APLIMAT - JOURNAL OF APPLIED MATHEMATICS
VOLUME 3 (2010), NUMBER 2
APLIMAT - JOURNAL OF APPLIED MATHEMATICS
VOLUME 3 (2010), NUMBER 2
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APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 3 (2010), NUMBER 2
DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS
MEMORY TERM BARBAGALLO Annamaria, MAUGERI Antonino: 13 FOR DYNAMIC OLIGOPOLISTIC MARKET EQUILIBRIUM PROBLEM BAŠTINEC Jaromír, DIBLÍK Josef, KHUSAINOV Denys, 25 RYVOLOVÁ Alena: STABILITY AND ESTIMATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS WITH CONSTANT COEFFICIENTS OF NEUTRAL TYPE CARKOVS Jevgenijs, PAVLENKO Oksana: AVERAGING, MERGER 35 AND STABILITY OF LINEAR DYNAMICAL SYSTEMS WITH SMALL MARKOV JUMPS DIBLÍK Josef, MORÁVKOVÁ Blanka: REPRESENTATION OF 45 SOLUTIONS OF LINEAR DISCRETE SYSTEMS WITH CONSTANT COEFFICIENTS, A SINGLE DELAY AND WITH IMPULSES SOME GENERALIZATIONS FAJMON Břetislav, ŠMARDA Zdeněk: 53 OF BANACH FIXED POINT THEOREM FARAGÓ István, KOROTOV Sergey, SZABÓ Tamás: 61 NON-NEGATIVITY PRESERVATION OF THE DISCRETE NONSTATIONARY HEAT EQUATION IN 1D AND 2D FILIPE José António, FERREIRA Manuel Alberto M., COELHO 83 M., PEDRO Maria Isabel: CHAOS, ANTI-CHAOS AND RESOURCES: DEALING WITH COMPLEXITY ADOMIAN DECOMPOSITION FILIPPOVA Olga, ŠMARDA Zdeněk: 91 METHOD FOR CERTAIN SINGULAR INITIAL VALUE PROBLEM HLAVIČKOVÁ Irena: BISECTION METHOD FOR FINDING INITIAL 99 DATA GENERATING BOUNDED SOLUTIONS OF DISCRETE EQUATIONS
ILAVSKÁ Iveta, NAJMANOVÁ Anna, OLACH Rudolf: 107 OSCILLATION OF NONLINEAR NEUTRAL DIFFERENTIAL SYSTEMS Kecskés Norbert: ON EXISTENCE OF SPECIAL SOLUTIONS OF 113 CERTAIN COUPLED SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS COMPUTER SIMULATIONS OF LINEAR KOLÁŘOVÁ Edita: 119 STOCHASTIC DIFFERENTIAL EQUATIONS MATVEJEVS Andrejs, SADURSKIS Karlis: AUTOREGRESSIVE 127 MODELS OF RISK PREDICTION AND ESTIMATION USING MARKOV CHAIN APPROACH THE STEP METHOD FOR A FUNCTIONAL- MURESAN Anton S.: 135 DIFFERENTIAL EQUATION FROM PRICE THEORY MURESAN Viorica: A FREDHOLM-VOLTERRA INTEGRO- 147 DIFFERENTIAL EQUATION WITH LINEAR MODIFICATION OF THE ARGUMENT THE SOLOW-SWAN GROWTH PŘIBYLOVÁ Lenka, VALENTA Petr: 159 MODEL WITH BOUNDED POPULATION ASYMPTOTIC PROPERTIES OF DELAYED SVOBODA Zdeněk: 167 EXPONENTIAL OF MATRIX VOLTERRA MATRIX INTEGRO-DIFFERENTIAL ŠMARDA Zdeněk: 173 EQUATIONS NEW ASPECTS IN Al ZOUKRA Kristine, SCHMIDT Werner H.: 179 PARAMETER IDENTIFICATION
APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 3 (2010), NUMBER 2
MODELING AND SIMULATION
APPLICATION OF NON-EUKLIDIAN KINDLER Eugene, KRIVY Ivan: 185 METRICS IN DISCRETE EVENT SIMULATION STABILITY ANALYSIS OF STATE-SPACE MODELS IN KVAPIL David: 189 MATLAB TALHOFER Václav, HOŠKOVÁ Šárka, KRATOCHVÍL Vlastimil, 199 HOFMANN Alois: MATHEMATICAL MODEL FOR SPATIAL DATA PROPERTIES ASSESSMENT
APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 3 (2010), NUMBER 2
ALGEBRA AND ITS APPLICATIONS
CHVALINA Jan, CHVALINOVÁ Ludmila: REALIZABILITY OF THE 211 ENDOMORPHISM MONOID OF A SEMI-CASCADE FORMED BY SOLUTION SPACES OF LINEAR ORDINARY N-TH ORDER DIFFERENTIAL EQUATIONS MYŠKOVÁ Helena: SOLVABILITY CONCEPTS FOR INTERVAL 225 SYSTEMS IN MAX-PLUS ALGEBRA NOVÁK Michal: THE NOTION OF SUBHYPERSTRUCTURE OF ”ENDS 237 LEMMA”–BASED HYPERSTRUCTURES SEIBERT Jaroslav: SOME QUANTITIES RELATED TO THE DISTANCE 249 MATRIX OF A SPECIAL TYPE OF DIGRAPHS. SZALAY István: GIANT BODIES PIECE BY PIECE 272
