0 x 0 i.e. (r, x)T is a solution for (9). Conversely, if (r, x)T ∈ Rm+n is a solution for (9) then ( ( r + Ax = b AT r + AT Ax = AT b ⇒ , AT r = 0 AT r = 0 i.e. x is a solution of (8). (ii) As in (i) we obtain that zLS from (11) is among the solutions of the augmented system (9). Then, let z = (r, x)T ∈ Rm+n be an arbitrary solution of (9). As before it results that

x ∈ LS S (A; b) ⇔ Ax = PR(A)(b), thus r = b − Ax = PN(AT)(b), (12) where we have also used the orthogonal decomposition b = PR(A)(b) + PN(AT)(b). Then, we successively get, by also using (10) and (12)

~~2 T 2 2 2 k z k =k (r, x) k =k PN(AT)(b) k + k x k ≥~~

~~2 2 T 2 2 k PN(AT)(b) k + k xLS k =k (rLS , xLS ) k =k zLS k and the proof is complete.~~

~~Proposition 1.3. If M is the (m + n) × (m + n) symmetric matrix of the augmented system (9), i.e. " # IA M = , (13) AT 0 then 2 gk2(M) = O((gk2(A)) ). (14) 148 Aurelian Nicola, Constantin Popa~~

Proof. The first part of the proof is adapted from [1]. Let λ ∈ σ(M) be an eigenvalue and 0 , z = (s, x)T ∈ Rm+n a corresponding eigenvector, i.e. Mz = λz. Thus, by using (13) we obtain s + Ax = λs, AT s = λx. (15) If we multiply the first equation in (15) from the left with AT and we then replace AT s by λx from the second one, we get AT Ax = (λ2 − λ)x, (16) which tells us that, if x , 0 then λ2 − λ ∈ σ(AT A). (17) If x = 0, then s , 0 and from the first equality in (15) we obtain s = λs ⇔ λ = 1. (18) But, according to (2) we have T 2 2 σ(A A) = {σ1, . . . , σr , 0}. (19) Then, using (17) - (19) we obtain r 1 1 σ(M) = { ± + σ2, i = 1,..., r} ∪ {1, 0} (20) 2 4 i where σ1 ≥ · · · ≥ σr > 0 are the singular values of A and the eigenvalues 1 and 0 have the multiplicity m − r and n − r, respectively. In the second part of the proof we evaluate gk2(M) according to Remark 1.1, equation (6). From (20) and because MT M = M2 we get r 1 1 + + σ2 > 1, ∀i = 1,..., r (21) 2 4 i and r 1 1 σ2 2 i 2 − + σi = q = O(σi ). (22) 2 4 1 1 2 2 + 4 + σi It will then result from (2) that  r 2  r 2 1 1  1 1  λ∗ (MT M) =  − + σ2 , λ (MT M) =  + + σ2 . (23) min 2 4 r  max 2 4 1 Then, according to (6) we obtain q s 1 1 2 T + + σ λmax(M M) 2 4 1 gk2(M) = ∗ T = q = λmin(M M) 1 2 1 4 + σr − 2 Kaczmarz Extended versus augmented system solution in image reconstruction 149 q q 1 1 2 1 1 2 ( 2 + 4 + σ1)( 2 + 4 + σr ) . (24) 2 σr But, for any x > 0 we have r 1 1 1 1 + x > + + x2 > + x (25) 2 4 2 which combined with (24) gives us

2 1 1 (1 + σ ) (1 + σ )(1 + σ ) ( + σ1)( + σr) 1 1 1 > 1 r > gk (M) > 2 2 > . (26) 2 2 2 2 2 σr σr σr 4 σr Because of (3) and the fact that we may suppose without restricting the generality of the problem that σ1 = O(1), from (26) we get (14) and the proof is complete. 2. KACZMARZ AND KACZMARZ EXTENDED ALGORITHMS First proposed by its author in [5], Kaczmarz’s projection method has been developed by many others in various directions (see [1, 2, 6, 7, 9] and references therein). n n If we consider the applications fi(b; ·), F(b; ·): R −→ R , deﬁned by hx, A i − b f (ω; b; x) = x − ω i i A , F(ω; b; x) = ( f ◦ · · · ◦ f )(ω; b; x), (27) i 2 i 1 m k Ai k where, for the moment ω , 0 and b ∈ Rm are considered as ﬁxed parameters in fi(ω; b; ·), then Kaczmarz’s iteration with relaxation parameter (ω − K for short) for the problem (1) can be written as follows. n Algorithm ω − K. Initialization: x0 ∈ R Iterative step: xk+1 = F(ω; b; xk), k ≥ 0. (28) Proposition 2.1. (see [2, 6]) If the problem (1) is consistent, for any ω ∈ (0, 2) and 0 n k x ∈ R the sequence (x )k≥0 generated by (28) converges and k 0 lim x = PN(A)(x ) + xLS ∈ S (A; b). (29) k→∞ Remark 2.1. Let " # r M = bˆ, bˆ = [b, 0]T (30) x be the (always consistent) augmented system (9) (with M from (13)). The above Proposition 2.1 tells us that the ω-Kaczmarz algorithm (28) applied to the augmented system (30), with the initial approximation (r0, x0)T = (0, 0)T converges to its minimal norm solution, zLS from (11). 150 Aurelian Nicola, Constantin Popa

Unfortunately, in the inconsistent case for (1) (which appears in practical applica- k tions) the above Kaczmarz’s sequence (x )k≥0 from (28) still converges, but its limit is at a certain distance from the set LS S (A; b) (see e.g. [8]). For overcoming this diﬃculty, one of the authors proposed in [7] the following extension of the algorithm ω − K. Algorithm Kaczmarz Extended with Relaxation Parameters (KERP). n 0 Initialization: α, ω ∈ (0, 2); x0 ∈ R , y = b; Iterative step: yk+1 = Φ(α, yk), (31) bk+1 = b − yk+1, (32) xk+1 = F(ω; bk+1; xk), (33) with ( j = 1,..., n) hy, A ji Φ(α, y) = (ϕ ◦ · · · ◦ ϕ )(α; y), ϕ (α; y) = y − α A j. (34) 1 n j k A j k2 Proposition 2.2. For any problem of the form (1), any α, ω ∈ (0, 2) and x0 ∈ Rn the k sequence (x )k≥0 generated by the algorithm KERP (31)-(33) converges and k 0 lim x = PN(A)(x ) + xLS ∈ LS S (A; b). (35) k→∞ k As proved in [6], the sequence (x )k≥0 obtained by applying ω − K algorithm to k T k k (28) can be constructed as x = A y , where the sequence (y )k≥0 is constructed by applying the SOR iterative method (see [10] ) to the system AAT y = b. (36) Then, we expect for ω − K algorithm a behaviour as for the SOR iteration applied to (36), i.e. an asymptotic convergence rate depending on T T 2 gk2(AA ) = gk2(A A) = (gk2(A)) . (37) Thus, if ω−K will be applied to the augmented system (9), we then expect an asymptotic convergence rate depending on (see (37) and (14)) T T 2 4 gk2(MM ) = gk2(M M) = (gk2(M)) = O((gk2(A)) ). (38) On the other hand, because the step (31) of the algorithm KERP is an α − K like iteration applied to the system AT y = 0 and the step (33) is an ω − K iteration applied to the system Ax = bk+1, according to (37) we expect for the whole KERP method (31)- (33) a convergence rate depending on the condition numbers in (37), i.e. ﬁnally of 2 order O((gk2(A)) ). Then, relations (37) and (38), together with the above comments tell us that we expect that the KERP algorithm applied to the initial problem (1) will be faster and will produce better reconstruction than the ω − K algorithm applied to the augmented system (9). Kaczmarz Extended versus augmented system solution in image reconstruction 151 3. NUMERICAL EXPERIMENTS We have used in our experiments the mitochondrian phantom from the paper [3] (see also [4] for a complete description of a phantom). It contains the exact picture xex ∈ R3969, i.e. with a 63 × 63 pixels resolution from Figure 1 left, a scanning matrix A : 1378 × 3969 and a measurements right hand side b ∈ R1378. The associated reconstruction problem is an inconsistent formulation as (1). We applied 100 iterations with the algorithms from Section 2 (with α = ω = 1), as it is described there.

~~Fig. 1. Exact image, ω-Kaczmarz for (30) , KERP for (1)~~

~~Fig. 2. Top-left: distance, top-right: relative error, bottom-left: standard deviation, bottom-right: normal equation residual~~

The reconstructions are presented in Figure 1. Figure 2 shows the wellknown four error measures used in image reconstruction (see e.g. [4]): standard deviation, distance, relative error and normal equation residual, deﬁned below. ex ex ex T x = (x1 ,..., x3969) - the mitochondrian phantom 152 Aurelian Nicola, Constantin Popa

k k k T x = (x1,..., x3969) - the current approximation 3969 3969 ex 1 P ex k 1 P k x¯ = 3969 xi ;x ¯ = 3969 xi - mean values of the exact image and i=1 i=1 current approximation, respectively s 3969P √ 1 k k 2 Standard deviation = (xi − x¯ ) 3969 i=1 v u 3969 tu P ex k 2 (xi −xi ) i=1 Distance = 3969 P ex ex 2 (xi −x¯ ) i=1

~~3969 P ex k |xi −xi | i=1 Relative error = 3969 P ex xi i=1 1 T k Normal equation residual = 63 k A (Ax − b) k The results presented in Figure 2 show the better behaviour of KERP algorithm.~~

~~References~~

[1] Bjork A., Numerical methods for least squares problems, SIAM Philadelphia, 1996. [2] Censor Y., Stavros A. Z., Parallel optimization: theory, algorithms and applications, ”Numer. Math. and Sci. Comp.” Series, Oxford Univ. Press, New York, 1997. [3] Censor, Y., Elfving T., Herman G. T., Nikazad T., On diagonally-relaxed orthogonal projection methods, SIAM J. Sci. Comput., 30(2007/08), 473–504. [4] Herman, G. T., Image reconstruction from projections. The fundamentals of computerized tomography, Academic Press, New York, 1980. [5] Kaczmarz S., Angenaherte Auﬂosung von Systemen linearer Gleichungen, Bull. Acad. Polonaise Sci. et Lettres A (1937), 355–357. [6] Natterer F., The Mathematics of Computerized Tomography, John Wiley and Sons, New York, 1986. [7] Popa C., Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deﬃcient least-squares problems, BIT, 38(1)(1998), 151–176. [8] Popa C., Zdunek R., Kaczmarz extended algorithm for tomographic image reconstruction from limited-data, Math. and Computers in Simulation, 65(2004), 579–598. [9] Tanabe K., Projection Method for Solving a Singular System of Linear Equations and its Appli- cations, Numer. Math., 17(1971), 203–214. [10] Young D. M., Iterative solution of large linear systems, Academic Press, New York, 1971. ROMAI J., 6, 1(2010), 153–154

A CRITERION FOR PARAMETRICAL COMPLETENESS IN THE 5-VALUED NON-LINEAR ALGEBRAIC MODEL OF INTUITIONISTIC LOGIC Mefodie Rat¸˘a,Ion Cucu Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova, Chi¸sin˘au,Republic of Moldova [email protected] Abstract A.V. Kuznetsov [2, p.28] put the problem to ﬁnd out conditions for parametrical completeness of any system of formulas in the Intuitionistic Propositional Logic. In the present paper we solve a more weak problem. We ﬁnd out conditions permitting to determine the parametrical completeness in the logic of 5-valued non-linear pseudo- boolean algebra. We give the suitable solution in terms of 11 parametrical pre-complete classes of formulas.

~~Keywords: Intuitionistic Logic, parametrical expressibility, parametrical completeness, pre-complete system, pseudo-boolean algebra. 2000 MSC: 03B45.~~

1. INTRODUCTION A.V. Kuznetsov [2] introduced the concept of parametrical expressibility and considered the problem of parametrical completeness in the Intuitionistic Logic ([2, p.28], problem 16). For solving this problem it is natural to ﬁnd out conditions for parametrical completeness in some intermediate more simple logics that approximate the Intuitionistic logic. In the present paper we establish the necessary and suﬃcient conditions for the parametrical completeness of an arbitrary system of formulas in the logic of 5-valued pseudo-boolean algebra with two incomparable elements. Formulas (of propositional logic) are constructed from variables p, q, r by means of logical operations: & (conjunction), ∨ (disjunction), ⊃ (implication), ¬ (negation). Using the mark , and reading it as “means” we introduce some designations for the formulas: 1 (p ⊃ p), 0 (p & ¬p), (F v G) (F ⊃ G)&(G ⊃ F) (equivalence). The Intuitionistic and classical propositional calculi are based on the mentioned notion of formula. We determine the logic of some calculus as the set of all formulas deducible in considered calculus.

