Nikolai Azbelev −−− the Giant of Causal

Efim A. Galperin

Universite du Quebec a Montreal, Canada

One of the greatest of all times, Nikolai Viktorovich Azbelev has died. He is one of those great scholars who rise to lasting prominence after leaving this world of fast market values in . He left immortal ideas that are changing mathematics. I met him 20 years ago at the first world Congress of Nonlinear Analysts (1992) in Tampa, USA, and later in Ariel and Athens. We talked a lot about differential equations with deviating arguments, the branch of mathematics not duly recognized at the time and, I am afraid, not clearly understood right now. The classical theories of ODEs and PDEs well presented in all textbooks, monographs and high level articles in mathematics are physically invalid, and Nikolai Azbelev felt it strongly although he could not tell it from the podium of a congress. Even now, it is not common to talk about it. To put things straight, consider the 2nd law of Newton usually formulated in textbooks as follows: ma = mx’’ = F(t, x, v), v = x’ = lim[x(t+ dt) – x(t)]/dt as dt→ 0. Here the mass m is presumed to be constant. For m ≠ const, Georg Buquoy proposed (1812) another formula: mdv + (v – w)dm = F(t, x, v)dt where v – w is the relative velocity with which dm is ejected from a moving body. Both formulas are however non-causal, thus physically invalid, since at any current moment t, the value x(t + dt), dt > 0, does not exist and cannot be known (measured) at a future moment t + dt > t not yet realized. The Newton-Leibnitz right time and partial derivatives produce non-causal equations of rigid frozen evolutions as if the future values x(t + dt) were actually known in the real physical processes. But it is the consideration of the left time derivatives v* = lim[x(t) – x(t− dt)]/dt as dt→ 0, or delayed arguments in the right-hand side as v = x’(t − δ1), v’ = x’’(t − δ2), etc., with δi > 0, dt < δi all i (Azbelev) that presents the causal ODEs and PDEs as valid descriptions of the realistic physical processes that allow us to use controls dependent on the higher order left or delayed derivatives in the right-hand side F(t, x, v, v’, v’’, v’’’, ...) in order to control the motion and effectively alleviate actual disturbances and uncertainties always present in nature. For example, the autopilot control systems for airplanes are now constructed without the use of acceleration assisted control, according to the current textbook formula for the 2nd law of Newton. Such autopilots are dangerous if applied at take-off, at landing, or in bad weather, in which cases the pilot has to take controls since he feels the acceleration and sudden changes of velocity of the plane even if the Pitot tubes fail in flight as already happened on May 31, 2009 in the Airbus A330 flight between Rio de Janeiro and .

It is these ideas of causality and their implementation in mathematics and engineering that present the everlasting legacy of Nikolai Viktorovich Azbelev for further progress in science and technology.

Functional Differential Equations and Applications 2012 On LQR and control of structures using canonical representations

G. Agranovich Department of Electrical and Electronics Engineering

I. Halperin Department of Electrical and Electronics Engineering Department of Civil Engineering

Y. Ribakov Department of Civil Engineering Faculty of Engineering, Ariel University Center of Samaria, Ariel, 40700,

AMS Subject Classi cation: 93C05, 93C15, 93C95, 49K15, 70E55 Keywords and Phrases: modal control, LQR, optimal structural control, algebraic Ric- cati’s equation.

According to the seismic modal approach, optimal modal control design should derive a control signal a¤ecting only the dominant modal coordinates. However, existence of an optimal state feedback, a¤ecting only a selected set of modes, is questionable. This study proves the existence of optimal modal feedback for the case of in nite horizon LQR and control methods. It also formulates a modal design method, based on a corresponding se- lection of LQR weighting and output matrices. New state-space similarity transformations are introduced. E¢ciency of the theoretical results is demonstrated in a numerical example, presenting a modal design of a typical 4 story residential building

Functional Differential Equations and Applications 2012 A binomial identity via di¤erential equations

D. Aharonov and U. Elias

Technion, Haifa, Israel

In the following we discuss a well known binomial identity. Many proofs by di¤erent methods are known for this identity. Here we present another proof which uses linear ordinary di¤erential equations of the rst order.

Functional Differential Equations and Applications 2012 A Numerical method to solve the axisymmetric static Maxwell equations in singular domains

F. Assous Ariel University Center, 40700 Ariel, Israel.

I. Raichik Bar Ilan University, 52900, Ramat Gan, Israel.

We propose a new numerical method to solve the axisymmetric static Maxwell equations in singular domains, as for example a non convex polygonal domain ! belonging to the meridian half-space (r; z). In these conditions, computing for instance the static magnetic eld B = (Br;Bz) consists in nding the divergence-free solution to

@Br @Bz curlB := = f : @z @r together with an ad hoc boundary condition. This problem is singular in the sense that for a non-convex axisymmetric domain !, the space of solutions (says W ) is not a subspace of the H1. Nevertheless, W can be decomposed into two subspaces, i.e. 1 W = WR  WS, where WR is a regular subspace, that is a subspace of H in which one can easily compute a numerical solution. The di¢culty comes from the singular subspace WS, that is a nite-dimensional subspace, the depending on the number of reentrant corners of the domain !. Moreover, a basis function wS of WS can be characterized as the solution to

curl wS = PS 2 !; div wS = 0 2 !; wS   = 0 2 @!:

2 1 Here, the right-hand side PS is singular, that it belongs to L but not to H , and this solves 2 2 @ PS @ PS 1 @PS PS  PS := + + = 0 in !; 0 @r2 @z2 r @r r2 PS = 0 on @!;

Hence, the key point is to compute PS, which can not be solved by a standard nite element method, which would give PS = 0. In this talk, we propose a new method to e¢ciently compute PS and consequently wS. It consists in decomposing the domain ! into 2 subdo- mains, and to derive an ad hoc variational formulation, in which the interface conditions are imposed through a method deduced from a Nitsche approach. Examples to illustrate our method will be shown in the talk.