APLIMAT - JOURNAL OF APPLIED MATHEMATICS
LIST OF REVIEWERS
Andrade Marina, Professor Auxiliar ISCTE - Lisbon University Institute, Lisboa, Portugal
Ayud Sani Muhammed NGO, University of Cape Cost , Accra
Barbagallo Annamaria, Dr. University of Catania, Italy
University of Economics Prague, Jindřichův Hradec, Czech Bartošová Jitka, RNDr., PhD. Republic
Bayaraa Nyamish, Ulaanbaatar, Mongolia
Beránek Jaroslav, Doc. RNDr. CSc. Masaryk University, Brno, Czech Republic
Brueckler Franka Miriam, DrSc. University of Zagreb, Zagreb, Poland
Budíková Marie, RNDr., Dr. Masarykova univerzita, Brno, Czech Republic
Budkina Natalja, Dr. math Riga, Latvia Technical university, Riga, Latvia
Buy Dmitriy B., PhD. Kyiv Taras Shevchenko National University, Kyiv
Conversano Elisa Architettura, Roma, Italy
Csatáryová Mária, RNDr., PhD. University of Prešov, Prešov, Slovak Republic
Daniele Patrizia, Associate Professor University of Catania, Italy,
Slovak University of Technology in Bratislava, Slovak Dobrakovová Jana, Doc., RNDr., CSc. Republic
Doležalová Jarmila, Doc., RNDr., CSc. VŠB - TU, Ostrava, Czech Republic
Dorociaková Božena, RNDr., PhD. University of Žilina, Žilina, Slovak Republic
Doupovec Miroslav, doc. RNDr.. CSc. University of Technology, Brno, Czech Republic
Ferreira Manuel Alberto M., Professor ISCTE- Lisbon University Institute , Lisboa, Portugal Catedrático
Filipe José António, Prof. Auxiliar ISCTE - Lisbon University Institute, Lisboa, Portugal
9 Francaviglia Mauro, Full Prof. University of Calabria, Calabria, Italy
Franců Jan, prof. RNDr., CSc. University of Technology, Brno, Czech Republic
Fulier Jozef, prof. RNDr. CSc. UKF Nitra, Slovak Republic
Habiballa Hashim, RNDr., PaedDr., PhD. University of Ostrava, Ostrava, Czech Republic
Hennyeyová Klára, doc. Ing. CSc. Slovak University of Agriculture, Nitra, Slovak Republic
Hinterleitner Irena, PhD. University of Technology, Brno, Czech Republic
Hošková Šárka, Doc. RNDr., PhD. University of Defence, Brno, Czech Republic
Hunka Frantisek, Doc. (Assoc. Prof) University of Ostrava, Ostrava, Czech Republic
Chvosteková Martina, Mgr. Slovak Academy of Science, Bratislava, Slovak Republic
Jan Chvalina, Prof. DrSc. University of Technology, Brno, Czech Republic
Janková Mária, Mgr. Slovak Academy of science, Bratislava, Slovak Republic
Jukic Ljerka University of Osijek, Osijek, Croatia
Slovak University of Technology in Bratislava, Slovak Kalina Martin, Doc. RNDr., CSc. Republic
Kapička Vratislav, Prof. RNDr., DrSc. Masaryk university, Brno, Czech Republic
Klobucar Antoaneta, University in Osijek, Osijek, Croatia
Klůfa Jindřich, prof. RNDr., CSc. University of Economics, Prague, Czech Republic
Slovak University of Technology in Bratislava, Slovak Kováčová Monika, Mgr., PhD. Republic
M. Bel University Banská Bystrica, Poprad, Slovak Kulčár Ladislav, Doc. RNDr. CSc. Republic
Brno, Czech Republic University of Technology, Brno, Kureš Miroslav, Assoc. Prof. Czech Republic