~~153 154 Mefodie Rat¸˘a,Ion Cucu~~

It is known that the pseudo-boolean algebras [3] represent the algebraic interpre- tations of intuitionistic logic. Let consider the 5-valued non-linear pseudo-boolean algebra Z5 = < {0, ρ, σ, ω, 1};& , ∨ , ⊃, ¬ > where 0 < ρ < ω < 1, 0 < σ < ω, ρ & σ = 0, ρ ∨ σ = ω. It is clear that the algebra Z3 = < {0, ω, 1};& , ∨, ⊃, ¬ > is a subalgebra of Z5. We deﬁne the logic LZ5 of the algebra Z5 as the set of all formulas true on Z5, i.e. the set of formulas identically equal to the greatest element 1 of this algebra. A F formula is called directly expressible via the Σ system of formulas if it is possible to obtain F from variables and formulas of Σ using a ﬁnite number of weak substitution rule. A formula F is said to be parametrically expressible (p. expressible) in a L logic in terms of a Σ system of formulas, if there exist numbers l and s, variables π, π1, . . . , πl not occurring in F, pairs of formulas Ai, Bi (i = 1, s) directly expressible in L in terms of Σ and formulas D1,..., Dl, not containing the variables π, π1, . . . , πl, such that the following relations take place: L ` (F ∼ π) ∼ (A1 ∼ B1)& ... &(As ∼ Bs)[π1 | D1],..., [πl | Dl] L ` (A1 ∼ B1)& ... &(As ∼ Bs) ⊃ (F ∼ π) . A Σ system of formulas is said to be parametrically complete (p. complete) in a L logic if all formulas of L language are p. expressibly in L in terms of Σ. One says that a formula F(p1,..., pn) preserves the predicate R(x1,..., xm) on algebra A if, for any elements αi j ∈ A (i = 1, m; j = 1, n), the truth of propositions

~~R[α11, α21, . . . , αm1],..., R[α1n, α2n, . . . , αmn] implies R F[α11, α12, . . . , α1n],..., F[αm1, αm2, . . . , αmn] .~~

~~Let consider the following predicate on Z5 algebra: g(x) = y, where g(0) = 0; g(ρ) = σ; g(σ) = ρ; g(ω) = g(1) = 1.~~

Theorem. In order that a system Σ of formulas be p. complete in the logic LZ5 it is necessary and suﬃcient that Σ be p. complete in the logic LZ3 and there exists a formula F of Σ which does not preserve the predicate g(x) = y on the algebra Z5. References

[1] Cucu I.V., Rat¸a˘ M.F. Parametrical completeness in the logic of simplest pseudo-boolean algebra with two incomparable elements, Bul. Acad. of Sci. of Moldova, 1(7)(1992), 46–51 (in Russian). [2] Kuznetsov A.V. On tools for discovery of nondeducibility or nonexpressibility, Logical Deduc- tion, Moscow, Nauka, 1979, 5-33 (in Russian). [3] Rasiova H., Sikorski R., The mathematics of Metamathematics, Warsawa, (1963). ROMAI J., 6, 1(2010), 155–166

VECTOR OPTIMIZATION PROBLEMS Cristina Stamate Institute of Mathematics, Ia¸si [email protected] Abstract We consider vector optimization problems for multifunctions, defined with infimal and supremal efficient points in locally convex spaces ordered by convex, pointed closed cones with nonempty interior. We introduce and study the solutions for these problems using the algebraic and topological results for the efficient points. We also present the links between our problems and two special problems, the scalar and the approximate problems.

~~Keywords: order vector spaces, convex cones, eﬃcient points, saddle points, Lagrangian 2000 MSC: 90C29, 49J35.~~

1. INTRODUCTION A general vector optimization problem with multifunctions can be presented in the following form: [ (VP) E f f F(x) x∈G where F : X ⇒ Z is a multifunction between two vector spaces X, Z, G ⊂ X and E f f denotes one’s of the efficient points set, the minimum and maximum Pareto points set, usually. These problems was intensively studied in the last 20 years taking into account the interest generated in mathematical economics. Thus lagrangean and conjugate duality results were obtained as well as weak and strong duality theorems and scalar- ization results. Nevertheless, the authors continue the study of the Problem (VP) by replacing the Pareto efficient points with other efficient points, since the minimum and maximum Pareto sets are nonempty under certain strong restriction concerning the set. Thus, in [19], [12], we can find the ε-Lagrangean multiplicators, ε-weak saddle points, ε-duality problem studied in connection with the approximate efficient points. Given that, we can find in the literature several kinds of infimal and supremal points (see [9], [10], [16]) which are natural generalization for the real infimum and supremum. We consider in the following the optimization problem obtained with these kind of efficient points and will be called the INF (or SUP) problem. The essential property of these efficient points set is that it is nonempty for a large class of sets and thus the problem has often nonempty values but it was not easy to find a suitable notion of a solution for this problem since an infimum point is not really a

155 156 Cristina Stamate point of the set. The ﬁnal solution was the consideration of a sequence as a solution for such vector problem as we will see in the section 3. The paper is structured in 4 parts. The second section presents the principal properties of the inﬁmal and supremal points which will be used in what follows. The third part presents the notion of the solution for our problem and the connections between this solution and some notion introduced in [2]. The last section gives a study of some optimization problems closely related with our problem, i.e. the approximate problem (Pa) and the scalar problem (P∗).

2. BACKGROUND TO THE EFFICIENT POINTS It is a well known fact that the Pareto and the weak Pareto efficiencies were intensively used for the study of the optimal problems in the literature of mathematical economics. The special conditions imposed for the existence of such points led to the study of other kinds of efficient points, the so called supremal and infimal points. These points are natural generalizations for ordinary concepts of infimum and supremum known in real analysis. Firstly, these notions were given in vector lattices (see [22]) or in a multi-dimensional Euclidian space using the lower and upper bounds (see [3]) or efficiencies for different sets (see [5], [10], [8]). Nieuwenhuis, firstly introduced [9] the supremal and infimal points in Banach spaces ordered by a closed convex cone with nonempty interior and Tanino [17] presented such points in a linear topological space ordered by a nonempty interior cone. In the general form, these notions were given by Postolica [11] and the above discussed points can be found in the weak efficiencies introduced here. Let us recall in the sequel the principal definitions of the infimal and supremal points. We consider the space Z ordered by a closed, convex, pointed cone Z+. We will write a ≤ b if b − a ∈ Z+ and a < b if b − a ∈ Z+ \{0}. As usually, we can consider a smallest element denoted −∞ and a biggest element denoted +∞ and Z¯ = Z ∪ {+∞} ∪ {−∞}. The complementary set of A will be denoted Ac = Z \ A. p p We’ll denote R+ the positive cone of the euclidian space R . The closure (resp. the interior and the boundary) of the set A ⊂ Z will be denoted cl A (resp. Int A, Fr A) and if IntZ+ , ∅ then K = IntZ+ ∪ {0}. In this case, the efficiencies considered with respect to the cone K will be called weak efficiencies and will be denoted wEFF A. As usually, a fundamental system of neighborhoods for a point x will be denoted by V(x). If the interior of the cone Z+ is nonempty, we can consider a fundamental system of neighborhoods for −∞ denoted V(−∞) (resp. for +∞ denoted V(+∞)) given by the sets V = [−∞, a) = {x ∈ Z | x < a} ∪ {−∞} (resp. V = (a, +∞] = {x ∈ Z | x > a} ∪ {+∞}).

~~Deﬁnition 4. 1)(Pareto minimum) MINA = {a ∈ A | a ≯ b, ∀b ∈ A} (MAXA = {a ∈ A | a ≮ b, ∀b ∈ A}); Vector optimization problems 157~~

2)[22][2] INFA = {y ∈ Z | y ≤ a, ∀a ∈ A, and if y0 ≤ a, ∀a ∈ A ⇒ y0 ≤ y} (SUPA = {y ∈ Z | y ≥ a, ∀a ∈ A, and if y0 ≥ a, ∀a ∈ A ⇒ y0 ≥ y}); 3)[11] INFA = MIN cl A (SUPA = MAX cl A); p 4)[5][16] INFA = MIN cl(A + R+) p (SUPA = MAX cl(A + R+)); 5)[8][17] INFA = wMIN cl(A + Int Z+) (SUPA = wMAX cl(A + Int Z+)); 0 6)[9] INFA = {y ∈ Z | y − a < IntZ+, ∀a ∈ A, and if y − y ∈ IntZ+ ⇒ a ∈ 0 A, y − a ∈ IntZ+} 0 0 (SUPA = {y ∈ Z | y − a < −IntZ+, ∀a ∈ A, and if y − y ∈ −IntZ+ ⇒ a ∈ A, y − a ∈ −IntZ+}); 0 7)[11] INFA = {y ∈ Z¯ | y − a < Z+ \{0}, ∀a ∈ A, and if y − y ∈ Z+ \{0} ⇒ ∃a ∈ 0 A, y − a ∈ Z+ \{0}} 0 (SUPA = {y ∈ Z¯ | y − a < −Z+ \{0}, ∀a ∈ A, and if y − y ∈ −Z+ \{0} ⇒ ∃a ∈ 0 A, y − a ∈ −Z+ \{0}}). Remark 3. a) Definition 4 5) may be given in a general form INFA = MIN cl(A + Z+). b) The author denotes the infimal points set given in Definition 4 7) by INF1A. This is called the proximal infimum points set. An other infimal points set may be found in [10], the infimum set given by INFAg = {y ∈ Z¯ | y ≯ a ∀a ∈ A} = c (A+Z+ \{0}) ∪{−∞}. Using this notations, let us remark that INF1A = MAX INFAg and the set given in Definition 4 6) is wINF1A. c) In [9], the author proves that the infimal set given in Definition 4 6) has the property that INFA = Fr (A + Z+) (see Theorem I-17). d) Obviously, the notions presented in Definition 4 1)-6) may be given in the same form for sets from Z¯. In what follows we are interested about the relationships between these types of infimal points, more exactly between (1) INFA = Fr (A + Z+) (2) INFA = MAX INFAg (w2) INFA = wMAX wINFAg (3) INFA = MIN cl(A + Z+)(w3) INFA = wMIN cl(A + Z+). Proposition 1. Let A ⊂ Z¯. Under the previous notations, we have: (2) ⇒ (1), (3) ⇒ (1) and if IntZ+ , ∅ then (1) ⇐⇒ (w2) ⇐⇒ (w3) Remark 4. In general, the implications (1) ⇒ (2), (1) ⇒ (3), (2) ⇒ (3) and (3) ⇒ 2 2 2 (2) are not true. Indeed, let consider Z = R , Z+ = R+ and A = {(x, y) | (x − 1) + (y − 2 1) < 1}. The point a = (0, 1) ∈ Fr (A + Z+), a ∈ INFAg but a < b = (0, 2) ∈ INFAg , so (1) ; (2). Also, b ∈ Fr (A + Z+) but b < MIN cl(A + Z+) since a ∈ cl(A + Z+) and a < b. Thus (1) ; (3). In the same time, we can see that a ∈ MIN cl(A + Z+) but a < MAX INFAg since b > a and @a0 ∈ A such that a0 < b. By this way, (3) ; (2). 158 Cristina Stamate

~~Now, if we take B = A∪{b}, we’ll have that b ∈ MAX INFBg but b < MIN cl(B+Z+) and thus (2) ; (3).~~

Proof. (of the proposition) For proving (2) ⇒ (1), let x ∈ MAX INFAg and suppose c that x < Fr (A + Z+). Since Fr (A + Z+) = Fr (A + Z+ \{0}) = Fr (A + Z+ \{0}) = Fr INFAg , we have x < Fr INFAg . Since x ∈ INFAg , we must have x ∈ Int(INFAg ) and consequently there exists V ∈ V(x), V ⊂ INFAg . Since x ∈ cl (x + Z+ \{0}), we ﬁnd y ∈ V ∩ (x + Z+ \{0}). Thus, y ∈ INFAg and y > x, which contradict the fact that x ∈ MAX INFAg . Now, let x ∈ MIN cl(A + Z+) and suppose that x ∈ Int(A + Z+). This implies that ∃V ∈ V(x), V ⊂ A + Z+. Consequently, we ﬁnd ε ∈ Z+ \{0} such that x − ε ∈ V ⊂ A + Z+ ⊂ cl(A + Z+) which contradict the choices of x ∈ MIN cl(A + Z+). Thus, (3) ⇒ (1). If the interior of the cone Z+ is nonempty, then we consider ε ∈ IntZ+ and this will yields to (w3) ⇒ (1). Following Remark 19c) we get (1) ⇐⇒ (w2). Similarly to implication (2) ⇒ (1), we get (w2) ⇒ (1) and thus, (w2) ⇐⇒ (1). We’ll prove in what follows that (1) ⇒ (w3). Let x ∈ Fr(A + Z+) and suppose that x < wMINcl(A + Z+). This means that there exists y ∈ cl(A + Z+), y − x ∈ −IntZ+. Thus, x−IntZ+ ∈ V(y) and so x−IntZ+ ∩(A+Z+) , ∅ which gives that x ∈ A+IntZ+ ⊂ Int(A + Z+). But initially, we chose x ∈ Fr(A + Z+) and this contradiction shows that our implication is valid. Finally, the proposition equivalents are shown.

It is a known fact that the existence of the Pareto minimal points involves some special conditions concerning the set or the cone. A necessary and suﬃcient condition for this fact appears in [7], Theorem 3.4:

Theorem 2.1. Assume that Z+ is a convex cone and A ⊂ Z, A , ∅. MINA , ∅ if and only if A has a nonempty strongly Z+-complete section. Let us recall that a set U ⊂ Z is K-complete (resp., strongly K-complete) if it has c c no cover of the form {(xα − clK) , α ∈ I} (resp., {(xα − K) , α ∈ I}) with {xα, α ∈ I} being a decreasing net in A. A section of A is a set Ax = A ∩ (x − K) (see [7]). Following this, we deduce necessary and suﬃcient conditions for the existence of the inﬁmal (supremal ) points given in Proposition 1.

Theorem 2.2. a) A nonempty set A ⊂ Z has a nonempty infimal points set (2) if and c only if −(A + Z+) has a nonempty strongly Z+-complete section. b) A nonempty set A ⊂ Z has a nonempty infimal points set (3) if and only if cl(A+Z+) has a nonempty strongly Z+-complete section. For the weak infimal or supremal points we can prove a more explicit result. Fol- lowing Proposition 1, the infimal points sets (1), (w2) and (w3) are coincident with the general notion of weak infimum points set presented in Definition 4 7). For this reason, in what follows we’ll use the notations presented in Remark 19b). Vector optimization problems 159

Theorem 2.3. Let A ⊂ Z¯ be a nonempty set of a locally convex space Z ordered by a convex, pointed, closed cone with nonempty interior. Then, ∅ , wINF1A ⊂ Z if and only if wINFAg , {−∞}. In this case, the following ”domination” properties (wDP) does holds:

~~A ⊆ (wINF1A + K) ∪ {+∞} wINF A = (wINF1A − K) ∪ {−∞}~~

Proof. Obviously, if ∅ , wINF1A ,⊂ Z, then wINFAg , {−∞} since wINF1A ⊂ wINFAg . Now, let a ∈ A, a , ∞ andy ˜ ∈ wINFAg \ {−∞}. Since K is a generating cone, we find z , a, , y˜ such that z − a ∈ −K and z − y˜ ∈ −K which implies z ∈ wINFAg . Let denote z(t) = z + t(a − z), t > 0 and we remark that for t > 1, z(t) ∈ A + K \{0}. If we consider inf{t > 0 | z(t) ∈ A + K \{0}} = t¯, we have t¯ ≤ 1 andz ¯ = z + t¯(a − z) < A + K \{0} which impliesz ¯ ∈ wINFAg . We’ll prove in what follows thatz ¯ ∈ wINF1A. Let consider µ ∈ IntZ+; we can find ε > 0 such 0 0 0 that µ − ε (a − z) ∈ IntZ+, for all 0 < ε < ε and for this ε there exists ε such that z+(t¯+ε0)(a−z) ∈ A+K\{0}. Thus,z ¯+µ = z+t¯(a−z)+µ = z+(t¯+ε0)(a−z)−ε0(a−z)+µ ∈ A + K \{0}. Consequently,z ¯ ∈ wINF1A , ∅ andz ¯ − a = (t¯ − 1)(a − z) ∈ K which gives that A ⊂ (wINF1A + K) ∪ {+∞}. Similarly, if we consider z ∈ wINFAg \{−∞} and a ∈ A, we can find a0 ∈ A+K\{0}, 0 0 0 a − z ∈ IntZ+ and the elementz ¯ = z + t¯(a − z) with t¯ = inf{t > 0 | z + t(a − z) ∈ A + K \{0}} will be an element from wINF1A which have the property thatz ¯ − z ∈ K and thus the equality wINF A = (wINF1A − K) ∪ {−∞} does holds.