Functional Differential Equations and Applications 2012 On estimates of alternating-sign solutions to nonlinear di¤erential equations

I. Astashova

Moscow State University of Economics, Statistics and Informatics, Lomonosov State University ast@di¢ety.ac.ru

AMS Subject Classi cation: 34C10, 34C11 Keywords and Phrases: nonlinear ordinary di¤erential equation of higher order, uniform estimates of solutions. Uniform estimates and qualitative behavior of solutions to quasi-linear ordinary di¤er- ential equations of the higher order are described. In particular, to the equation

n 1 (n) (j) k y + X aj(x) y + p(x) jyj sgn y = 0 (1) j=0 with n  1; real (not necessary natural) k > 1; and continuous functions p(x) and aj(x); uniform estimates for positive solutions with the same domain ([1]) are obtained. For alternating-sign solutions the uniform estimates are obtained to the equation

n 1 (n) (j) k y + X aj(x) y + p(x) jyj = 0 (2) j=0 and for some special cases of equation (1).

References

[1] Astashova, I.V. Uniform estimates for positive solutions of quasi-linear ordinary di¤er- ential equations, (English. Russian original) Izv. Math. 72, No. 6, (2008), 1141–1160; from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 6, (2008), 85–104.

Functional Differential Equations and Applications 2012 Oscillation criterion for one class of discrete equations

J. Baštinec, J. Diblík

Brno University of Technology,Brno Czech Republic, [email protected], [email protected]

AMS Subject Classi cation: 39A10, 39A11. Keywords and Phrases: Discrete delayed equation, oscillating solution, positive solution, asymptotic behavior.

We consider the delayed (k + 1)-order linear discrete equation

x(n) = p(n)x(n k) (1) where n 2 Za1 := fa; a + 1;::: g, a 2 N := f1; 2;::: g is xed, x(n) = x(n + 1) x(n), p: Za1 ! R, k 2 N. A solution x = x(n): Za1 ! R of (1) is positive (negative) on Za1 if x(n) > 0 (x(n) < 0) for every n 2 Za1. A solution x = x(n): Za1 ! R of (1) is oscillating on Za1 if it is not positive or negative Z a1 Z on a11 for an arbitrary 2 a1. Let us de ne the expression lnq t, q  1, by lnq t = ln(lnq 1 t), ln0 t  t. The equation (1) is known to have a positive solution if the sequence p(n) satis es an . Our aim is to show that, in the case of the opposite inequality for p(n),

k k 1 k k k p(n)   + + +    + ; k + 1 k + 1 8n2 8(n ln n)2 8(n ln n : : : ln n)2    q  assuming  > 1, all solutions of the equation (1) are oscillating for n ! 1.

Acknowledgement: This research was supported by the grants P201/10/1032 and P201/11/0768 of the Czech Grant Agency (Prague) and by the project FEKT-S-11-2(921) Brno University of Technology.

Functional Differential Equations and Applications 2012 On global stability for some nonlinear functional di¤erential systems

Leonid Berezansky

Ben Gurion University of the Negev Beer-Sheva, Israel

2000 MSC: 34K20

For the vector functional di¤erential equatuion dx A t x t F x t : dt = ( ) ( ) + ( )( ) (1) where F is a nonlinear causal (Volterra) operator we discuss some new global stability results. There results are applied for nonautonomous Mackey-Glass models.

Functional Differential Equations and Applications 2012 Integro-di¤erential equations in high energy scattering

S.Bondarenko

Department of Physics, Ariel University Center, Ariel, Israel

We will talk about the special class of integro-di¤erential equations arising in high-energy physics. These equations we can de ne as a generalization of Lotka-Volterra equations at the case of non-local interactions between the functions when two-value boundary conditions are given. We discuss numerical solutions of our equations for some special case and properties of the solutions on an example of oversimpli ed model of the same equations in zero transverse .

Functional Differential Equations and Applications 2012 On the best constants in the solvability conditions of the periodic problem

Eugene Bravyi 

Perm State Technical University, Perm,

AMS Subject Classi cation: 34K06, 34K10. Keywords and Phrases: Functional di¤erential equations, boundary value problems.

In 1995 A. Lomtatidze and S. Mukhigulashvili obtained the following result: the periodic

x (t) = (T x)(t) + f(t); t 2 [0;!]; x(0) = x(!); x_(0) =x _(!); has a unique solution for all f 2 L[0;!] and for all linear bounded positive operators T : C[0;!] ! L[0;!] with a given norm kT k = T > 0 i¤ T  16=!. Here we consider the periodic problem for a perturbed (a > 0) equation with negative feedback:

x (t) + a2x(t) = (T x)(t) + f(t); t 2 [0;!]; x(0) = x(!); x_(0) =x _(!):

This problem has a unique solution for all f 2 L[0;!] and for all linear bounded positive operators T : C[0;!] ! L[0;!] with a given norm kT k = T i¤

a! a! 16 4 cot( 4 ) for 0 < a! < 2; T   a! a! N ! ( 8 jsin( 2 )j for 2 < a! 6= 2k; k 2 :

Supported by Grant 10-01-96054-r-ural-a of The Russian Foundation for Basic Research

Functional Differential Equations and Applications 2012 On mathematical model for the treatment of chronic myelogenous (myeloid) leukemia (CML)

Svetlana Bunimovich

Ariel University Center of Samaria, Ariel, Israel

We propose and analyze a mathematical model for the treatment of chronic myelogenous (myeloid) leukemia (CML), a cancer of the blood. We introduce combined treatment of CML based on Imatinib therapy and Immunotherapy. Imatinib therapy is a molecular targeted therapy that inhibits the cell, involved in the chronic CML pathogenesis. Immunotherapy based on interferon alfa-2a e¤ects the cancer cells mortality and leads to improvement out- come of the combined therapy. The system of di¤erential equations was used to model the interaction between CML cancer cells and e¤ector cells of the immune system in the human body. We introduced biologically motivated time-varying delays in the treatment terms. The proposed model belongs to a special class of nonlinear nonautonomous systems of ordinary di¤erential equations (ODEs). The analysis of the described system shows the existence of a unique global positive solution, existence of a unique nontrivial equilibrium, explicit local and global stability conditions for the nontrivial equilibrium.

Functional Differential Equations and Applications 2012 Multiplicity results for the periodic Hill’s equation

Alberto Cabada

Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain. [email protected]

AMS Subject Classi cation: 34B15, 34B16, 34B27. Keywords and Phrases: Hill’s equation; periodic boundary value problem; second order singular equation.

In this talk we present some existence and multiplicity results for the periodic boundary value problem

x00(t) + a(t) x(t) =  g(t) f(x) + c(t); x(0) = x(T ); x0(0) = x0(T ); where  is a positive parameter. The function f : (0; 1) ! (0; 1) is allowed to be singular at x = 0 and the related Green’s function does not change its sign. Moreover we will make a survey on the known results that imply that the Green’s function is nonnegative on its square of de nition.