Kvasz Ladislav, Doc. PhD. (Assoc. Prof) University Komenskeho, Bratislava, Slovak Republic
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Marcella Giulia Lorenzi, prof. Università della Calabria, Calabria, Italy
10 Technical University of Cluj- Napoca, Cluj-Napoca, Lungu Nicolae, Prof. Romania
Majstorović Snježana J.J.Strossmayer University, Osijek, Croatia
Mendel University in Brno, Czech Republic, Brno, Czech Mařík Robert, doc. PhD. Republic
Matvejevs Andrejs, Ing., DrSc. Riga, Latvia Technical university, Riga, Latvia
Michálek Jiří, RNDr, CSc. Czech Academy of Sciences, Prague, Czech Republic
Mikeš Josef, Prof. RNDr., DrSc. Palacky University, Olomouc Czech Republic
Myšková Helena, RNDr., PhD. Technical University of Košice, Košice
Nemoga Karol, doc. RNDr., PhD. Slovak Academy of Sciences, Bratislava, Slovak Republic
Mathematical Institute of the Academy, Brno, Czech Neuman František, Prof., RNDr., DrSc. Republic
Slovak University of Technology in Bratislava, Slovak Omachelová Milada, Mgr., PhD Republic
Paun Marius, Prof. Dr. Transilvania University of Brasov, Brasov, Romania
Pavlenko Oksana, Dr. math Riga, Latvia Technical university, Riga, Latvia
Plavka Ján Technical University Košice, Košice, Slovak Republic
Plch Roman, RNDr., PhD. Masaryk University, Brno, Czech Republic
Pokorný Michal, Prof. University of Zilina, Zilina, Slovak Republic
Potůček Radovan, RNDr., PhD. University of Defense, Brno, Czech Republic
University of Hradec Králové, Hradec Králové, Czech Půlpán Zdeněk, Prof., RNDr., PhD.r., CSc. Republic
Račková Pavlína, PhD.r., PhD. University of Defense, Brno, Czech Republic
Rus Ioan A., Professor Babes-Bolyai University, Cluj-Napoca, Romania
University of Žilina, Slovak Republic, Žilina, Slovak Růžičková Miroslava, RNDr. Doc. CSc. Republic
Segeth Karel, Prof., RNDr., CSc. Academy of Sciences, Praha, Czech Republic
Stankovičová Iveta, Ing., PhD. Univerzita Komenského, Bratislava, Slovak Republic
11 Sterba Jan, Ing. University of Economics in Bratislava, Slovak Republic
Svatošová Libuše, Prof. Ing., CSc. Czech University of Life Sciences, Praha, Czech Republic
Slovak University of Technology in Bratislava, Slovak Šooš Ľubomír, Prof. Ing., PhD. Republic
Inštitút Matematiky a Informatiky, Banská Bystrica, Slovak Šuch Ondrej, PhD. Republic
Tarabella Leonello, Researcher Research Area pf CNR, Pisa, Italy
Terek Milan, prof., Ing., PhD. University of Economics, Bratislava, Slovak Republic
Slovak University of Technology in Bratislava, Slovak Trokanova Katarina, Doc. Republic
Tseytlin Georgiy E., PhD. Institute of Program System , Kyiv, Ukraine
Slovak University of Technology, Bratislava, Slovak Urban František, Doc. Ing., CSc. Republic
Vanžurová Alena, Doc. RNDr., CSc. Palacký University, Olomouc, Czech Republic
Slovak University of Technology, Bratislava, Slovak Velichová Daniela, Doc. RNDr., CSc. Republic
Wikovsky Viktor, Doc. RNDr., CSc. Slovak Academy of Sciences, Bratislava, Slovak Republic
Wimmer Gejza, Mgr. Slovak Academy of Sciences, Bratislava, Slovak Republic
12 MEMORY TERM FOR DYNAMIC OLIGOPOLISTIC MARKET EQUILIBRIUM PROBLEM
BARBAGALLO Annamaria, (I), MAUGERI Antonino, (I)