~~Remark 5. If A ⊂ Z, the equivalence ”wINF1A , ∅ if and only if wINFAg , ∅” is Theorem I-18 [9].~~

3. SOLUTIONS FOR (VP0) In what follows we’ll consider F : X ⇒ Z¯, C ⊂ X and the vector optimization problems [ (VP0) INF1 F(x) x∈C [ (wVP0) wINF1 F(x) x∈C S S As usually, an element from INF1 F(x)(wINF1 F(x)) will be called a value x∈C x∈C of (VP0) (respectively (wVP0)) and following Theorem 2.3, the set of values for (wVP0) is identically {−∞} or is a nonempty subset of Z which satisfies (wDP) properties. Generally, for a vector optimization problem (VPS), a solution for the problem is a point x0 ∈ C with the property F(x0) ∩ E f f F(x) , ∅. For the problem x∈C S (VP0), this definition is not suitable since if F(x0) ∩ INF1 F(x) , ∅ then F(x0) ∩ x∈C 160 Cristina Stamate S MIN F(x) , ∅. Thus, using this definition, we will study in fact the existence of x∈C S solutions for the problem MIN F(x) which may not exist even the values set for x∈C the problem (VP ) is nonempty. 0 S In [19] we can find an approximative vector problem i.e. (VPε) ε MIN F(x). x∈C As usually, a solution for this problem will be a point x0 ∈ C with the property ε S F(x0) ∩ MIN F(x) , ∅. x∈C Let recall that for ε > 0, ε MINA = {a ∈ A | a ≮ b − ε, ∀b ∈ A}. We remark that ε if INF1A , ∅ then for all ε > 0, MINA , ∅. Taking into account this remark, we may think to define the solution of the problem (VP0) using the solutions for the approximative problems, (VPε). Thus we can ε S consider that x0 ∈ C is a solution for (VP0) if F(x0) ∩ MIN F(x) , ∅ for x∈C all ε > 0. A problem occur now if F is a single valued map since in this case, ε S the previous condition F(x0) ∩ MIN F(x) , ∅ for all ε > 0 conclude to the S x∈C fact that F(x0) ∩ MIn F(x) , ∅ and we reduce our study again to the problem S x∈C MIN F(x). x∈C By this reasons, we will consider in Definition 3.1 the solution of our problem as a net of approximative efficient points. Firstly, we’ll present a general notion of approximative efficient points. Definition 5. Let V ⊂ V(0) and A ⊂ Z. The V-minimum points set of A will be denoted V MINA and it will be given by

~~V MINA = {y ∈ A | (y + V) ∩ INFA , ∅}~~

~~Remark 6. If Z+ is a generating cone then [ [ ε MINA = V MINA ε>0 V∈V(0) .~~

ε V Indeed, if y ∈ MINA then y ∈ ε MINA where Vε = V ∪ {ε} with V ∈ V(0). Now, if y ∈V MINA where V ∈ V(0) and v ∈ V such that y + v ∈ INFA we can ﬁnd ε > 0 with v > −ε (Z+ is a generating cone ). Thus, y − ε ∈ INFA which implies y ∈ε MINA and the equality follows.

Deﬁnition 6. a) A generalized sequence (aα)α∈I ⊂ Z ((I, ) is a directed set) is con- 0 0 vergent to y ∈ Z¯ if for each V ∈ V(0), ∃α ∈ I such that aα ∈ y + V, ∀α α . In this case we denote aα →α Y

~~b) Let (Aα)α∈I ⊂ Z. lim inf Aα = {y ∈ Z | ∃aα ∈ Aα, aα →α y}. α Vector optimization problems 161~~

In what follows we’ll consider V(0) a fundamental system of convex, symmetric and barrelled neighborhoods. This is a directed set if we consider the preorder given by V V0 if and only if V ⊂ V0. Proposition 2. Let A ⊂ Z be a nonempty set. Then,

lim inf wV MINA = wMIN A¯ V . ¯ ¯ ¯ Proof. Let y ∈ wMIN A = A∩wINF1 A. For all V ∈ V(0) we’ll find µV ∈ (α+V)∩A. ¯ V Thus α ∈ (µV − V) ∩ wIn f1A = (µV + V) ∩ wINF1A, which say that µV ∈ w MINA V ¯ and µV ⇒V y. We obtain that y ∈ Limin f w MINA and finally wMIN A ⊆ Limin f wV MINA. V V Now, let consider y ∈ Limin f w MINA. We find aV ∈ w MINA, aV ⇒V y and since aV ∈ A we get y ∈ A¯. Let suppose that y < wINF A; in this case we find 0 a ∈ (y − IntZ+) ∩ A. Since x = 1/2(a + y) + IntZ+ ∈ V(y) we find V ∈ V(0) such that 0 for all V ⊂ V , aV ∈ x+IntZ+. Also, there exists U ∈ V(0) such that x+U ⊂ a+IntZ+. 0 For V = V ∩ U we have x + V ⊂ a + IntZ+ and aV + V ⊂ x + V + IntZ+ ⊂ a + IntZ+. V But aV +V∩wINF A , ∅ and this contradiction proves the inclusion lim infV w MINA ⊆ wMIN A¯ and finally we get the equality.

Proposition 3.1. Let A ⊂ Z, x ∈ A+IntZ+ and y ∈ wINF A. Then, [x, y]∩wINF1A , ∅. Proof. Since Z = (A + K \{0}) ∪ (A + K \{0})c, following Proposition 1 we get Z = (A + K \{0}) ∪ wINF A = (A + K \{0}) ∪ IntwINF A ∪ Fr wINF A = (A + K \{0}) ∪ IntwINF A ∪ wINF1A. Thus [x, y] = ([x, y] ∩ (A + K \{0})) ∪ ([x, y] ∩ IntwINF A) ∪ ([x, y] ∩ wINF1A). If we suppose [x, y] ∩ wINF1A = ∅ then [x, y] = ([x, y] ∩ (A + IntZ+)) ∪ ([x, y] ∩ IntwINF A). Following the hypothesis, y < A + IntZ+ and thus the both sets [x, y] ∩ (A + IntZ+) and [x, y] ∩ IntwINF A will be nonempty, open and disjoint sets. But [x, y] is a point wise set and then, this contradiction prove the proposition.

~~Deﬁnition 3.1. We’ll call solution for the problem (VP0) a net (xV )V∈V(0) from X such V that for all V ∈ V(0), xV ∈ MIN ∪x∈C F(x). Shortly, we’ll write (xV ).~~

~~Remark 3.1. If the interior of the ordering cone Z+ is nonempty, then (xV )VS∈V(0) ⊂ X is a solution for (wVP0) if and only if for all V ∈ V(0), (F(xV )+V)∩wINF1 F(x) , x∈C ∅.~~

Indeed, following Deﬁnition 5, if for all V ∈ V(0), (F(xV )+V)∩wINF1∪x∈C F(x) , ∅, then (x ) ⊂ X is a solution for (wVP ). Now, let V ∈ V(0), α ∈ F(x ) and v ∈ V V∈V(0) S 0 S V S V such that α+v ∈ wINF F(x). If α ∈ wINF F(x), then α ∈ wINF1 F(x) x∈C x∈C x∈C 162 Cristina Stamate

w S S and (F(xV ) + V) ∩ INF1 F(x) , ∅. If α < wINF F(x), following Proposition x∈C S x∈C 3.1 we get [α, α + V] ∩ wINF1 F(x) , ∅. Since V is a barrelled set we have x∈C S [α, α + v] ⊂ F(xV ) + Vand thus (F(xV ) + V) ∩ wINF1 F(x) , ∅. x∈C In [2] we find the notion of ”asymptotically weakly Pareto optimizing” (a.w.p.) sequence used for the characterization of the vector convex functions having the property that approximate necessary first order weakly-efficiency condition implies approximate weakly efficiency, a generalization of the asymptotically well-behaved functions from the scalar case. Thus, if X, Z are Banach spaces and F : X ⇒ Z¯ is a set-valued map, a sequence (xn) in dom FSwill be called asymptoticallyS weakly Pareto optimizing (a.w.p) if dist(F(xn), wMIN F(x)) → 0 when −∞ < F(x) and F(xn) → −∞, else. x∈X x∈X It is not difficult to see that a generalization for this notion in a locally convex space is the following:

Deﬁnition 3.2. (xV ) is a asymptotically weakly Pareto optimizing sequence if for all 0 00 0 V ∈ VS(0), there exists V ∈ V(0) suchS that for all V V we have F(xV00 ) ∩ wMIN F(x) + V , ∅ when −∞ < F(x) and F(xV ) → −∞, else. x∈X x∈X Now we’ll present the links between the solution of a vector optimization problem and an a.w.p. sequence. Proposition 3.2. Let X, Z be locally convex spaces and let F : X ⇒ Z¯ be a set-valued map. If (xV ) is a solution for the (VP0) problem, then there exists a subsequence (xV0 ) of (xV ) which is an a.w.p. sequence. Conversely, if (xV ) is an a.w.p. sequence and (VP0) has a solution, then there exists a subsequence (xV0 ) of (xV ) which is a solution for the (VP0) problem.

Proof. Let (xV ) be a solution for the (VP0) problem. S V S Obviously, −∞ < F(x). Thus, F(xV ) ∈ MIN F(x). Following Proposition3.1, x∈X x∈X 00 0 for all V ∈ V(0),S there exists a subsequence (xV0 ) of (xV ) such that for all V V , F(xV00 ) ∩ wMIN F(x) + V , ∅. Thus, (xV ) is an a.w.p. sequence. x∈X S Now, let (xV ) be an a.w.p. sequence. If (VP0) has a solution, then −∞ < F(x) x∈X and thus for all V ∈ V(0), there exists V0 ∈ V(0) such that for all V00 V0 we have 00 S 0 F(xV ) ∩ wMIN F(x) + V , ∅. Thus,for all V ∈ V(0), there exists V ∈ V(0) x∈X 00 0 00 S such that for all V V we have F(xV ) ∩ wINF F(x) + V , ∅. Consequently, x∈X V S F(xV0 ) ∈ MIN F(x) and thus, (xV0 ) is a subsequence of (xV ) which is a solution x∈X for (VP0).

To conclude this section, let’s give a characterization for the (VP0)’s solution using the Pareto approximative subdiﬀerentials. Let recall that for a multifunction F : X → Vector optimization problems 163

~~V Z and V ⊂ Z, the Pareto V-subdiﬀerential of F at x0 denoted by ∂≯ F(x0) is~~

~~V ∂≯ F(x0) = {T ∈ L(X, Z) | ∃y0 ∈ F(x0) and v ∈ VT(x − x0) ≯ y − y0 + v}~~

~~V We remark that T ∈ ∂≯ F(x0) if and only if [ (T(x − x0) + F(x0) + V) ∩ INF F(x) , ∅ x∈X It is not diﬃcult to give now the proposed characterization.~~

~~V Proposition 3.3. (xV ) is a solution for (VP0) if and only if 0 ∈ ∂≯ F(xV ) for all V ∈ V(0).~~

4. SCALAR AND APPROXIMATIVE OPTIMIZATION PROBLEMS. In this section we are interested to see the links between our problem and some ”simpliﬁed” problems, the scalar and the linear approximative problems. ∗ ∗ For x ∈ Z+ \{0} we’ll consider the scalar problems S ∗ (Px∗ ) : inf x ◦ F(x) Sx∈X S (P∗): inf x∗ ◦ F(x) ∗ ∗ y ∈Z+\{0} x∈C V S ∗ (Px∗ ) : inf x ◦ F(x) where KV = {x ∈ C | F(xV ) + V ⊆ F(x) + Z+} x∈KV

~~Deﬁnition 4.1. 1) (xV ) will be a solution for (Px∗ ) if [ ∗ ∗ x (F(xV ) + V) ∩ INF x ◦ F(x) , ∅. x∈C~~

~~∗ ∗ 2)(xV , xV ) will be a solution for (P ) if [ ∗ ∗ xV (F(xV ) + V) ∩ INF xV ◦ F(x) , ∅ x∈C~~

~~V 3)xV will be a solution for (Px∗ ) if [ ∗ ∗ x (F(xV ) + V) ∩ INF x ◦ F(x) , ∅ x∈KV~~

∗ ∗ Proposition 4.1. If (xV , xV ) is a solution for (P ) then (xV ) is a solution for (wVP0). ∗ ∗ If (SxV ) is a solution for Px∗ for some x ∈ Z+ \{0} then (xV ) is a solution for (wVP0). If F(x) + Z+ \{0} is a convex set and (xV ) is a solution for (VP0) then there exists x∈C ∗ ∗ ∗ ∗ xV ∈ Z+ \{0} such that (xV , yV ) is a solution for (P ). 164 Cristina Stamate