This work was partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.

References

[1] A. Cabada and J. A. Cid, Existence and Multiplicity of Solutions for a Periodic Hill’s Equation with Parametric Dependence and Singularities. Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 pages, 2011.

Functional Differential Equations and Applications 2012 p–Laplacian discrete boundary value problems on bounded and unbounded intervals

Alberto Cabada Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain. [email protected]

AMS Subject Classi cation: 39A10, 39A12, 47J30, 58E05 Keywords and Phrases: Di¤erence equations, Discrete p–Laplacian, Variational methods, Heteroclinic solutions. In this talk we present some applications of the critical point theory to deduce the existence of solutions for certain discrete problems. We will speak about the existence of some  > 0 for which there are nontrivial solutions of the non autonomous equation px(k 1) =  f(k; x(k)); (1) coupled with one of the following boundary value conditions x(0) = x(T + 1) = 0; (2) or x(0) = 0; x(1) = 1: (3) Here p > 1 is a given real number, and p 2 px(k 1) =  jx(k 1)j x(k 1)  'p(x(k 1)); is the classical p – Laplacian operator. Concerning the problem (1) – (2), under suitable assumptions on the function f, in [1] it is proved the existence of a positive  for which this problem admits at least three solutions. The proof follows from critical point theory. Problem (1) – (3) has been considered in [2]. In this case it is showed that, by using some sign assumptions in function f, there is a positive solution for all  > 0. The solution is attained as the limit of a sequence of solutions of related homogeneous Dirichlet problems in bounded intervals.

References

[1] A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value prob- lems, J. Math. Anal. Appl. 356 (2009), 418–428. [2] A. Cabada and S. Tersian, Existence of heteroclinic solutions for discrete p – laplacian problems with a parameter, Nonlinear Anal. Real World Appl. (2011). doi:10.1016/j.nonrwa.2011.02.022 This work was partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2010- 15314.

Functional Differential Equations and Applications 2012 Smooth solutions of some linear functional di¤erential equations

Valery Cherepennikov

Melentiev Energy System Institute of Sib. Dep. RAS. Irkutsk. Russia

AMS Subject Classi cation: 34K06, 34K10.

The paper considers initial and boundary value problems for the following scalar func- tional di¤erential equations: x_(t) = a(t)x(t ) + f(t);   0 const; x_(t) + p(t)_x(t 1) = a(t)x(t 1) + b(t)x(t) + f(t); x_(t) = a(t)x(t (1 "t)) + b(t)x(t) + f(t); x_(t) = a(t)x(t 1) + b(t)x(t + 1) + d(t)x(t) + f(t); x_(t) = a(t)x(t 1) + b(t)x(t=s) + d(t)x(t) + f(t); s > 1: Here t 2 R, the coe¢cients of equations are represented in the form of polynomials. One can investigate other equations as well. Smooth solutions to these problems are studied by the method of polynomial quasiso- lutions (PQ-solutions) [1, 2]. The method is based on the representation of the unknown N n function in the form of polynomial x(t) = Pn=1 xnt , whose substitution into each of the N equations results in the residuals (t)s = O(t ). The paper investigates the problems of existence of PQ-solutions of di¤erent degrees, an algorithm for nding unknown coe¢cients xn as well as exact formulas for the residuals, which allow one to characterize the measure of disturbance for the considered problems. The results obtained are illustrated by examples.

References

[1] V.B. Cherepennikov. Analytic solutions of some functional di¤erential equations linear systems, Nonlinea Analysis, Theory, Methods & Applications. 30/5, (1997), 2641-2651. [2] V.B. Cherepennikov, P.G. Ermolaeva, Polynomial quasisolutions of linear di¤erential dif- ference equations, Opuscula Mathematica, 26/3, AGH Univ. of Science and Technology, Kracow, (2006), 47-57.

Functional Differential Equations and Applications 2012 Classical solutions on a cylindrical domain to quasilinear hyperbolic functional di¤erential equations

W. Czernous

Institute of Mathematics, University of Gdansk, Gdansk, Poland

AMS Subject Classi cation: 35R10, 35L45. Keywords and Phrases: partial functional di¤erential equations, classical solutions, local existence, bicharacteristics, cylindrical domain, well-posedness.

We break the tradition of setting the domain as a Cartesian product of real intervals, and we give a new set of conditions on the possibly unbounded domain with Lipschitz di¤erentiable boundary. Well-posedness is then relying on a variant of normal vector condition. Consequently, negative invariance of a neighbourhood of is shown, which enables us to use the method of bicharacteristics. With local assumptions on equation coe¢cients, we prove local existence and continuous dependence on data of classical solutions on initial boundary value problem. Regularity of solutions matches this of the domain, and the proof uses the Banach xed- point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-di¤erential equations. Nevertheless, presented existence result is new also in the case without functional dependence.

Functional Differential Equations and Applications 2012 Oscillation criterion for one class of discrete equations

J. Diblík, M. R°uµziµcková, Z. Šutá

Zilinaµ University, Zilina,µ Slovak Republic,

AMS Subject Classi cation: 39A10, 39A11. Keywords and Phrases: Discrete equation, delay, asymptotical convergence, increasing solution.

A system of s discrete equations

y(n) = (n)[y(n j) y(n k)] is considered where k and j are integers, k > j  0, (n) is a real s  s square de ned for n  n0 k, n0 2 Z with non-negative elements ij(n), i; j = 1; : : : ; s such s T s that Pj=1 ij(n) > 0, y = (y1; y2; : : : ; ys) : fn0 k; n0 k + 1;::: g ! R and y(n) = y(n + 1) y(n) for n  n0. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions, i.e., for every solution y(n) de ned for all su¢ciently large n, there exists a nite limit limn y(n). Moreover, the asymptotic convergence of all solutions is equivalent to the existence!1 of one asymptotically convergent solution with increasing coordinates. A relation to the so-called critical case known for scalar equations will be discussed as well. Acknowledgement: This research was supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).