Abstract. The aim of the paper is to provide a more realistic model for the dynamic oligopolistic market equilibrium problems. In particular, a long-term memory is introduced and the corresponding variational inequality model is discussed in order to study the problem in presence of delay. Moreover, existence and regularity results are proved. Key words and phrases. Oligopolistic market equilibrium problem, evolutionary vari- ational inequality, existence and continuity, memory term. Mathematics Subject Classification. Primary 49J40, 49K40, 91B62
1 Introduction
The theory of elastic body was pioneered by Boltzman [9, 10] who gave a first mathematical formulation to hereditary phenomena, where the deformations of a body are studied along with the history of the deformations under which it was subjected in the past. Later Volterra [22, 23, 24] gave his contribution to the theory of elasticity, introducing some hereditary coefficients in form of integral term in the constitutive equations for an elastic body with memory. Starting from the 1960s, see [11, 13], the principle of fading memory was advanced, suggesting that a body is able to recollect only its recent past and thus all the history before can be neglected. As a consequence, the memory term represents only the history in the time interval (0,t)and all the previous events cannot affect the body behavior. Since then, notable applications in different fields have been studied. In economics we many refer to [8], where the dynamics of market adjustment processes are described via a Volterra integral term. More recently, the integral memory term has been used in order to represent some physical characteristics of the quantities involved in mechanical and engineering problems. Aplimat - Journal of Applied Mathematics
For instance, it may describe the relaxation tensor in viscoelastic contact models as in [1], or the conductivity of an electrolyte in electrochemical machining as in [14, 21]. Inspired by these applied problems, in [18] the authors suggest to introduce the memory term in the framework of network equilibria, thus leading to a refinement of the model. In fact, they explicitly incorporate the contribution of flows from the initial time to the observation time t, which causes the presence of the memory term. Hence we are able to analyze how the current equilibrium solution is affected by past equilibria. The same suggestion can concern the dynamic oligopolistic market equilibrium problem. This problem due to Cournot in [12], recently, has been studied in the dynamic case. The dynamic framework has been developed by [6]. In this paper, the variational formulation has been proved, from which the existence and regularity of the dynamic oligopolistic market equilibrium solution has been derived. The continuity of solution allows to provide a computational procedure to compute the equilibrium solution. In the subsequently paper [7], applying the infinite-dimensional duality results developed in [17], the existence of the Lagrange variables, which allow to describe the behaviour of the market, is provided. Furthermore, some sensitivity results has been proved showing that small changes of the solution happen in correspondence of small changes of the profit function.