∗ ∗ Proof. Let (xV , xV ) be a solution for (P ). This means that for all V ∈ V(0), there exists v ∈ V and y ∈ F(x ) such that x∗ (y ) + v) ≤ x∗ ◦ y for all y ∈ F(x), x ∈ C. S V V V V V S Thus yV + v < F(x) + IntZ+ which implies that F(xV ) + V ∩ wINF F(x) i.e. x∈C x∈C (xV ) is a solution for (wVP0). The proof is similar for the second assertion. The last part of the proposition follows using the Hahn-Banach separation theorem. Previous result can be recast as follows: ∗ ∗ S ∗ Proposition 4.2. If there exists x ∈ Z+ \{0} such that inf x ◦ F(x) > −∞ then S x∈C (wVP0) has solution. Conversely, if F(x)+IntZ+ is convex and wVP0 has solution, x∈C ∗ ∗ then there exists x ∈ Z+ \{0} such that (Px∗ ) has solution. S ∗ S Remark 4.1. If we denote by v(Px∗ ) = inf x ◦F(x) and by v(wVP0) = wINF1 F(x) x∈C x∈C ∗ the values of (Px∗ ) respectively of (wVP0), we have that v(Px∗ ) = inf x ◦ v(wVP0). Indeed, following Theorem 2.3 and the deﬁnition of the ”INF1” set we have S S ∗ ∗ wINF1 F(x) + IntZ+ = F(x) + IntZ+ and thus x ◦ v(wVP0) + (0, ∞) = x ◦ x∈C x∈C S ∗ ( F(x)) + (0, ∞). Consequently, v(Px∗ ) = inf x ◦ v(wVP0). x∈C ∗ # V Proposition 3. Let x ∈ Z+. If for each V ∈ V(0), xV is a solution for (Px∗ ) then (xV ) is a solution for (VP0). The equivalence does hold if F is a single valued map. Remark 7. We can consider a more general problem by replacing x∗ with a positive multifunction G : Z → Y (here Y is a locally convex space orderedS by a cone Y+) with G(Z+ \{0}) ⊆ Y+ \{0}. The problems will be (PG): INF1 G ◦ F(x) respectively, x∈C S V (PG): INF1 G ◦ F(x). If for each V ∈ V(0), xV is a solution for (PG) then (xV ) x∈KV is a solution for (PG). The equivalence does hold if G ◦ F = MING ◦ F.

~~In what follows we’ll prove that, even (uV ) is not a solution for our problem, we can ﬁnd a solution (vV ) in a ”neighborhood”. More exactly, we have the following result.~~

~~Proposition 4.3. For all (uV ) which is not a solution for (VP0), there exists a solution (vV ) such that F(vV ) ∩ (F(uV ) + V − Z+ \{0}) , ∅.~~

Proof. Let consider K(u) = {v | F(v) ∩ (F(u) + V − Z+ \{0}) , ∅}. Since (uV ) is not a solution , K(uV ) , ∅ and we denote Y(v) = F(v) ∩ (F(u) + V − Z+ \{0}). For ∗ ∗ x ∈ Z+ \{0}, there exists vV ∈ K(u) such that ∗ ∗ ∗ inf x ◦ Y(v) + sup x (V) > x (yV ) v∈K(u) where yV ∈ Y(vV ). Consequently, we can ﬁndv ˜ ∈ V such that ∗ ∗ ∗ inf x ◦ Y(v) + x (˜v) > x (yV ). v∈K(u) Vector optimization problems 165

~~We’ll prove that (vV ) is a solution. Let suppose that this is not true and thus [ F(vV ) + V ⊆ F(x) + Z+ \{0}. x∈X~~

This inclusion implies that for al v ∈ V there exists xV such that yV +v ∈ F(xV )+Z+{0}. Since yV ∈ Y(vV ) we get F(xV )∩(F(uV )+V −Z+ \{0}) , ∅ which say that xV ∈ K(uV ) and yV + v ∈ Y(xV ) + Z+ \{0}. Let consider now v = −v˜; we get ∗ ∗ ∗ ∗ ∗ inf x ◦ Y(v) > x (yV ) + x (−v˜) ≥ inf x ◦ Y(xV ) ≥ inf x ◦ Y(v) v∈K(u) v∈K(u) which is a contradiction. Thus (vV ) is a solution and F(vV ) ∩ (F(uV ) + V − Z+ \{0}) , ∅. Remark 4.2. It is not difficult to see that our scalar multivalued problem (SMP) ∗ given by inf x ◦F(x) is equivalent with a scalar single valued problem (SP), m(x)x∈X x∈X ∗ where m(x) = inf x ◦ F(x) i.e. the problemS (SMPS) and (SP) has the same solutions and the same values. Indeed, since inf Ai = inf inf Ai for Ai ⊂ R, we deduce that i∈I i∈I the both problems has the same values. Now, if (xε) is a solution for (SP) we have S S ∗ S ∗ m(xε) < inf m(x) + ε = inf inf x ◦ F(x) + ε = inf x ◦ F(x) + ε. We can find x∈X x∈X x∈X 0 ∗ 0 S ∗ ε > 0 and yε ∈ F(xε) such that x (yε) < m(xε) + ε < inf x ◦ F(x) + ε. Thus, (xε) x∈X is a solution for (SMP). Conversely, let (xε) be a solution for (SMP). Thus, there ∗ S S ∗ exists yε ∈ x ◦ F(xε) such that inf m(x) + ε = inf x ◦ F(x) + ε > yε ≥ m(xε). x∈X x∈X We conclude that (xε) is a solution for (SP). We remark also that F is convex if and only if m is convex. Another simplified problem which allows to us some information concerning the (VP0) problem is the (VPa). More exactly, if F is a subdifferentiable multifunction, the problem (VPa) is the following [ (VPa): wINF1 S (x) x∈X where S (x) = {T x | T ∈ ∂≤F(x)}.

~~Proposition 4. If (xV ) is a solution for (VPa) then (xV ) is a solution for (VP0).~~

Proof. Let suppose that (xV ) is a solution for (VPSa) and is not a solution for (VP0). In this case will exists V such that F(xV ) + V ⊆ F(x) + IntZ+. Let consider T ∈ x∈X ∂≤F(xV ). There exists yV ∈ F(xV ) such that T(x − xV ) ≤ y − yV , ∀y ∈ F(x), ∀x ∈ X. 0 Let v ∈ V and following our assumption, we can ﬁnd x ∈ X such that yV + v ∈ 0 0 0 F(x ) + IntZ+. Thus, T(x − xV ) ≤ y − yV − v + v, ∀y ∈ F(x ) which give to us that 0 S T(x − xV ) ≤ v. Consequently, T(xV ) + V ⊆ T(x) + IntZ+ for all T ∈ ∂≤F(xV ) x∈X which contradict the fact that (xV ) is a solution for (VPa). 166 Cristina Stamate References

[1] G.R. Bitran, Duality for nonlinear multiple-criteria optimization -problems, J.O.T.A., 35, 3(1981), 367-401. [2] S.Bolintineanu, Approximate efficiency and scalar stationarity in unbounded nonsmooth convex vector optimization problems, J. Optimization Theory and Applications, 106, 2(2000), 265-296. [3] S. Brumelle, Duality for multiple objective convex programming, Math. Oper. Res., 6(1981), 159-172. [4] H.W.Corley, Existence and Lagrangian duality for maximizations of set-valued functions, J.O.T.A., 54(1987), 489-501. [5] C. Gros, Generalization of Fenchel’s duality theory for convex optimization, European J. Oper. Res., 2(1978), 368-376. [6] G. Isac, V. Postolica, The best approximation and optimization in locally convex spaces, Verlag Peter Lang, Frankfurt am Main, Germany, 1993. [7] P.Loridan, ε-solutions in vector minimization problem, J.O.T.A., 43(1984). [8] H. Kawasaki, A duality theorem in multiobjective nonlinear programming, Math. Oper. Res., 7(1982), 95-110. [9] J.W.Nieuwenhuis, Supremal points and generalized duality , Math. Oper. Statist., Ser. Optimiza- tion, 11(1980). [10] J.Ponstein, On the dualization of multiobjective optimization problems, Univ. Groningen Econom. Inst. Rep. 88, 1982. [11] V. Postolica, Vectorial optimization programs with multifunctions and duality, Ann.Sci.Math. Quebec, 10(1986). [12] W.D. Rong, Y.N.Wu, ε-Weak minimal solutions of vector optimization problems with set-valued maps, J.O.T.A., 106(2000). [13] E.E. Rosinger, Duality and alternative in multiobjective optimization, Proc. of Amer.Math. Soc., 64(1977), 307-312. [14] Y. Sawargi, T. Tanino, H. Nakayama, Theory of multiobjective optimization, Academic Press Inc. Orlando, 1985. [15] C.Stamate, Vector subdifferentials, PhD these, Limoges, 1999. [16] T.Tanino, On supremum of a set in a multi-dimensional space, J.M.A.A., 130(1988), 386-397. [17] T.Tanino, Conjugate duality in vector optimization, J.M.A.A., 167(1992). [18] T. Tanino, Y. Sawaragi, Duality theory in multiobjective programming, Journal of Optimization Theory and Applications, 27(1979), 509-529. [19] K. Yokoyama Epsilon approximate solutions for multiobjective programming problems, J.M.A.A., 203(1996). [20] C. Zalinescu,˘ Programare matematica in spatii normate infinit dimensionale, Ed. Academiei Romane, 1998. [21] J. Zowe, The saddle-point theorem of Kuhn and Tucker in ordered vector spaces, Journal of Math. Anal. and Appl., 57(1977), 41-55. [22] J. Zowe, A duality theorem for a convex programming problem in order complete vector lattices, Journ. of Math. Anal. and Appl., 50(1975), 273-287. ROMAI J., 6, 1(2010), 167–178

CONVERGENCE RATE ESTIMATION FOR A ILL-POSED HEAT PROBLEM Nguyen Huy Tuan, Pham Hoang Quan Department of Mathematics, SaiGon University, HoChiMinh city, VietNam tuanhuy [email protected] Abstract In this paper we consider backward heat conduction problem that appears in some applications. As the problem is ill-posed, a new regularization method is applied to construct regularized solutions which are stably convergent to the exact ones with Holder¨ estimates. This work extends earlier results by Clark and Oppenheimer [3] and some other authors [5–6, 10–13].

~~Keywords: backward heat problem, ill-posed problem, nonhomogeneous heat, contraction principle. 2000 MSC: 35K05, 35K99, 47J06, 47H10.~~

1. INTRODUCTION In this paper we consider the problem of ﬁnding the temperature u(x, t), (x, t) ∈ (0, π) × [0, T], such that   uxx = ut, (x, t) ∈ (0, π) × (0, T),   u(0, t) = u(π, t) = 0, t ∈ (0, T), (1)   u(x, T) = g(x), x ∈ (0, π), where g(x) is given. The problem is called the backward heat problem, the backward Cauchy problem, or the ﬁnal value problem. As it is known, the problem is severely ill-posed; i.e., solutions do not always exist, and in the case of existence, these do not depend continuously on the given data. Physically, g can only be measured and measurement errors may appear such that we would actually have as data some function g ∈ L2(0, π), for which kg − gk ≤ where the constant > 0 represents a bound on the measurement error, while k.k denotes the L2-norm. Problem (1) was investigated by Clark and Oppenheimer [3], Denche and Bessila [5], ChuLiFu [4, 9, 8] Tauten- hahn[32], Trong and his group [26, 28], et al. However, in those papers, the error estimates are established in logarithmic form only, i.e.,

~~1 ku(., t) − v(., t)k ≤ C , r > 0. (2) 1 1 ln( r ) where C1 is a coeﬃcient depending on u. The aim of this paper is to provide a new regularization method to establish some~~

~~167 168 Nguyen Huy Tuan, Pham Hoang Quan~~

~~Holder¨ estimates such as~~

ku(., t) − v(., t)k ≤ Ck, k > 0. (3) where C is a coeﬃcient depending on u, while k does not depend on u or t. It is easy to see that k converges to zero more quickly than the logarithmic terms. The remainder of the paper is organized as follows. In Section 2, we establish the approximate problem and state our results concerning the norm of the diﬀerence between an exact solution u of Problem (1) and the approximation solution u. The proofs of our results are given in Section 3.

~~2. NEW ERROR ESTIMATES Starting from the ideas mentioned in the paper of Clark and Oppeinheimer [3], we consider the following approximate problem~~

,a ,a ut − uxx = 0, (x, t) ∈ (0, π) × (0, T), (4) u,a(0, t) = u,a(π, t) = 0, t ∈ [0, T], (5) X∞ −Tm2 ,a e u (x, T) = gm sin(mx), x ∈ (0, π), (6) (a−1)Tm2 −Tm2 m=1 α()e + e where Z 2 π gm = g(x) sin(mx) (7) π 0 and α() is a small parameter depending on . For brevity we denote α() by α. The real number a ≥ 1 is a constant. The case a = 1 is considered in [3]. The major reason to choose a ≥ 1 is explained in Section 3, Remark 2. We shall prove that the (unique) solution u,a of (4)–(6) is given by

~~X∞ (T−t)m2 ,a e u (x, t) = gm sin(mx) 0 ≤ t ≤ T. (8) aTm2 m=1 1 + αe~~

~~Let v,a be the solution of problem (4)–(6) with noisy data g(.)~~

~~X∞ e(T−t)m2 v,a(x, t) = g sin(mx), 0 ≤ t ≤ T. (9) aTm2 m m=1 1 + αe Convergence rate estimation for a ill-posed heat problem 169~~

Lemma 2.1. Let n, x, α, > 0, 0 ≤ a ≤ b, then na e − a i) ≤ x b , 1 + xenb 1 T ii) ≤ , ∈ (0, eT), α + e−αT ln(T/) (10) 1 T x iii) ≤ , ∈ (0, eT), x ≥ 1, + e−xT ln(T/)

e−na n(b − a) iv) ≤ , n ≥ 1. 1 + αe−nb b−a α ln( α ) Lemma 2.2. The problem (1) has a unique solution u if and only if X∞ 2Tm2 2 e gm < ∞. (11) m=1 Theorem 2.1. Let g(x) ∈ L2(0, π). The problem (4)–(6) has a unique weak solution ,a 2 2 1 1 1 u ∈ C([0, T]; L (0, π) ∩ L (0, T; H0(0, π)) ∩ C (0, T; H0(0, π)), namely (8). The solution of problem (4)–(6) given by (8) depends continuously on g in L2(0, π). Remark 2.1. As shown in the Introduction, several regularizations were defined in the literature. The stability of their solutions with respect to the variation of the final value (at t = T) is controlled by inequalities with coefficients of different order of T magnitude. For instance, in [7, 10, 26], the order of magnitude is e . In [3, 27], the authors give some better stability estimates than the latter discussed methods. They t −1 show that the stability estimate is of order M T . In [30, 31], the authors improve the T previous results by a better estimation of the stability order, that is A = C1 T . (1+ln( ) − 1 From (23), if we set α = then the stability order is B = C2 a . It is easy to see that for a > 1 then B C2 1− 1 T lim = lim a 1 + ln( ) = 0. →0 A C1T →0 It is easy to see that the order of the error is less than the above order of stability estimate. This proves the advantages of our method.