Functional Differential Equations and Applications 2012 Maximum principles and stability of delay di¤erential equations

Alexander Domoshnitsky

Ariel University Center, Ariel, Israel, [email protected]

In this talk we discuss maximum principles for systems of functional di¤erential equa- tions. A connection of maximum principles with nonoscillation and positivity of the Cauchy functions is demonstrated. The method to compare only one component of the solution vector of linear functional di¤erential systems, which does not require heavy sign restric- tions on their coe¢cients, is proposed. Necessary and su¢cient conditions of the positivity of elements in a corresponding row of the Cauchy and Green’s matrices are obtained in the form of theorems about di¤erential inequalities. Tests of the exponential stability of functional di¤erential systems are obtained on this basis. The main idea of our approach is to construct a rst order functional di¤erential equation for one of the components of the solution vector and then to use assertions about positivity of its Green’s functions.

Functional Differential Equations and Applications 2012 Stability of equilibrium of di¤erential systems with quadratic right-hand side

Irada Dzhalladova

Kyiv National Economical University, Ukraine [email protected]

AMS Subject Classi cation: 93D05, 34D05 Keywords and Phrases: Di¤erence Lotka-Volterra equations, stability analysis, di¤erence systems with quadratic right-hand side, stability estimates.

Nonlinear systems of di¤erence Lotka-Volterra equations with quadratic right-hand non- linearity and asymptotically stable linear part are considered. One of the basic methods for stability analysis of trivial solution of nonlinear systems are methods of linearization and stability studying based on stability results for linear approximation system. If the trivial solution of linear approximation system is asymptotically stable, then trivial solution of the initial nonlinear system will be also stable in a su¢ciently small neighborhood of the equi- librium. Di¤erence systems with the quadratic right-hand side are considered in the report. The systems are presented in the uniform vector-matrix form. An algorithm of estimation of stability region in the phase space of trivial equilibrium of the system is proposed.

Functional Differential Equations and Applications 2012 Time evolution of spin exchange with a time delay

D. Gamliel, Department of Medical Physics, Ariel University Center of Samaria, Israel,

A. Domoshnitsky, Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel

R. Shklyar Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel

AMS Subject Classi cation: 34K06,34C26 Keywords and Phrases: Magnetic resonance, delay di¤erential equations, Lambert func- tion, Cauchy matrix

The NMR (nuclear magnetic resonance) spectrum of a spin system is a¤ected by exchange processes in the system, and thus enables investigation of the exchange process. In a previous work we considered a generalization of spin exchange in which the jump process takes a non- negligible time, so the system is described by di¤erential equations with a time delay. In this paper some characteristics of the solutions are studied, rst by using the Lambert function and then by using the Cauchy matrix approach.

Functional Differential Equations and Applications 2012 Normal forms and asymptotic solution of nonhomogeneous, nonautonomous quasilinear di¤erential equations

Ya. Goltser

Ariel University Center of Samaria, Ariel, Israel

We present a method for asymptotic expansion of the solution of the quasi-linear paramet- rically perturbed systems. Subject under consideration: i) Formal equivalent systems, ii) Normal forms of quasilinear asymptotic expansion of the solution, iv) Cases that the linear homogeneous part of di¤erential system has exponential di- chotomy or exponential trichotomy.

Functional Differential Equations and Applications 2012 Fredholm type theorem for systems of functional-di¤erential equations with positively homogeneous operators

Robert Hakl

Institute of Mathematics, Academy of of the Czech Republic, branch in Brno, Ziµ µzkova 22, 616 62 Brno, Czech Republic

AMS Subject Classi cation: [2000]34K10 Keywords and Phrases: Functional-di¤erential equations, boundary value problems, Fredholm type theorems Consider the system of functional-di¤erential equations

ui0 (t) = pi(u1; : : : ; un)(t) + fi(u1; : : : ; un)(t) for a. e. t 2 [a; b](i = 1; : : : ; n) (1) together with boundary conditions

`i(u1; : : : ; un) = hi(u1; : : : ; un)(i = 1; : : : ; n): (2)

n Here, pi; fi : C[a; b]; R 2 [a; b]; R are continuous operators satisfying Carathéodory condition, i.e., they are bounded on every ball by an integrable function, and `i; hi : n C[a; b]; R ! R are continuous functionals which are bounded on every ball by a con- stant. Furthermore, we assume that pi and `i satisfy the folowing condition: there exist positive real numbers ij and i such that ijjm = im whenever i; j; m 2 f1; : : : ; ng, and for every c > 0 and uk 2 [a; b]; R (k = 1; : : : ; n) we have

i1 in cpi(u1; : : : ; un)(t) = pi(c u1; : : : ; c un)(t) for a. e. t 2 [a; b];

 i1 in c i `i(u1; : : : ; un) = `i(c u1; : : : ; c un):

By a solution to (1), (2) we understand an absolutely continuous vector-valued function n n (ui)i=1 :[a; b] ! R satisfying (1) almost everywhere in [a; b] and (2).

Functional Differential Equations and Applications 2012 Partial stability of linear stochastic functional di¤erential equations and N.V. Azbelev’s w-transforms

Ramazan I. Kadiev1 and Arcady Ponosov2

1Dagestan Scienti c Center, Russian Academy of Sciences, Makhachkala 367005, Russia 2Department of Mathematical Sciences and Technology Norwegian University of Life Sciences, NO-1432 Ås, Norway

AMS Subject Classi cation: 34K50, 34D20. Keywords and Phrases: Partial stability, stochastic di¤erential equations, aftere¤ect, semimartingales, integral transforms.

The concept of partial Lyapunov stability was introduced by A. M. Lyapunov himself. Informally speaking, partial stability means that only a part of the variables in a system of di¤erential equations is Lyapunov stable, while the remaining variables may behave arbi- trarily. A mainstream framework of studying partial stability is based on modi cations of the direct (second) Lyapunov method. In our analysis we demonstrate how partial Lyapunov stability for linear stochastic func- tional di¤erential equations can be formulated in terms of a di¤erent kind of stability: partial input-to-state stability. We also show how N.V.Azbelev’s W-transformations can be used to study partial input-to-state stability. As a result of applying this technique, e¢cient conditions of asymptotic and exponential partial Lyapunov stability for general and speci c stochastic functional di¤erential equations are o¤ered.