2 Dynamic Oligopolistic Market Equilibrium
Here we describe, for the reader’s convenience, a dynamic spatial oligopolistic market equi- librium problem, which constitutes an example of imperfect competition and can be viewed as a prototypical game theoretic problem, operating under the Nash equilibrium concept of noncooperative behaviour. Let us consider m firms Pi, i =1, 2,...,m,andn demand markets Qj, j =1, 2,...,n, that are generally spatially separated. Assume that the homogeneous commodity, produced by the m firms and consumed at the n markets, is involved during a period of time [0,T], T>0. Let pi(t), t ∈ [0,T], i =1, 2,...,m, denote the nonnegative commodity output produced by firm Pi at the time t ∈ [0,T]andletqj(t), t ∈ [0,T], j =1, 2,...,n, denote the demand for the commodity at demand market Qj at the same time t ∈ [0,T]. Let xij(t), t ∈ [0,T], i =1, 2,...,m, j =1, 2,...,n, denote the nonnegative commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0,T]. Group the production output into m n a vector-function p :[0,T] → R+ , the demands into a vector-function q :[0,T] → R+, and the mn commodity shipments into a matrix-function x :[0,T] → R+ . Assuming that we are not in presence of production and demand excesses the following feasibility conditions must hold for every i and j and a.e. in [0,T]: n pi(t)= xij(t), (1) j=1 m qj(t)= xij(t). (2) i=1 Hence, the quantity produced by a firm, at the time t ∈ [0,T], must be equal to the sum of the commodity from that firm to all the demand markets, at the same t ∈ [0,T], and the demand
14 volume 3 (2010), number 2 Aplimat - Journal of Applied Mathematics at a demand market, at the time t ∈ [0,T], must be equal to the sum of all the commodity shipments to that demand market, at the same t ∈ [0,T]. Furthermore, assuming that the nonnegative commodity shipment between the supply market Pi and the demand market Qj has to satisfy time-dependent constrains, namely that:
xij(t) ≤ xij(t) ≤ xij(t), ∀i =1, 2,...,m, j =1, 2,...,n, a.e. in [0,T], (3) where x, x ∈ L2([0,T], Rmn), with x(t) < x(t)a.e.in[0,T], are considered known with non-negative values. In this paper we consider the functional setting of the Hilbert space L2([0,T], Rmn). Hence, the set of feasible vectors x ∈ L2([0,T], Rmn)is K = x ∈ L2([0,T], Rmn): 0≤ x(t) ≤ x(t) ≤ x(t), a.e. in [0,T] . (4)
This set is convex, closed and bounded in the Hilbert space L2([0,T], Rmn). Furthermore, associate with each firm Pi a production cost fi and let us consider the more general situation where the production cost of the quantity pi a firm i may depend upon the entire production pattern and the time, namely we assume that
fi = fi(t, p(t)). (5) Similarly, allow the demand price for the commodity at a demand market to depend, in general, upon the entire consumption pattern and the time, namely we assume that
dj = dj(t, q(t)), (6) where we have denoted with p and q the vector-functions given by (1) and (2). Then, we have the following mappings:
2 m 2 m f :[0,T] × L ([0,T], R+ ) → L ([0,T], R+ ), 2 n 2 n d :[0,T] × L ([0,T], R+) → L ([0,T], R+).
Let cij, i =1, 2,...,m, j =1, 2,...,n, denote the transaction cost at the same time t ∈ [0,T], which includes between firm Pi and demand market Qj. Here we permit the transaction cost to depend, in general, upon the entire shipment pattern and the time, namely,
cij = cij(t, x(t)), (7) so we have 2 mn 2 nm c :[0,T] × L ([0,T], R+ ) → L ([0,T], R+ ).
The profit vi(t, x(t)), t ∈ [0,T], i =1, 2,...,m, of firm Pi at the same time t ∈ [0,T] is then n n vi(t, x(t)) = dj(t, q(t))xij(t) − fi(t, p(t)) − cij(t, x(t))xij(t). (8) j=1 j=1 Now consider the dynamic oligopolistic market mechanism, in which the m firms supply the commodity in a noncooperative fashion, each one trying to maximize its own profit at the time t ∈ [0,T]. We seek to determine a nonnegative commodity distribution pattern x for which the m firms will be in a state of equilibrium as defined below.
volume 3 (2010), number 2 15 Aplimat - Journal of Applied Mathematics
Definition 2.1 A commodity shipment distribution x∗ ∈ K is a dynamic oligopolistic market equilibrium if and only if for each i =1, 2,...,m and j =1, 2,...,n and a.e. in [0,T] we have:
∗ ∗ vi(t, x (t)) ≥ vi(t, xi(t), x i (t)), ∀x ∈ K,a.e.in[0,T], (9)
∗ ∗ ∗ ∗ ∗ wherewedenotebyx i =(x1,...,xi−1,xi+1,...,xm), and we set ∗ ∗ ∗ ∗ ∗ vi(t, xi(t), x i (t)) = vi(t, x1(t),...,xi−1(t),xi(t),xi+1(t),...,xm(t)). In the Hilbert space L2([0,T], Rk), let us recall that