Theorem 2.2. Assume that v,a(., t) deﬁned in (9), is the unique solution of Problem (4)–(6), corresponding to the noisy data g(.) and that problem (1) has a unique so- 1 1 2 lution u(., t) ∈ C([0, T]; H0(0, π)) ∩ C ((0, T); L (0, π)). Let g be a function satisfying condition (11). a) If α = a, then, for every t ∈ [0, T], ,a t ku(., t) − v (., t)k ≤ (C1 + 1) T . (12) 170 Nguyen Huy Tuan, Pham Hoang Quan

aT b) If α = T+β for β > 0, then, for every t ∈ [0, T], !−1 aT t+β ,a T+β ku(., t) − v (., t)k ≤ C2 ln( aT ) + . (13) T+β c) Assume that there exist a positive number k ∈ [0, (a − 1)T) such that X∞ 2(T+k)m2 2 e gm < ∞. m=1 aT If α = T+k , then, for every t ∈ [0, T], ,a t+k ku(., t) − v (., t)k ≤ (C3 + 1) T+k . (14)

~~C1, C2, C3 are numbers deﬁned by~~

~~ku(., 0)k ≤ C1,~~

aTku (., 0)k ≤ C , xx 2 (15) r ∞ P π 2(T+k)m2 2 2 e gm ≤ C3. m=1 Remark 2.2. 1. In t = 0, the error (12) does not converges to zero when → 0. This is of the same order as in [3, 32]. The error (13) becomes, for t = 0, !−1 aT β ,a T+β ku(., 0) − v (., 0)k ≤ C2 ln( aT ) + . (16) T+β which converges to zero with logarithmic rate. It is of the same order as some results that are mentioned in Introduction. This often occurs in the boundary error estimate for ill-posed problems. 2. We emphasize once more that the error (14) (k > 0) is of Holder¨ type for all p t ∈ [0, T]. It is easy to see that the convergence to 0 of , (0 < p) is more rapid 1 −q than the that of ln( ) (q > 0) when → 0. Hence, the convergence rate in this paper is better than that of the regularizations proposed in some recent papers such as Clark and Oppenheimer [3], Denche and Bessila [5], ChuLiFu [4, 9, 8] Tautenhahn [32], Trong and his group [26, 28], et al. This proves that our regularizing method is better. 3. For t = 0 in (14), we get ,a k ku(., t) − v (., t)k ≤ (C3 + 1) T+k . (17) The rate of convergence at t = 0 is k/(T+k). Since 0 < k ≤ (a − 1)T, the fastest a−1 convergence is a . Hence it can approach for a large. This proves that (17) is sharp and best known estimate. Convergence rate estimation for a ill-posed heat problem 171 3. PROOFS OF THE MAIN RESULTS. Proof of Lemma 2.1. i) We have ena ena = nb a 1− a 1 + xe (1 + xenb) b (1 + xenb) b ena ≤ a (1 + xenb) b − a ≤ x b . ii) See proof in page 4 [28]. iii) We have 1 x = + e−xT x + xe−xT x ≤ x + e−xT T ≤ x ln(T/) iv) e−na n = 1 + αe−nb αnena + nen(a−b) n ≤ αn + en(a−b) n(b − a) ≤ . b−a α ln( α ) 1 1 2 Proof of Lemma 2.2. If problem (1) has a solution u ∈ C([0, T]; H0(0, π))∩C ((0, T); L (0, π)), then u can be written as X∞ −(t−T)m2 u(x, t) = e gm sin(mx). m=1 This implies that Tm2 um(0) = e gm. (18) Then X∞ 2 2Tm2 2 ku(., 0)k = e gm < ∞. m=1 If we get (11), then deﬁne v(x) be as the function X∞ Tm2 2 v(x) = e gm sin mx ∈ L (0, π). m=1 172 Nguyen Huy Tuan, Pham Hoang Quan

~~Consider the problem   ut − uxx = 0,   u(0, t) = u(π, t) = 0, t ∈ (0, T), (19)   u(x, 0) = v(x), x ∈ (0, π).~~

~~It is clear that (19) is the direct problem so it has a unique solution u. We have~~

~~∞ X 2 u(x, t) = e−tm < v(x), sin mx > sin mx. (20) m=1 Let t = T in (20), we have~~

~~∞ P −Tm2 Tm2 u(x, T) = e e gm sin mx m=1 P∞ (21) = gm sin mx = g(x). m=1 Hence, u is the unique solution of (1).~~

Proof of Theorem 2.1. The proof is divided into three steps. ,a 2 2 1 1 1 Step 1. If u ∈ C([0, T]; L (0, π)) ∩ L (0, T; H0(0, π)) ∩ C (0, T; H0(0, π)) satisﬁes (8) then u is solution of (4)–(6). We have

~~X∞ (T−t)m2 ,a e u (x, t) = gm sin(mx) aTm2 m=1 1 + αe X∞ e−tm2 = gm sin(mx) −Tm2 (a−1)Tm2 m=1 e + αe for 0 ≤ t ≤ T. We can verify directly that~~

,a 2 1 1 2 1 u ∈ C([0, T]; L (0, π) ∩ C ((0, T); H0(0, π)) ∩ L (0, T; H0(0, π))). .a ∞ 1 In fact, u ∈ C ((0, T]; H0(0, π)). Moreover, one has ,a ut (x, t) X∞ −m2e−tm2 = gm sin(mx) −Tm2 (a−1)Tm2 m=1 e + αe 2 X∞ = − m2hu,a(x, t), sin mxi sin(mx) π m=1 ,a = uxx (x, t) Convergence rate estimation for a ill-posed heat problem 173 and X∞ −Tm2 ,a e u (x, T) = gm sin(mx). −Tm2 (a−1)Tm2 m=1 e + αe Hence u,a is the solution of (4)–(6). 1 1 2 Step 2. The Problem (4)–(6) has at most one solution C([0, T]; H0(0, π))∩C ((0, T); L (0, π)). A proof of this statement can be found in [2, Theorem 11]. Step 3. The solution of problem (4)–(6) given by (8) depends continuously on g in L2(0, π). Let u and v be two solutions of (4)–(6) corresponding to the ﬁnal values g and h respectively. We prove that

~~t−T ku(., t) − v(., t)k ≤ α aT kg − hk. (22)~~

~~From the deﬁnitions of u and v we have X∞ e(T−t)m2 u(x, t) = gm sin(mx) 0 ≤ t ≤ T, aTm2 m=1 1 + αe~~

~~X∞ e(T−t)m2 v(x, t) = hm sin(mx) 0 ≤ t ≤ T, aTm2 m=1 1 + αe where Z Z 2 π 2 π gm = g(x) sin(mx)dx, hm = h(x) sin(mx)dx. π 0 π 0 By using (10), we obtain~~

~~2 X∞ (T−t)m2 2 π e ku(., t) − v(., t)k = (gm − hm) 2 aTm2 m=1 1 + αe X∞ π 2t−2T 2 ≤ α aT |g − h | 2 m m m=1 2t−2T 2 = α aT kg − hk .~~

~~Therefore t−T ku(., t) − v(., t)k ≤ α aT kg − hk. (23) This completes the proof of the theorem.~~

~~Proof of Theorem 2.2. Let u,a be the solution deﬁned by (8) with exact data g. By using the triangle inequality, we get~~

~~ku(., t) − v,a(., t)k ≤ ku(., t) − u,a(., t)k + ku,a(., t) − v,a(., t)k. (24) 174 Nguyen Huy Tuan, Pham Hoang Quan~~

~~For the term ku,a(., t) − v,a(., t)k, by using (23), we obtain~~

~~ ,a t−T t−T ku (., t) − v (., t)k ≤ α aT kg (.) − g(.)k ≤ α aT . (25)~~

1 a) Assume that the Problem (1)-(3) has an exact solution u ∈ C([0, T]; H0(0, π)) ∩ C1((0, T); L2(0, π)). Then u can be written as X∞ −(t−T)m2 u(x, t) = e gm sin(mx). m=1 This implies that −(t−T)m2 um(t) = e gm. (26) From (8), we get (T−t)m2 ,a e u (t) = gm. (27) m 1 + αeaTm2 Combining (26) and (27) and using (10), we obtain   (T−t)m2  2 e  ,a  −(t−T)m  |um(t) − um (t)| = e − 2  gm 1 + αeaTm

~~aTm2 e (T−t)m2 = α e |gm| (1 + αeaTm2 )~~

~~α (T−t)m2 = e |gm| (α + e−aTm2 )~~

~~αe−tm2 = |um(0)| (α + e−aTm2 )~~

~~t ≤ α aT |um(0)|.~~

~~This implies that~~

π X∞ ku(., t) − u,a(., t)k2 = |u (t) − u,a(t)|2 2 m m m=1 X∞ 2t π 2 ≤ α aT |u (0)| . 2 m m=1 Hence ,a t ku(., t) − u (., t)k ≤ C1α aT . (28) Convergence rate estimation for a ill-posed heat problem 175

~~Combining (24), (25), (28) and α = a, we have~~

~~ t t−T ku(., t) − v (., t)k ≤ C1α aT + α aT . t ≤ T (C1 + 1). b) From (28), we obtain~~

~~,a 1 t−T ku(., t) − v (., t)k ≤ C + α aT . 2 ln(aT/α) 1 t+β ≤ C + T+β 2 ln(aT/α) !−1 aT t+β T+β = C2 ln( aT ) + . T+β c) With (28), we have~~

~~,a α (T−t)m2 |um(t) − u (t)| = e |gm| m (α + e−aTm2 ) −(t+k)m2 αe (T+k)m2 ≤ e gm (α + e−aTm2 )~~

~~t+k (T+k)m2 ≤ α aT e |gm| . 176 Nguyen Huy Tuan, Pham Hoang Quan~~

~~This implies that~~

π X∞ ku(., ., t) − u,a(., ., t)k2 = |u (t) − u,a(t)|2 2 m m m=1 X∞ 2t+2k π 2(T+k)m2 2 ≤ α aT e g 2 m m=1 X∞ 2t+2k π 2(T+k)m2 2 = α aT e g . 2 m m=1 Hence ,a t+k ku(., t) − u (., t)k ≤ C3α aT . (30) aT By combining (24), (25), (30) and taking α = T+k , we have

~~,a t+k t−T ku(., t) − v (., t)k ≤ C3α aT + α aT . t+k t+k ≤ C3 T+k + T+k t+k = (C3 + 1) T+k .~~

~~This completes the proof of Theorem 2.~~

~~References~~

[1] K. A. Ames, L. E. Payne, Continuous dependence on modeling for some well-posed perturbations of the backward heat equation, J. Inequal. Appl., 3(1999), 51-64. [2] A. S. Carraso, Logarithmic convexity and the “slow evolution” constraint in ill-posed initial value problems, SIAM J. Math. Anal., 30, 3(1999), 479-496. [3] G. W. Clark, S. F. Oppenheimer, Quasireversibility methods for non-well posed problems, Elect. J. Diff. Eqns., 1994, 8(1994), 1-9. [4] Chu-Li Fu, Zhi Qian, Rui Shi, A modified method for a backward heat conduction problem, Applied Mathematics and Computation, 185(2007), 564-573. [5] M. Denche, K. Bessila, A modified quasi-boundary value method for ill-posed problems, J.Math.Anal.Appl, 301(2005), 419-426. [6] L. C. Evans, Partial Differential Equation, American Math. Soc., Providence, Rhode Island, Volume 19, 1997. [7] R.E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal., 6, 2(1975), 283-294. [8] Feng Xiao-Li, Qian Zhi, Fu Chu-Li, Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region, Math. Comput. Simulation, 79, 2(2008), 177-188. [9] Fu Chu-Li, Xiong Xiang-Tuan, Qian Zhi, Fourier regularization for a backward heat equation., J. Math. Anal. Appl., 331, 1(2007), 472-480. Convergence rate estimation for a ill-posed heat problem 177