Functional Differential Equations and Applications 2012 Representation of solutions for linear stationary systems with one delay

D. Khusainov Taras Shevchenko National University of Kyiv, Ukraine Y. Bastinec, G. Piddubna Brno University of Technology, Czech Repablic [email protected]

Systems with commuting matrices. The Cauchy problem is considered for a system x_ (t) = Ax (t) +Bx (t ) +f (t) ; x (t) = ' (t) ;   t  0; with matrices A and B satisfying the commutativity condition AB = BA. A special matrix function, called a delayed exponential, is used.

De nition. The delayed exponential function exp fB; tg is a matrix function de ned as  ; 1 < t < 

exp fB; tg = I;   t < 0 2 k 8 t 2 (t ) k [t (k 1)] < I + B 1! + B 2! +    + B k! ; (k 1)  t < k where k = 0; 1; 2; :::,: and  is a zero matrix. By using this function, a solution of the Cauchy problem can be written as

x (t) = exp fA (t t0)g exp fB1; t g ' () 0

+ exp fA (t  s)g exp fB1; t  sg ['0(s) A'(s)] ds Z t

+ exp fA (t  s)g exp fB1; t  sg f(s)ds: Z0

General systems. For general system a fundamental matrix of solutions X0 (t) for homoge- neous systems of linear equations can be represented as follows

n 1 X0(t) = I + 'k(t); (n 1) < t  n; k X=0 k+1 i (t k) 1 i (t k) k where 'k(t) = k! A (k+i+1)! B (A + B): A solution of the Cauchy problem for i=0 inhomogeneous system is P 0 t

x(t) = X0(t)'() + X0(t  s)'0(s)ds + X0 (t  s) f (s) ds: Z Z0

Functional Differential Equations and Applications 2012 Two-point boundary balue problems for second order linear di¤erential equation

Roman Koplatadze

Department of Mathemetics of State University Tbilisi, Georgia

AMS Subject Classi cation: 34K10.

Keywords and Phrases: Boundary value problem; Singular di¤erential equation.

Consider the problem

u00 + p(t)u = f(t); (1) u(0) = ; u(1) = ; (2) where p 2 C((0; 1); R), f 2 C([0; 1]; R), ; 2 R. The existence of a solution of the singular di¤erential equation (1), satisfying the condi- tion (2) is established.

Functional Differential Equations and Applications 2012 Sharp real-part theorems for derivatives of analytic functions

Gershon Kresin Department of Computer Science and Mathematics Ariel University Center of Samaria, Israel [email protected]

It is assumed that the boundary values of the real part of analytic functions in the unit disk

p D and the upper half-plane C+ are in L . Representations for the sharp coe¢cient in an estimate of the modulus of the n-th deriv- ative of analytic functions in D and C+ are obtained. The maximum of a bounded factor in the representation of the sharp coe¢cient for analytic functions in D is found. Thereby, a pointwise estimate of the modulus of the n-th derivative of an analytic function in D with a best constant is given. The representation for the sharp coe¢cient in the estimate of the modulus of the n-th derivative of analytic functions in C+ is concretized for some n and p. In particular, for p = 1 and for derivatives of odd order of analytic functions in C+, an explicit formula for the sharp coe¢cient is found. Also, explicit formulas for the sharp coe¢cient in the estimate of the modulus of the rst derivative of analytic functions in D and C+ are derived. In the case of analytic functions in D under the assumption that p 2 (1; 1), the coe¢cient is represented as the product of monotonic functions of jzj. A limit relation for the sharp coe¢cient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in C+. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary.

Functional Differential Equations and Applications 2012 Control in di¤erence-di¤erential equations with distributed parameters

Oleksandra Kukharenko

Taras Shevchenko National University of Kyiv, Ukraine e-mail: [email protected]

AMS Subject Classi cation: 35Q93, 35K20, 35R10 Keywords and Phrases: Control problem, delay wave equation, steady state control.

The control problem for di¤erence-di¤erential wave equation with distributed parameters is considered in the presented paper. Preliminary, an equation without delay is considered. The rst boundary value problem is solved using the method of separation of variables. Next, delay wave equation is considered. The rst boundary value problem is solved using the Fourier method and special functions, called the delay sine and cosine functions. Finally, the steady state control problem is solved.

Functional Differential Equations and Applications 2012 On positivity of the Green’s operator of two-point boundary value problem for functional-di¤erential equation

S. Labovskiy Moscow State University of Economics, Statistics and Informatics Moscow, Russia

Positive de niteness of the quadratic functional l 2 2 E(u) = (u0 + qu )dx 0 Z under boundary conditions u(0) = u(l) = 0 is equivalent to positiveness of the Green function for the Euler equation

u00 + qu = 0; x 2 [0; l]; under boundary conditions . Both a¢rmations are equivalent to the nonoscillation of the equation . For the deviating equation l u00 + u(t)dtr(x; t) = 0 0 Z the mentioned above a¢rmations are not equivalent. Under certain symmetry condition the equation is the Euler equation for the quadratic functional l l l 2 E(u) = u0 dx + dx u(x) u(t)dtr(x; t): 0 0 0 Z Z Z It may be positive de nite, while the Green function of the problem , can change sign. In many articles this question was considered for delay equations, see for example, [?],[?],[?]. Here we consider the question about positivity of the Green function of the problem under nondecreasing condition of the function r(x; t) with respect to t. For general case a necessary and su¢cient condition of positivity of the Green function in terms of solv- ability of auxiliary boundary value problems is obtained. These problems are considered for the same di¤erential equation with boundary conditions

u(0) = 0; u0(0) = 0 and u(l) = 1; u0(l) = 1:

1

Functional Differential Equations and Applications 2012

Functional Differential Equations and Applications 2012 Representation formula for solution of a functional equation with volterra operator

Elena Litsyn

Department of Mathematics, Ben Gurion University of the Negev,Beer Sheva, 84105, Israel. Supported in part by the KAMEA program.

The following functional equation is under consideration,

Lx = f (1) with a linear continuous operator L, de ned on the Banach space X0( 0; 0; 0; Y0) of 0 functions x : 0 ! Y0 and having values in the Banach space X2( 2; 2; 2; Y2) of 2 functions x : 2 ! Y2. The peculiarity of X0 is that the convergence of a sequence 0 2 0 2 xn X0; n = 1; 2; :::; to the function x X0 in the norm of X0 implies the convergence 0 ! 0 2 xn(s) x (s); s 0, 0-almost everywhere. The assumption on the space X2 is that it is an ideal space. The suggested representation of solution to (1) is based on a notion of the Volterra property together with a special presentation of the equation using an isomorphism between X0 and the direct product X1( 1; 1; 1; Y1)  Y0 (here X1( 1; 1; 1; Y1) is the 1 Banach space of measurable functions x : 1 ! Y1). The representation X0 = X1  Y0 leads to a decomposition of L : X0 ! X2 for the pair of operators Q : X1 ! X2 and A : Y0 ! X2. A series of basic properties of (1) is implied by the properties of operator Q.