[10] H. Gajewski, K. Zaccharias, Zur regularisierung einer klass nichtkorrekter probleme bei evolu- tiongleichungen, J. Math. Anal. Appl., 38 (1972), 784-789. [11] J. Hadamard, Lecture note on Cauchy’s problem in linear partial differential equations, Yale Uni Press, New Haven, 1923. [12] D.N. Hao, A mollification method for ill-posed problems, Numer. Math., 68(1994), 469-506. [13] Y. Huang, Q. Zhneg, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations, 203, 1(2004), 38-54. [14] V. K. Ivanov, I. V. Mel’nikova, F. M. Filinkov, Differential-Operator Equations and Ill-Posed problems, Nauka, Moscow, 1995 (Russian). [15] F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math, 13, (1960), 551-585. [16] M. Jourhmane, N.S. Mera, An iterative algorithm for the backward heat conduction problem based on variable relaxation factors, Inverse Probl. Eng., 10(2002), 293-308. [17] R. Lattes,` J.-L. Lions, Méthodede Quasi-réversibilitéet Applications, Dunod, Paris, 1967. [18] K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems, Symposium on Non-Well Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, 316(1973), Springer-Verlag, Berlin , 161-176. [19] I. V. Mel’nikova, S. V. Bochkareva, C-semigroups and regularization of an ill-posed Cauchy problem, Dok. Akad. Nauk., 329(1993), 270-273. [20] I. V. Mel’nikova, Q. Zheng and J. Zheng, Regularization of weakly ill-posed Cauchy problem, J. Inv. Ill-posed Problems, 10, 5(2002), 385-393. [21] N.S. Mera, L. Elliott, D.B. Ingham, D. Lesnic, An iterative boundary element method for solving the one dimensional backward heat conduction problem, Int. J. Heat Mass Transfer, 44(2001), 1937-1946. [22] L. E. Payne, Some general remarks on improperly posed problems for partial differential equations, Symposium on Non-Well Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, 316(1973), Springer-Verlag, Berlin, 1-30. [23] Quan, P. H. and Trong, D. D., A nonlinearly backward heat problem: uniqueness, regularization and error estimate, Applicable Analysis, 85, 6-7(2006), 641-657. [24] R.E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl, 47(1974), 563-572. [25] R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, in Trends in the Theory and Practice of Non-Linear Analysis, Elsevier 1983. [26] D. D. Trong, N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006, 4(2006), 1-10. [27] D. D. Trong, P. H. Quan, T.V. Khanh, N. H. Tuan, A nonlinear case of the 1-D backward heat problem: Regularization and error estimate, Zeitschrift Analysis und ihre Anwendungen, 26, 2(2007), 231-245. [28] D. D. Trong, N. H. Tuan, A nonhomogeneous backward heat problem: Regularization and error estimates, Electron. J. Diff. Eqns., 2008, 33(2008), 1-14. [29] D. D. Trong, N. H. Tuan, Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems , Electron. J. Diff. Eqns., 2008, 84(2008), 1-12. [30] D. D. Trong, N. H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation., Nonlinear Anal., 71, 9(2009), 4167-4176. 178 Nguyen Huy Tuan, Pham Hoang Quan

[31] N. H. Tuan, D. D. Trong, A new regularized method for two dimensional nonhomogeneous backward heat problem, Appl. Math. Comput, 215, 3(2009), 873-880. [32] T. Schroter,¨ U. Tautenhahn, On optimal regularization methods for the backward heat equation, Z. Anal. Anw., 15 (1996), 475-493. [33] B.Yildiz, M. Ozdemir, Stability of the solution of backwrad heat equation on a weak conpactum, Appl.Math.Comput., 111(2000), 1-6. [34] B.Yildiz, H. Yetis, A.Sever, A stability estimate on the regularized solution of the backward heat problem, Appl.Math.Comput., 135(2003), 561-567. ROMAI J., 6, 1(2010), 179–193

CONCURRENCY ANALYSIS FOR COLOURED PETRI NETS Cristian Vidra¸scu Faculty of Computer Science, University “Al. I. Cuza” of Ia¸si,Romania [email protected] Abstract The goal of this paper is to study the relationships between the concurrency-degrees of a coloured Petri net and those of its subnets.

~~Keywords: parallel/distributed systems, Petri nets, concurrency. 2000 MSC: 68Q85.~~

1. INTRODUCTION A Petri net is a mathematical model used for the specification and the analysis of parallel/distributed systems. An introduction about Petri nets can be found in [3]. The concurrency-degrees are a measurement of the concurrency in Petri nets, which was first introduced in [2]. It is useful to introduce a measure of concurrency for parallel/distributed systems, for answering a question like this: What is the meaning of the fact that in the system S 1 the concurrency is greater than in the system S 2? The problem of concurrency is studied for Petri nets, but, since the Petri nets are used as suitable models for real-world parallel or distributed systems, the results are applicable also to these systems. The basic idea is that the number of transitions which can fire simultaneously in a Petri net which models a given real system, can be used as an intuitive measure of the concurrency of that system. An advantage of studying the concurrency-degrees is that they can be computed for the model during the design of a system, and this will usually lead to an improved design. It is well-known that the behaviour of some distributed systems cannot be ade- quately modelled by classical Petri nets. Many extensions which increase the com- putational power and/or the expressive power of Petri nets have been thus introduced. One direction has led to high-level Petri nets. High-level Petri nets are a powerful language for system modelling and validation. They are now in widespread use for many different practical and theoretical purposes in various fields of software and hardware development. The step from low-level (i.e. classical) nets to high-level nets can be compared to the step from assembly languages to modern programming languages with an elaborated type concept. While in low-level nets there is only one kind of token and the state of a place is described

179 180 Cristian Vidra¸scu by an integer or a boolean value, in high-level nets each token can carry complex information. There are two basic kinds of high-level Petri nets: predicate/transition nets and coloured Petri nets, which are very similar, but they are defined and presented in two rather different ways. Predicate/transition nets are defined using the notation and concepts of many-sorted algebras. Coloured Petri nets are defined using types, variables and expressions. Coloured Petri nets were defined by K. Jensen (see [1]) shortly after the first kind of high-level nets (developed by H.J Genrich and K. Lautenbach), and they are more user-friendly as a system modelling language than are the predicate/transition nets. In this paper we make an analysis of concurrency for high-level Petri nets, using coloured Petri nets as the presentation formalism. More precisely, we study the relationships between the concurrency-degrees of a coloured Petri net and those of its subnets.

2. PRELIMINARIES We assume the basic terminology and notation about sets, relations and functions, vectors, multi-sets and formal languages to be known. Let us just briefly remind that a multi-set m, over aP non-empty set S , is a function m : S → N, usually represented as a formal sum: s∈S m(s)`s. In the previous sum, m(s)` is the multiplicity of the s element in that multi-set (see [1]). A multi-set will be sometimes identified with a |S |-dimensional vector. The operations and relations on multi-sets are defined component-wise.P S MS denotes the set of all multi-sets over S . The emptyP multi-set s∈S 0`s is denoted by ∅. The size of the multi-set m is defined as |m| = s∈S m(s). The multi-set m is called infinite iff |m| = ∞. Also, we will assume as known the basic terminology and notation about P/T-nets, the classical Petri nets. For details the reader is referred to [3]. The concurrency- degrees are a measurement of the concurrency in Petri nets, which was first introduced in [2] for P/T-nets. A more general definition of concurrency-degrees for them, which takes into consideration the self-concurrency (i.e. the case of the transitions concurrently enabled with themselves), and a finer notion, namely the concurrency- degrees w.r.t. a set of transitions, was presented in [5]. In the sequel we establish the basic terminology, notation, and results concerning coloured Petri nets in order to give the reader the necessary prerequisites for the understanding of this paper (for details the reader is referred to [1]).

2.1. COLOURED PETRI NETS We present the formal deﬁnition of non-hierarchical CP-nets (abbreviation used for coloured Petri nets) given by Kurt Jensen in [1] (that book contains also the deﬁni- tion of hierarchical CP-nets, which are nets obtained by composing non-hierarchical CP-nets, possibly on more levels of composition). Concurrency analysis for coloured Petri nets 181

Remark 2.1. A non-hierarchical CP-net is formally defined as a n-tuple, as we will see below. However, the only purpose of this is to give a mathematically sound and unambiguous definition of CP-nets and their semantics. Any particular net, created by a modeller (either manually, or using the software tool Design/CPN for modelling and analysis of CP-nets), will always be specified in terms of a graphical representation, called CPN diagram (see [1]). This approach is analogous to the definition of directed graphs and non-deterministic finite automata. Formally, they are defined as pairs G = (V, A) and as 5-tuples A = (S, Σ, δ, s0, F), respectively, but usually they are represented by drawings containing sets of nodes, arcs and inscriptions.

To give the abstract definition of CP-nets it is not necessary to fix the concrete syntax in which the modeller writes the net expressions, and thus it will be assumed that such a syntax exists together with a well-defined semantics – making it possible in an unambiguous way to talk about: i) The elements of a data type (or sort), T. The set of all elements of T (i.e., the domain of T) is denoted by the type name T itself. ii) The type of a variable, v – denoted by Type(v). iii) The type of an expression, expr – denoted by Type(expr). iv) The set of variables occurring in an expression, expr – denoted by Var(expr). v) A binding, b, of a set of variables, V – associating with each variable v ∈ V an element b(v) ∈ Type(v). vi) The value obtained by evaluating an expression, expr, in a binding, b – denoted by expr . Var(expr) is required to be a subset of the variables of b, and the evaluation is performed by substituting for each variable v ∈ Var(expr) the value b(v) ∈ Type(v) determined by that binding. Therefore, the syntax and semantics which are used for net expressions specify a many-sorted universal algebra (i.e., a many-sorted signature Σ and a many-sorted algebra over signature Σ), together with the set of terms (i.e. expressions) constructed over this algebra using a set of sorted variables, and with a first-order many-sorted logic with equality (over signature Σ) used for evaluating the expressions. An expression without variables is called closed expression. It can be evaluated in all bindings, and all evaluations give the same value – which it will often be denoted by the expression itself (i.e. we simply write “expr” instead of the more pedantic “expr ”). Now we are ready to remind the definition of non-hierarchical CP-nets (see [1]). We use B to denote the boolean type (containing the elements {false, true}, and having the standard operations from propositional logic), and when V is a set of variables, we use Type(V) to denote the set of types {Type(v) | v ∈ V}. Some motivations and explanations of the individual parts of the definition are given immediately below the definition, and it is recommended that these are read in parallel with the definition.

Definition 2.1. A non-hierarchical CP-net, is a 9-tuple CPN = (S, T, A, N, Σ, C, G, E, I) satisfying the following requirements: (a) S is a finite set of places. (b) T is a finite set of transitions. (c) A is a finite set of arcs, such that: S ∩ T = S ∩ A = T ∩ A = ∅. (d) N : A → S × T ∪ T × S is a node function, describing the nodes of an arc. (e) Σ is a finite set of non-empty types, called colour sets. (f) C : S → Σ is a 182 Cristian Vidra¸scu colour function. (g) G : T → {expr | expr expression} is a guard function, satisfying the requirement: ∀t ∈ T : Type(G(t)) = B ∧ Type(Var(G(t))) ⊆ Σ . (h) E : A → {expr | expr is expression} is an arc expression function, satisfying the requirement: Type(E(a)) = C(s(a))MS ∧ Type(Var(E(a))) ⊆ Σ , for all a ∈ A, where s(a) is the place of N(a). (i) I : S → {expr | expr closed expression} is an initialization function, satisfying ∀s ∈ S : Type(I(s)) = C(s)MS . Notation 2.1. We use X = S ∪ T to denote the set of all nodes (i.e. places and transitions). Moreover, we define a number of functions describing the relationship between neighbouring elements of the net structure (the name of the function begins with a capital letter when its value is a set of elements; sometimes the same name is used for several functions/sets, but from the argument(s) it will always be clear which one is referred to): i) s : A → S maps each arc, a, to the place from the pair N(a). ii) t : A → T maps each arc, a, to the transition from the pair N(a). iii) o : A → X maps each arc, a, to the source of a, i.e. the first component of N(a). iv) d : A → X maps each arc, a, to the destination of a, i.e. the second component of N(a). v) A : S × T ∪ T × S → P(A) maps each ordered pairs of nodes, (x1, x2), to the set of its connecting arcs, i.e. A(x1, x2) = {a ∈ A | N(a) = (x1, x2)}. vi) A : X → P(A) maps each node, x, to the set of its surrounding arcs, i.e. A(x) = {a ∈ A | ∃x0 ∈ X : N(a) = (x, x0) ∨ N(a) = (x0, x)}. vii) In : X → P(X) maps each node, x, to the set of its input nodes, i.e. In(x) = {x0 ∈ X | ∃a ∈ A : N(a) = (x0, x)}. viii) Out : X → P(X) maps each node, x, to the set of its output nodes, i.e. Out(x) = {x0 ∈ X | ∃a ∈ A : N(a) = (x, x0)}. ix) X : X → P(X) maps each node, x, to the set of its surrounding nodes, i.e. X(x) = In(x) ∪ Out(x). All the previous functions can be extended, in the usual way, to take sets as input (then they all return sets and thus all the function names are written with a capital letter). Having defined the structure of CP-nets, we are now ready to consider their behaviour – but first we introduce the following notation: Notation 2.2. (i) Var(t) denotes the set of variables of transition t, i.e. Var(t) = {v|v ∈ Var(G(t)) ∨ ∃a ∈ A(t): v ∈ Var(E(a))}. (ii) E(x1, x2)Pdenotes the expression of (x1, x2), i.e. ∀(x1, x2) ∈ S × T ∪ T × S : E(x1, x2) = a∈A(x1,x2) E(a) (the sum indicates addition of expressions; it is well- defined because all expressions have as type the same multi-set). Next we remind the meaning of a transition binding. Intuitively, a binding of a transition t is a substitution that replaces each variable of t with a colour of the correct type and such that the guard evaluates to true. Definition 2.2. A binding of a transition t is a function b defined on the set Var(t), such that: (i) ∀v ∈ Var(t): b(v) ∈ Type(v). (ii) G(t) = true. By B(t) we denote the set of all bindings for t. Next we present the notions of token elements, binding elements, markings and steps. Concurrency analysis for coloured Petri nets 183

Deﬁnition 2.3. (i) A token element is an ordered pair (s, c), where s ∈ S and c ∈ C(s). The set of all token elements is denoted by TE.A marking is a multi-set over TE. The set of all markings is denoted by M, i.e. M = TEMS . The initial marking M0 is the marking obtained by evaluating the initialization expressions: ∀(s, c) ∈ TE : M0(s, c) = (I(s))(c). (ii) A binding element is an ordered pair (t, b), where t ∈ T and b ∈ B(t). The set of all binding elements is denoted by BE.A step is a non-empty and ﬁnite multi-set over BE. The set of all steps is denoted by Y. Notation 2.3. BE(t) = {t}×B(t) denotes the set of all binding elements corresponding to transition t ∈ T, and TE(s) = {s}×C(s) the set of all token elements corresponding to place s ∈ S .

Remark 2.2. There is a unique correspondence between a marking M ∈ TEMS and a function M¯ defined on S such that M¯ (s) ∈ C(s)MS , given by: (M¯ (s))(c) = M(s, c), ∀s ∈ S, ∀c ∈ C(s). This permits us often to represent markings as functions defined on S . Analogously, there is a unique correspondence between a step Y and a function Y¯ defined on T such that Y¯(t) ∈ B(t)MS is finite for all t ∈ T and non-empty for at least one t ∈ T. Hence, we often represent steps as functions defined on T. Notation 2.4. Let CPN be a CP-net, and Y ∈ Y a step. The multi-sets (functions) Y−, Y+ : TE → N and the function ∆Y : TE → Z are defined by: Y−(s, c) = P P E(s, t) (c) , Y+(s, c) = E(t, s) (c) (t,b)∈Y (t,b)∈Y and ∆Y(s, c) = Y+(s, c) − Y−(s, c) , ∀(s, c) ∈ TE. Now we can present the formal definition of the behaviour of a CP-net: Definition 2.4. The behaviour of a CP-net CPN is given by the firing rule, which consists in: (i) the enabling rule: a step Y is enabled in a marking M (or Y is fireable − from M), abbreviated M[YiCPN , iff ∀s ∈ S : Y (s) ≤ M(s) ; (ii) the computing 0 rule: if M[YiCPN, then Y may occur at the marking M yielding a new marking M , 0 0 − + abbreviated M[YiCPN M , defined by: ∀s ∈ S : M (s) = M(s) − Y (s) + Y (s) . Let us notice that, in this way, for every step Y of the CP-net CPN we have defined 0 − a binary relation on TEMS , denoted by [YiCPN, such that: M[YiCPN M ⇔ Y ≤ M 0 and M = M + ∆Y. The notation “[.iCPN” will be simplified to “[.i” anytime the CP-net can be understood from the context. Definition 2.5. Let the step Y be enabled in the marking M. When (t, b) ∈ Y, we say that t is enabled in M for the binding b. We also say that (t, b) is enabled in M, and so is t. When (t1, b1), (t2, b2) ∈ Y and (t1, b1) , (t2, b2) we say that (t1, b1) and (t2, b2) are concurrently enabled, and so are t1 and t2. When Y(t, b) ≥ 2 we say that (t, b) is concurrently enabled with itself. And when |Y(t)| ≥ 2 we say that t is concurrently enabled with itself. 184 Cristian Vidra¸scu

Definition 2.6. A finite occurrence sequence is a sequence of markings and steps M1[Y1iM2[Y2iM3 ... Mn[YniMn+1 such that n ∈ N, and Mi[YiiMi+1 for all 1 ≤ i ≤ n. M1 is called the start marking, Mn+1 is called the end marking, and the non-negative integer n is called the length of the sequence. Analogously, an infinite occurrence sequence is a sequence of markings and steps M1[Y1iM2[Y2iM3 ... such that Mi[YiiMi+1 for all i ≥ 1. M1 is called the start marking of the sequence, which is said to have infinite length. Definition 2.7. A marking M0 is reachable from a marking M in the CP-net CPN iff there exists a finite occurrence sequence having M as start marking and M0 as end marking. The set of all reachable markings from M is denoted by [MiCPN, or by RS (CPN, M).

3. CONCURRENCY-DEGREES FOR CP-NETS In this section we present a measure of concurrency for CP-nets, introduced in [4] (a tech. report in Romanian), which was obtained by extending to CP-nets the measures defined for classical Petri nets in [5]. Definition 3.1. Let CPN be a CP-net and M an arbitrary marking of CPN. A step Y is called a maximal step enabled at the marking M iff Y is enabled at M and there exists no step Y0 enabled at M with Y0 > Y. Notation 3.1. i) BE(M) denotes the set of all binding elements enabled in M, i.e. BE(M) = {(t, b)∈ BE | M[(t, b)i} = {Y ∈Y | M[Yi ∧ |Y| = 1}. ii) Y(M) denotes the set of all steps enabled at M in the net CPN, i.e. the set of all multi-sets of binding elements (concurrently) enabled at M: Y(M) = {Y ∈ Y | M[YiCPN}. iii) Ymax(M) denotes the set of all maximal steps enabled at M in CPN, i.e. the set of all maximal multi-sets of binding elements (concurrently) enabled at M: Ymax(M) = {Y ∈Y(M) | ∀ Y0 ∈Y : Y0 > Y ⇒ Y0

Remark 3.1. Let T0 = {t ∈ T | In(t) = ∅} be the set of the transitions with no input nodes in the CP-net CPN. If T0 = ∅, then it is easy to remark that the sets Y(M) and Ymax(M) are finite, for any marking M. Otherwise, if T0 , ∅, then, for any marking M of CPN, the set Y(M) is infinite, and Ymax(M) = ∅. Indeed, there exists a transition t0 ∈ T0, because T0 , ∅, and therefore (t0, b) ∈ Y(M), for any binding b ∈ B(t). Thus, Y(M) , ∅, and, if Y ∈ Y(M) is an arbitrary step enabled at M, then also Yk = Y + k`(t0, b) ∈ Y(M), for all k ∈ N. Thus, Y(M) is infinite. Moreover, for any step Y ∈ Y(M), the set Y(M) contains an infinite strictly increasing sequence of steps converging to the limit Y∗ : BE → N ∪ {∞}, with ( ∞ , if In(t) = 0 Y∗(t, b) = , for all (t, b) ∈ BE. Y(t, b) , otherwise

~~Obviously, there exists no maximal step enabled at M, thus Ymax(M) = ∅. Concurrency analysis for coloured Petri nets 185~~

Intuitively, the notion of concurrency-degree at a marking M of a CP-net represents the maximum (i.e. the supremum) number of binding elements concurrently enabled at M. Deﬁnition 3.2. Let CPN be a CP-net and M an arbitrary marking of CPN. The concurrency-degree at M of the net CPN is deﬁned by:

~~d(CPN, M) = sup{ |Y| | Y ∈ Y(M)} . (1)~~

Remark 3.2. Directly from definitions and Remark 3.1 it follows that ( max{ |Y| | Y ∈ Y (M)} , if T = ∅ ∀ M ∈M : d(CPN, M) = max 0 +∞ , otherwise and it does not depend on the initial marking of the net. Definition 3.3. i) The inferior concurrency-degree of a CP-net CPN is defined by:

~~− d (CPN) = min{ d(CPN, M) | M ∈ [M0iCPN} . (2) ii) The superior concurrency-degree of the net CPN is deﬁned by:~~

+ d (CPN) = sup{ d(CPN, M) | M ∈ [M0iCPN} . (3) iii) If d−(CPN) = d+(CPN), then this number is called the concurrency-degree of the net CPN and it is denoted by d(CPN). Remark 3.3. Directly from definitions we have i) 0 ≤ d−(CPN) ≤ d+(CPN) ≤ ∞ . ii) The inferior concurrency-degree of a CP-net CPN, d−(CPN), represents the minimum number of binding elements (transitions) maximal concurrently enabled at any reachable marking of CPN. In other words, at any reachable marking M of the net CPN there exist d−(CPN) binding elements (transitions) concurrently enabled at M. iii) The superior concurrency-degree of a CP-net CPN, d+(CPN), represents the supremum number of binding elements (transitions) maximal concurrently enabled at any reachable marking of CPN. In other words, at any reachable marking M of the net CPN there exist at most d+(CPN) binding elements (transitions) concurrently enabled at M. iv) The concurrency-degree of the net CPN means that at any reachable marking M there exist d(CPN) binding elements (transitions) concurrently enabled at M, and there is no reachable marking M0 of CPN with more than d(CPN) binding elements (transitions) concurrently enabled at M0. As we could see from Remark 3.1, sometimes it can be useful to ignore some transitions of a net and to study the behaviour of the net w.r.t. the remaining transitions. Or we could ignore only some binding elements and study the behaviour of 186 Cristian Vidra¸scu the net w.r.t. the remaining binding elements. Therefore, we introduced the notion of concurrency-degree of a CP-net w.r.t. a subset of binding elements, and w.r.t. a subset of transitions. Definition 3.4. Let CPN be a CP-net and BE0 ⊆ BE a subset of binding elements. A step over BE0, Y, is a step satisfying Y(t, b) = 0, for all (t, b) ∈ BE − BE0 (practically, 0 Y is a non-empty and finite multi-set over the subset BE ). Therefore, by Y|BE0 = Y ∩ 0 0 BEMS we will denote the set of all steps over BE of the net CPN. Moreover, we will 0 denote by Y|BE0 (M) the set of all steps over BE enabled at M, and by (Y|BE0 )max(M) the set of all maximal steps over BE0 enabled at M. Definition 3.5. Let CPN be a CP-net, BE0 ⊆ BE a subset of binding elements, and M an arbitrary marking of CPN. The concurrency-degree w.r.t. BE0 at the marking M of the net CPN, denoted by d(CPN, BE0, M), is defined by replacing Y(M) with Y|BE0 (M) in (1). Definition 3.6. Let CPN be a CP-net and BE0 ⊆ BE a subset of binding elements. The inferior and superior concurrency-degree w.r.t. BE0 of the net CPN, denoted by d−(CPN, BE0) and d+(CPN, BE0), resp., are defined by replacing d(CPN, M) with d(CPN, BE0, M) in (2) and (3), resp. Moreover, if d−(CPN, BE0) = d+(CPN, BE0), then this number, denoted by d(CPN, BE0), is called the concurrency-degree of the net w.r.t. BE0. Remarks 3.2 and 3.3 hold similarly for these notions of concurrency-degrees w.r.t. a set of binding elements. Furthermore, we can speak about concurrency-degrees w.r.t. a subset of transitions, T 0 ⊆ T, of a CP-net CPN. We take the set of all binding elements corresponding to the transitions from the subset T 0, namely BE0 = ∪ {BE(t) | t ∈ T 0} = ∪ { {t} × B(t) | t ∈ T 0}, and we define all the above notions w.r.t. T 0 by using this set. 0 0 Thus, a step over T will be a step over BE , Y|T 0 will denote the set of all steps over 0 0 T , Y|T 0 (M) will denote the set of all steps over T enabled at M, and (Y|T 0 )max(M) will denote the set of all maximal steps over T 0 enabled at M. Definition 3.7. Let CPN be a CP-net, T 0 ⊆ T a subset of transitions, and M an arbitrary marking of CPN. The concurrency-degree of the net w.r.t. T 0 at M is defined as d(CPN, T 0, M) = d(CPN, ∪ {BE(t) | t ∈ T 0}, M). Definition 3.8. Let CPN be a CP-net and T 0 ⊆ T a subset of transitions of this net. The inferior and superior concurrency-degree w.r.t. T 0 of the net CPN, are defined as d−(CPN, T 0) = d−(CPN, ∪ {BE(t) | t ∈ T 0}) and d+(CPN, T 0) = d+(CPN, ∪ {BE(t) | t ∈ T 0}), resp. Moreover, if d−(CPN, T 0) = d+(CPN, T 0), then this number, denoted by d(CPN, T 0), is called the concurrency-degree w.r.t. T 0 of the net CPN. Remarks 3.2 and 3.3 hold similarly for these notions of concurrency-degrees w.r.t. a set of transitions. Concurrency analysis for coloured Petri nets 187 4. CONCURRENCY ANALYSIS FOR THE SUBNETS OF A NET We can take into consideration the problem of modularization for coloured Petri nets: a CP-net can be “decomposed” into several modules, i.e. subnets of it, which have in common some locations of the net; these locations play the role of “interface” (i.e., they are shared) between two or more modules. Using this setting, the study of the concurrency in the global net can be done by analyzing the concurrency of the subnets which form that net. Thus, it is useful to study the relationships between the concurrency-degrees of a CP-net and those of the subnets which compose that net. The following result shows the connection between the concurrency-degree w.r.t. the union of two disjoint sets of transitions and the concurrency-degrees w.r.t. each of those two sets:

~~Theorem 4.1. Let CPN be a CP-net and T1, T2 ⊆ T two disjoint sets of transitions. The following inequalities hold, for any marking M of the net:~~

~~d(CPN, T1 ∪ T2, M) ≤ d(CPN, T1, M) + d(CPN, T2, M) (4)~~

~~+ + + d (CPN, T1 ∪ T2) ≤ d (CPN, T1) + d (CPN, T2) (5)~~

Proof. i) Let CPN be a CP-net, T1, T2 ⊆ T two disjoint sets of transitions, and M an arbitrary marking of CPN. Let Y ∈Y|T1∪ T2 (CPN, M) be an arbitrary step over T1∪T2 enabled at M. Since T1 ∩ T2 = ∅, we can write: X X Y−(s) = E(s, t) = E(s, t) =

(t,b)∈Y t∈T1∪ T2,b∈BE(t) X X = E(s, t) + E(s, t) ,

t∈T1,b∈BE(t) t∈T2,b∈BE(t) − and thus, since M[YiCPN, by the step enabling rule it follows that M(s) ≥ Y (s), so we conclude that, for all places s: X X E(s, t) ≤ M(s) and E(s, t) ≤ M(s) .