Functional Differential Equations and Applications 2012 Positivity of solutions to neutral functional di¤erential equations

A.Maghakyan

Ariel University Center of Samaria, Ariel, Israel and Bar Ilan University, Ramat Gan, Israel

Conditions that solutions of the rst order neutral functional di¤erential equation

(Mx)(t)  x0(t) (Sx0)(t) (Ax)(t) + (Bx)(t) = f(t); t 2 [0;!]; ! ! are nondecreasing are obtained. Here A : C[0;!] L[01;!] , B : C[0;!] L[01;!] and S : ! L1[0;!] L[01;!] are linear continuous operators, A and B are positive operators, C[0;!] is the space of continuous functions and L[01;!] is the space of essentially bounded functions de ned on [0;!]: In results on positivity of neutral equations it was assumed that the operator S was a positive one. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.

Functional Differential Equations and Applications 2012 Unsolved mysteries of solutions to PDEs near the boundary

Vladimir Maz’ya

University, Department of Mathematics [email protected]

Department of Mathematical Sciences , UK [email protected]

Throughout its long history, specialists in the theory of partial di¤erential equations gained a deep insight into the boundary behaviour of solutions.Yet despite the apparent progress in this area achieved during the last century, there are fundamental unsolved problems and surprising paradoxes related to solvability, spectral, and asymptotic properties of boundary value problems in domains with irregular boundaries. I shall formulate some challenging questions arising naturally when one deals with unrestricted, polyhedral, Lipschitz graph, fractal and convex domains.

Functional Differential Equations and Applications 2012 New and old spectral criteria for the Schroedinger operator

Vladimir Maz’ya

University, Sweden Department of Mathematics [email protected]

Department of Mathematical Sciences University of Liverpool, UK [email protected]

The lecture is a survey of the conditions on the potential responsible for various spectral properties of the Schroedinger operator: positivity and strict positivity, semiboundedness, descreteness of the spectrum, form-boundedness, niteness and descreteness of the negative spectrum, etc.

Functional Differential Equations and Applications 2012 Particle growth in a subdi¤usive medium

Alexander Nepomnyashchy, Technion, Haifa, Israel

Vladimir Volpert Northwestern University, USA

During the last decades, the phenomenon of anomalous di¤usion has attracted much attention of researchers. A subdi¤usive transport has been observed in numerous physical and biological systems, speci cally in gels. We investigate the growth of a solid nucleus due to the subdi¤usive transport of a dissolved component towards the nucleus surface. The process is described by a subdi¤usive version of the Stefan problem. In planar and spherical cases, exact self-similar solutions of the problem have been found in terms of the Wright function. An instability of the particle growth, which is similar to the Mullins-Sekerka instability of a crystallization front in the case of a normal di¤usion, is revealed.

Functional Differential Equations and Applications 2012 Stability of solutions of a linear di¤erential delayed matrix system

D. Khusainov,J. Baštinec, G. Piddubna

T. Shevchenko National University of Kyiv, Kyiv, Ukraina Brno University of Technology, Brno, Czech Republic, [email protected], [email protected], [email protected]

AMS Subject Classi cation: 39A10, 39A11. Keywords and Phrases: Delayed di¤erential equation, Lyapunov’s second method of stability, asymptotic stability.

Let us consider the equation

X_ (t) = A0X(t) + A1X(t ); (1) with the initial condition X(t)  I;   t  0; where A0;A1 are square matrixes, I is unit matrix,  > 0 is constant delay. We will investigate the stability of delayed equation (1) with Lyapunov’s second method. Lyapunov’s functional is constructed in the form: V (x) = xT (t)Hx(t); where H is a sym- metric, positive de ne matrix.

Theorem 1 If there exist a symmetric, positive de nite matrix H, such that

max(H) min(C) 2jHA1j > 0; s min(H)

T where A0 H + HA0 = C, then zero solution x(t)  0 of the system (1) is asymptotically stable for any  > 0.

Acknowledgement: This research was supported by the grants P201/10/1032 and P201/11/0768 of the Czech Grant Agency (Prague) and by the project FEKT-S-11-2(921) Brno University of Technology.

Functional Differential Equations and Applications 2012 On the applicability of the theory of inventive problem solving to the functional di¤erential equations theory

M.A. Plaksin National Research University Higher School of Economics —- Perm Branch, Perm, Russia e-mail: [email protected]

V.P. Plaksina Perm National Research Polytechnic University, Perm, Russia e-mail: [email protected]

Key words: functional di¤erential equations, theory of inventive problem solving (TRIZ), laws of development of technical systems, modelling. AMS (MOS) Subject Classi cation: 34K05, 93A30.

The report discusses the methodology of abstract theory of functional di¤erential equa- tions (FDE) in terms of the theory of inventive problem solving (TRIZ). TRIZ has formulated a row of laws of the development of technical systems. These laws are objective and can be used for analyzing and forecasting the development of systems of di¤erent kinds. The report put forward a hypothesis about the possibility of applying the laws of devel- opment of technical systems for analyzing and forecasting the development of mathematical systems, in particular, the theory of abstract functional di¤erential equations. Examples of application of the laws are given (in particular, from the works of N.V. Azbelev and contemporary works of Perm Seminar on the FDE theory). There are the law increasing ideality, the law of the S-curve system development, the law of folding and unfolding, the law of ousting of the human from system, the laws increasing dynamism and controllability. The di¤erences between mathematical systems and the systems of other kinds are con- sidered.

Functional Differential Equations and Applications 2012 About one problem in theory of singular functional di¤erential equations.

I.M.Plaksina

Perm State Technical University, Perm, Russia

AMS Subject Classi cation: 34K06, 34K26, 47A53. Keywords and Phrases: functional di¤erential equations, singular equations, Fredholm property.

This article discusses some problems of functional di¤erential equations with singularity of special type. These are conditions of Fredholm property and of solvability.