~~t∈T1,b∈BE(t) t∈T2,b∈BE(t)~~

~~not First inequality means that the step denoted by Y|T1 = {(t, b) ∈ Y | t ∈ T1, b ∈~~

BE(t)} is a step over T1 ﬁreable at M in CPN, i.e. M[Y|T1 iCPN. By the deﬁni- tion of the concurrency-degree at a marking for CP-nets, this means that |Y|T1 | ≤ d(CPN, T1, M). Similarly, from the second inequality from above we deduce that the step denoted not by Y|T2 = {(t, b) ∈ Y | t ∈ T2, b ∈ BE(t)} is a step over T2 enabled at M in CPN, and therefore we have that |Y|T2 | ≤ d(CPN, T2, M). 188 Cristian Vidra¸scu

~~Obviously, Y = Y|T1 + Y|T2 , so we can conclude that~~

~~|Y| = |Y|T1 | + |Y|T2 | ≤ d(CPN, T1, M) + d(CPN, T2, M) .~~

Thus, we showed that |Y| ≤ d(CPN, T1, M) + d(CPN, T2, M) , for all Y which are steps over T1 ∪ T2 enabled at M in CPN. By taking the supremum in this inequality, after Y as a step over T1 ∪ T2 enabled at M in CPN, and using the definition of the concurrency-degree at a marking for CP-nets, we obtain the desired inequality: d(CPN, T1 ∪ T2, M) ≤ d(CPN, T1, M) + d(CPN, T2, M) . ii) Let CPN be a CP-net, and T1, T2 ⊆ T two disjoint sets of transitions. By the definition of the superior concurrency-degree w.r.t. a set of transitions, we have the + + inequalities: d(CPN, T1, M) ≤ d (CPN, T1) and d(CPN, T2, M) ≤ d (CPN, T2) , for any reachable marking. Since, by pct. i), the inequality (4) holds for any arbitrary marking M, and, thus, it holds particularly for the reachable ones, we conclude that d(CPN, T1 ∪ T2, M) ≤ + + d (CPN, T1) + d (CPN, T2) , for all M ∈ [M0iCPN . By taking the supremum after M ∈ [M0iCPN in the above inequality, and using the definition of the superior concurrency-degree for CP-nets, we obtain the desired + + + inequality: d (CPN, T1 ∪ T2) ≤ d (CPN, T1) + d (CPN, T2) .

Remark 4.1. Unfortunately, regarding the inferior concurrency-degree, neither an inequality like (5), nor one with an inverse sign, holds true (a counterexample was given in [4]). Despite the fact that for the inferior concurrency-degree there exists no upper bound like the ones which exist for the concurrency-degree at a marking and for the superior concurrency-degree, we can still specify a lower bound for the inferior concurrency-degree of Petri nets, namely:

~~Theorem 4.2. Let CPN be a CP-net and T1, T2 ⊆ T two disjoint sets of transitions. The following inequality holds:~~

~~− − − d (CPN, T1 ∪ T2) ≥ max{d (CPN, T1) , d (CPN, T2)} (6)~~

Proof. Let M be an arbitrary marking of the net CPN. Obviously, any step over T1 enabled at the marking M in CPN is also a step over T1 ∪ T2 enabled at M in CPN (being a multiset over T1 ∪ T2 having zero multiplicities for the elements from T2). Thus, by the deﬁnition of the concurrency-degree at a marking, we can deduce that d(CPN, T1 ∪ T2, M) ≥ d(CPN, T1, M) , for any arbitrary marking M, and thus, particularly, also for any M ∈ [M0iCPN . But, from the deﬁnition of the inferior concurrency-degree for CP-nets, it fol- − lows that d(CPN, T1, M) ≥ d (CPN, T1) , ∀ M ∈ [M0iCPN . Thus, we conclude that − d(CPN, T1 ∪ T2, M) ≥ d (CPN, T1) , ∀ M ∈ [M0iCPN . Concurrency analysis for coloured Petri nets 189

~~− Similarly it can be shown that d(CPN, T1∪T2, M) ≥ d (CPN, T2) , ∀ M ∈ [M0iCPN . Using these two inequalities, we deduce that~~

− − d(CPN, T1 ∪ T2, M) ≥ max{d (CPN, T1) , d (CPN, T2)} , for all M ∈ [M0iCPN . By taking the minimum after M ∈ [M0iCPN in the previous inequality, and using the deﬁnition of the inferior concurrency-degree, we get the desired inequality: − − − d (CPN, T1 ∪ T2) ≥ max{d (CPN, T1) , d (CPN, T2)} . Remark 4.2. The inequalities (4) and (5) from Theorem 4.1, as well as the inequality (6) from Theorem 4.2, can also be reformulated in terms of two disjoints sets of binding elements BE1 and BE2 (instead of the two disjoints sets of transitions T1 and T2), and the proofs are quite similar. Remark 4.3. Moreover, the inequalities (4) and (5) from Theorem 4.1, as well as the inequality (6) from Theorem 4.2, hold also for the generalized case of any ﬁnite union of pairwise-disjoint sets of transitions. (This rem can be easily proved by applying the inequalities mentioned above, reiteratively for unions of two disjoint sets of transitions.) A particular case of these generalized inequalities is represented by the case when the sets of transitions are all singletons (i.e., each set has only one element). In this case we obtain the following result, which expresses the relationship between the concurrency-degree w.r.t. a set of transitions and the concurrency-degrees w.r.t. each individual transition from that set: Corollary 4.1. Let CPN be a CP-net and T 0 ⊆ T a subset of transitions. Then the following inequalities hold, for any marking M: X d(CPN, T 0, M) ≤ d(CPN, {t}, M) , (7) t∈T 0 X d+(CPN, T 0) ≤ d+(CPN, {t}) , (8) t∈T 0 d−(CPN, T 0) ≥ max d−(CPN, {t}) . (9) t∈T 0 Proof. These inequalities follow as simple consequences from Theorem 4.1 and 4.2, by applying reiteratively (by |T 0| − 1 times) the corresponding inequalities from the two mentioned theorems.

Thus, an interesting question that arises is: when any of the inequalities (4), (5) and (6), or the analogous of inequality (5) for the inferior concurrency-degree of a CP-net becomes equality ? 190 Cristian Vidra¸scu

Let us notice the following fact: if CPN is a CP-net, T 0 ⊆ T a subset of transitions, and M1, M2 are two arbitrary markings of CPN such that M1(s) = M2(s), for any 0 0 location s ∈ In(T ), then any step over T enabled at M1 in CPN is also enabled at 0 M2 in CPN and viceversa. Thus, the set of steps over T enabled at M1 in the net 0 CPN is equal with the set of steps over T enabled at M2 in CPN, and, therefore, 0 0 d(CPN, T , M1) = d(CPN, T , M2). In other words, for CP-nets the concurrency-degree at a marking w.r.t. a set of transitions depends only on the components of that marking which correspond to the input places of the transitions from that set. This remark gives us a structural property of a CP-net which is a suﬃcient condition for some of the above mentioned equalities to hold:

Theorem 4.3. Let CPN be a CP-net, and T1, T2 ⊆ T two disjoint subsets of transitions. If its structure has the property that In(T1) ∩ In(T2) = ∅ , then the following equalities hold, for all markings M:

d(CPN, T1 ∪ T2, M)=d(CPN, T1, M) + d(CPN, T2, M) (10) − − − d (CPN, T1 ∪ T2) ≥ d (CPN, T1) + d (CPN, T2) (11) Proof. By inequality (4) from Theorem 4.1, for proving (10) it is suﬃcient to show that we have the inequality (4) with an invers sign in the hypothesis In(T1)∩ In(T2) = ∅ satisﬁed by the net CPN. Let Y1 be an arbitrary step over T1 enabled at M, and Y2 an arbitrary step over T2 enabled at M. Then, by the enabling rule of a step, we have X − not Y1 (s) = E(s, t) ≤ M(s) , ∀ s ∈ S , t∈T1,b∈BE(t) X − not and Y2 (s) = E(s, t) ≤ M(s) , ∀ s ∈ S . t∈T2,b∈BE(t)

Let Y = Y1 + Y2. Thus, Y is a step over (T1 ∪ T2). Then X − not − − Y (s) = E(s, t) = Y1 (s) + Y2 (s) , ∀ s ∈ S . t∈T1∪ T2,b∈BE(t) Considering the fact that t−(s) = 0 iﬀ s < In(t), for any t ∈ T and s ∈ S , and since, by hypothesis, In(T1) ∩ In(T2) = ∅, it follows that for any location s ∈ S , one and only one of the following three cases is possible: − − i) s ∈ In(T1). Then s < In(T2) and, thus, Y2 (s) = 0. So, we obtain that Y (s) = − Y1 (s) ≤ M(s). − − ii) s ∈ In(T2). Then s < In(T1) and, thus, Y1 (s) = 0. So, we obtain that Y (s) = − Y2 (s) ≤ M(s). iii) s ∈ S − (In(T1) ∪ In(T2)). Then s < In(T1) and s < In(T2), and, therefore, − − − Y1 (s) = Y2 (s) = 0. So, we obtain again that Y (s) = 0 ≤ M(s). Concurrency analysis for coloured Petri nets 191

In conclusion, we proved that Y−(s) ≤ M(s), ∀s ∈ S . This inequality means that Y is a step over (T1 ∪ T2) enabled at the marking M in CPN. By using the definition of the concurrency-degree at a marking w.r.t. a set of transitions, from this it follows that |Y| ≤ d(CPN, T1 ∪ T2, M) . Thus, since Y = Y1 + Y2, we have that |Y1| + |Y2| = |Y| ≤ d(CPN, T1 ∪ T2, M). Therefore, we proved that d(CPN, T1 ∪ T2, M) ≥ |Y1| + |Y2| , for any arbitrary step over T1 enabled at M in CPN, Y1, and any arbitrary step over T2 enabled at M in CPN, Y2. By succesively taking the supremum in the above inequality, after Y1 as a step over T1 enabled at M in CPN, and then after Y2 as a step over T2 enabled at M in CPN, and using the definition of the concurrency-degree at a marking for CP-nets, we obtain the desired inequality: d(CPN, T1 ∪ T2, M) ≥ d(CPN, T1, M) + d(CPN, T2, M) . To prove the second part of this theorem, let M ∈ [M0iCPN be an arbitrary reachable marking of the net CPN. Then, by the definition of the inferior concurrency- − − degree, we have that d(CPN, T1, M) ≥ d (CPN, T1) and d(CPN, T2, M) ≥ d (CPN, T2) . Thus, by applying the first part of this theorem, we obtain that d(CPN, T1 ∪ T2, M) = − − d(CPN, T1, M)+d(CPN, T2, M) ≥ d (CPN, T1)+d (CPN, T2) , for all M ∈ [M0iCPN. By taking the minimum after M ∈ [M0iCPN in the previous inequality, and using the definition of the inferior concurrency-degree, we get the desired inequality: − − − d (CPN, T1 ∪ T2) ≥ d (CPN, T1) + d (CPN, T2) .

Remark 4.4. We can formulate similar results to Corollary 4.1 and Theorem 4.3 (i.e. by reformulating them in terms of two disjoints sets of binding elements BE1 and BE2, instead of the two disjoints sets of transitions T1 and T2) to shows the connection between the concurrency-degree w.r.t. a subset of binding elements of a CP-net and the concurrency-degrees w.r.t. each individual binding element from that subset.

Remark 4.5. Obviously, equality (10) and inequality (11) from Theorem 4.3 hold also for the generalized case of any ﬁnite union of pairwise-disjoint sets of transitions. (This rem can be easily proved by applying (10), and respectively (11), reiteratively for unions of two disjoint sets of transitions.)

A particular case of these generalized equality and inequality for Petri nets is represented by the case when the sets of transitions are all singletons (i.e., each set has only one element). In this case we obtain the following result, which expresses a suﬃcient condition for having the above mentioned relationships between the concurrency-degree w.r.t. a set of transitions and the concurrency-degrees w.r.t. each individual transition from that set:

Corollary 4.2. Let CPN be a CP-net and T 0 ⊆ T a subset of transitions. If CPN 0 satisﬁes the property In(t1)∩In(t2) = ∅, for any t1, t2 ∈ T , then the following equality 192 Cristian Vidra¸scu holds for any marking M of the net CPN: X d(CPN, T 0, M) = d(CPN, {t}, M) , (12) t∈T 0 and, regarding the inferior concurrency-degree, the following inequality holds: X d−(CPN, T 0) ≥ d−(CPN, {t}) . (13) t∈T 0 Proof. These relations follow as simple consequences from Theorem 4.3, by applying reiteratively (by |T 0| − 1 times) the corresponding relations from that theorem.

Remark 4.6. Unfortunately, the condition In(T1)∩In(T2) = ∅ is not suﬃcient neither for having (5) with equality for the superior concurrency-degree, nor for having the similar equality for the inferior concurrency-degree. However, inequality (11), which holds in the hypothesis In(T1) ∩ In(T2) = ∅, represents an improvement of the lower bound given by inequality (6), for the inferior concurrency-degree of a CP-net.

5. CONCLUSION In this paper we presented the notion of concurrency-degrees for high-level Petri nets, using coloured Petri nets as the presentation formalism, and we made an analysis of concurrency for them, i.e. we studied the relationships between the concurrency- degrees of a coloured Petri net and those of its subnets. Since the CP-nets are used as suitable models for real parallel or distributed systems, the concurrency-degrees defined for CP-nets are an intuitive measure of the concurrency of the modelled systems, and, therefore, they have a practical impor- tance. For instance, they are useful for the evaluation of the models in the process of designing such a system: after making a model of that system as a CP-net, the study of the concurrency-degree of the model will give to the designers information about the concurrency of that system, allowing them to notice the inefficient components of the system, i.e. the components with bottlenecks w.r.t. parallelism/concurrency (because of the non-optimum use of the system’s resources, or because of other causes), and to make improvements of the model by remodelling those components in order to eliminate the causes of the bottlenecks. In this way, the evaluation of the model can produce useful feedback to the previous stages of the designing process, even until to the first stage of the specifications of the system, for making some improvements in that stage with the purpose of increasing the overall performance of the designed system. Some problems remain to be studied, for example: – finding the conditions for which the inequality (5) holds true with equality for the superior and resp. the inferior concurrency-degree of a CP-net; – making some case studies on models of real-world systems. Concurrency analysis for coloured Petri nets 193 References

[1] K. Jensen, Coloured Petri Nets. Basic Concepts, Analysis Methods and Practical Use, Volume 1, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1992. [2] T. Jucan, C. Vidras¸cu, Concurrency-degrees for Petri nets, Studia Universitatis Babes¸-Bolyai, Computer Science Section, XLIV(2)(1999), 3–15. [3] W. Reisig, Petri Nets. An Introduction, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin, 1985. [4] C. Vidras¸cu, Structural properties of Petri nets, Technical report TR 04–04, Faculty of Computer Science, Univ. “Al.I.Cuza” of Ias¸i, Dec. 2004. (in Romanian) [5] C. Vidras¸cu, T. Jucan, Concurrency-degrees for P/T-nets, Sci. Annals of Univ. “Al.I.Cuza” of Ias¸i, Computer Science Section, XIII(2003), 91–103.