Functional Differential Equations and Applications 2012 Stabilization through a control in integral form

A.Domoshnitsky and N.Puzanov Ariel University center of Samaria, Ariel, Israel

A.Maghakyan Ariel University center of Samaria, Ariel, Israel and Bar Ilan University, Ramat Gan, Israel

In this talk we demonstrate that the problem of controlling chaos, which is of great theoretical and practical importance, can be reduced to the stability analysis of integro-di¤erential equations. We consider a simple con guration of a magneto-elastic beam and a two magnet system, known as ”Moon’s beam”. Then we consider the cases of Lorenz and Rossler attractors. Sim- ple stabilization is proposed in all these cases. In order to obtain stability result we propose a method of reduction of integro-di¤erential equations to systems of ordinary di¤erential equations and then use their stability analysis.

Functional Differential Equations and Applications 2012 The Gardner method for symmetries

Alexander Rasin

Weizmann Institute, Rehovot, Israel

The Gardner method, traditionally used to generate conservation laws of integrable equa- tions, is generalized to generate symmetries. The method is demonstrated for the KdV. The method involves identifying a symmetry which depends upon a parameter; expansion of this symmetry in a (formal) power series in the parameter then gives the usual in nite hierarchy of symmetries. We show that the obtained symmetries commute.

Functional Differential Equations and Applications 2012 On the exponential stability of neutral functional di¤erential equations-prerequisites

Vladimir Rasvan¼ University of Craiova, Romania [email protected]

AMS Subject Classi cation: 34K40, 34K20, 39A06, 39A30. Keywords and Phrases: Neutral functional di¤erential equations, di¤erence operator, Perron condition, Persidskii theorem

This report is concerned with the exponential stability properties of the NFDE (neutral functional di¤erential equations) with time varying coe¢cients. The standard structure of the NFDE is considered, that strongly relies on the di¤erentiation of the di¤erence operator in the normal Cauchy form. The rst problem to be considered is the Perron condition for NFDE. While for RFDE (retarded functional di¤erential equations) the result is known as Perron Halanay theorem, for NFDE this result is conditioned by its validity for the di¤erence operator that is associated (V. R. Nosov, 1981). Starting from this fact and stimulated by a remark of J. Hale that the qualitative theory of time varying di¤erence operators is a task for the medium or remote future (at that time), there was considered rst the Perron condition for di¤erence equations in continuous time. The uniform asymptotic stability thus obtained is proved to be exponential by establishing a theorem of Persidskii type for the same di¤erence equations with continuous time. The results of the present report allow thus obtaining exponential stability for the time varying di¤erence operator. Since in the constant coe¢cient case it was proved that “a NFDE with stable D-operator is retarded (Sta¤ans, 1983), it is a challenge to see the persistence of this property in the time varying case.

Functional Differential Equations and Applications 2012 Positively invertible operators and solvability of boundary value problems for functional di¤erential equations

A. Rontó

Institute of Mathematics, Academy of Sciences of Czech Republic Ziµ µzkova 22, 61662 Brno, Czech Republi

We show how a number of general solvability conditions for various boundary value prob- lems for functional di¤erential equations can be obtained in a uni ed way by using order- theoretical considerations.

Functional Differential Equations and Applications 2012 Periodic successive approximations with interval halving

A. Rontó

Institute of Mathematics, Academy of Sciences of Czech Republic Ziµ µzkova 22, 61662 Brno, Czech Republi

The work concerns the periodic successive approximations method aimed at a constructive investigation of periodic solutions of systems of ordinary di¤erential equations with Lip- schitzian non-linearities. We show how, under quite general assumptions, a suitable interval halving and parametrization technique can weaken the convergence condition of the scheme so that the critical value for the Lipschitz constant is doubled.

Functional Differential Equations and Applications 2012 Multiplicity of solutions in nonlinear boundary value problems

F. Sadyrbaev

Daugavpils University, Latvia [email protected]

We treat mainly the two-point nonlinear boundary value problems for the second order ordinary di¤erential equations. The interrelation of the upper and lower functions, types of solutions and multiplicity of solutions will be discussed in the framework of the history of Riga research group working in the eld of BVPs for ODE.

Functional Differential Equations and Applications 2012 Boundary value problems for a super-sublinear asymmetric oscillator: exact number of solutions.

A. Gritsans, F. Sadyrbaev

Daugavpils University, Latvia [email protected]

Properties of asymmetric oscillator described by the equation

+ p q x = (x ) + (x) (1)

where p  1 and 0 < q  1 are studied.

A set of (; ) such that the problem (1); x(0) = 0 = x(1) (2); jx0(0)j = (3) has a nontrivial solution, is called spectrum. We give full description of spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (1); (2) is given.

Functional Differential Equations and Applications 2012 Homogenization of the di¤usion equation with nonlinear ‡ux condition on the interior boundary of a perforated domain - the in‡uence of the scaling on the nonlinearity in the e¤ective sink-source term

T.A. Shaposhnikova

Moscow State University, Moscow, Russia

We study the asymptotic behavior of solutions u" of the initial boundary value problem for n that parabolic equations in perforated domains "  R ; n  3, with nonlinear third type boundary condition @ u" + " (x; u") = " g(x) on the boundary of the cavities. It is supposed that the perforations are balls of radius " , = n=(n2), periodically distributed with period ". As " ! 0, the microscopic solutions can be approximated by the solution of an e¤ective equation on the domain . The e¤ective equation contains a new sink/source term repre- senting the macroscopic contribution of the processes on the boundary of the microscopic cavities.

Functional Differential Equations and Applications 2012 The life and work of

Tatyana Shaposhnikova

University, Sweden Department of Mathematics [email protected]

Jacques Hadamard lived a long life: he was born in 1865 and died in 1963. As a boy, he hated mathematics, had a passion for botany and played the violin. As an adult, he became a great without losing his childhood interests. He founded a famous seminar, traveled worldwide, fought for justice and human rights and at the same time made contri- butions which became landmarks in various domains of mathematics. He worked on function theory, number theory, partial di¤erential equations, geometry, analytical mechanics, elas- ticity, hydrodynamics, calculus of variations, algebra, psychology, etc. I’ll tell the story of his life, rich with joys and sorrows and survey his principal mathematical achievements.

Functional Differential Equations and Applications 2012 for higher order elliptic systems with BMO assumptions on the coe¢cients and the boudary

Tatyana Shaposhnikova

University, Sweden Department of Mathematics [email protected]

Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coe¢cients. The main result gives a sharp condition on the local mean oscillation of the coe¢cients of the di¤erential operator and the unit normal to the boundary (automatically satis ed if these functions belong to the space VMO) which guarantee that the solution operator associated with this problem is an isomorphism.

Functional Differential Equations and Applications 2012 On null-controllability of interconnected evolution systems with unbounded input operators

B. Shklyar

Holon Institute of Technology, Israel

Exact null-controllability conditions for linear control system consisting of two serially con- nected abstract control evolution systems with unbounded input operators are presented. Applications to interconnected Euler-Bernoulli beam equation with boundary smooth con- trol and heat equation with square -integrable distributed control are considered.

Functional Differential Equations and Applications 2012 Di¤erential Equations with Uncertain Coe¢cients

A. Domoshnitsky Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel [email protected]

R. Shklyar Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel [email protected]

In the system of di¤erential equations modelling real processes we usually do not know exactly their coe¢cients since they obtained as a result of corresponding measurements or regression procedures.The same can be noted also delays.A natural question is to estimate an in‡uence of "mistakes" in coe¢cients and delays on solution behavior.This topic is known in the literature as uncertain systems.Our main goal is to estimate the di¤erence between the solution of real system and a corresponding "model" system.

Functional Differential Equations and Applications 2012 System of di¤erence equations for de ning the distribution of geometric random variables of order k with Markovian parameter

E.Shmerling

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel [email protected]

AMS Subject Classi cation: 65Q10,60J10. Keywords and Phrases: di¤erence equations, geometric distribution of order k, Markov chain.

A system of di¤erence equations which enables one to obtain an explicit expression for the probability mass function (pmf) of geometric random variables of order k with success probability in each trial dependent on the outcome of the previous trial is derived. Software for calculating the values of the pmf is presented. An illustrative example is given.

Functional Differential Equations and Applications 2012 Geometric variables of order k with success probability dependent on outcomes of several previous trials as a solution of a system of di¤erence equations

E.Shmerling

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel [email protected]

AMS Subject Classi cation: 65Q10,60J10. Keywords and Phrases: di¤erence equations, geometric distribution of order k, Markov chain.

The problem of de ning the distribution of geometric random variables of order k with success probability in each trial dependent on outcomes of several previous trials is con- sidered. A method of solving this problem utilizing a system of di¤erence equations is presented. An example illustrating the advantages of the presented method is given.

Functional Differential Equations and Applications 2012 On oscillations of solutions to second-order delay di¤erential equations

Jiµrí Šremr

Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Czech Republic, [email protected]

AMS Subject Classi cation: 34K11

Keywords and Phrases: Linear second-order delay di¤erential equation, oscillatory solu- tion.

On the half-line R+ = [0; +1[ we consider the second-order delay di¤erential equation

u00(t) + p(t)u(t) = 0; (1) where p: R+ ! R+ is a locally integrable function and  : R+ ! R+ is a measurable function such that (t)  t for a. e. t  0; lim essf(s): s  tg = +1: t + ! 1 Some Wintner, Nehari, and Hille type oscillation criteria will be presented guaranteeing that every proper solution to equation (1) is oscillatory. The results obtained reduce to those well known for ordinary di¤erential equations in the case where  = id R+ .

Functional Differential Equations and Applications 2012 About the relation between the properties of the solutions of the di¤erential and the corresponding di¤erence equations

A. N. Stanzhitsky, O. V. Karpenko

Taras Shevchenko National University of Kyiv, Ukraine National Technical University of Kyiv "The Kyiv Polytechnic Institute", Ukraine [email protected], [email protected]

We consider the system of di¤erential equations in the space Rd dx = X(t; x) (1) dt and corresponding system of di¤erence equations

x(t + h) = x(t) + hX(t; x(t)) (2) where h > 0 is the step of the di¤erence equation. The conditions of correspondence between the qualitative properties of the solutions of equations (1) and (2) are studied (bounded, resistance, frequency etc) if h 0. Separately it is considered the question of oscillations for the second-order! equations.

Functional Differential Equations and Applications 2012 On the existence of an optimal initial element for neutral functional di¤erential equations

Tamaz Tadumadze

Tbilisi State University, Department of Mathematics, Tbilisi, Georgia

AMS Subject Classi cation: 34K40, 34K35, 49J21.

Keywords and Phrases: Neutral equations; Optimal initial element; Existence theorem.

For the neutral functional di¤erential equations

x_(t) = A(t)_x(t ) + f(t; x(t); x(t )); t 2 [t0; t1]; with the initial condition

x(t) = '(t); t 2 [t0 ; t0); x(t0) = x0; (x_(t) = (t); t 2 [t0 ; t0) existence theorems of an optimal initial element are proved. Here initial element implies the collection of delay parameters  2 [ 1;  2] and  2 [1; 2]; measurable initial functions '(t) 2 1 and (t) 2 2; initial moment t0 2 [t01; t02] and vector x0 2 X0. The theorems of existence for optimal control problems involving various functional di¤erential equations are given in [1-2].

References 1. T. A. Tadumadze, Some problems in the qualitative theory of optimal control. (Russian), Tbilisi State University Press, Tbilisi, 1983. 2. T. Tadumadze, On the existence of an optimal element for a delay control problem. Nonlinear Anal. 73 (2010), no. 1, 211—220.

Functional Differential Equations and Applications 2012 About positivity of Green’s functions of impulsive functional di¤erential equations

A. Domoshnitsky and I. Volinsky

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel [email protected] [email protected]

In this talk we consider the following boundary value problem

m x0(t) + pi(t)x(t  i(t)) = f(t); t 2 [a; b] (1) i=1 X

x(tj) = jx(tj 0); j = 1; :::; k: (2)

a < t1 < t2 < ::: < tk < b

x() = 0; 2 = [a; b] (3)

b

lx = '(s)x0(s)ds + x(a) = c; ' 2 L [a; b]; ; c 2 R: (4) 1 Za

Results of positivity of Green’s function of problem (1)-(4) are proposed.

Functional Differential Equations and Applications 2012 Approximate solution of impulsive functional di¤erential equations

A.Domoshnitsky, R.Yavich

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel

On the basic of Green’s Functions of impulsive boundary problems for auxiliary equation we propose method for approximate solution of boundary value problem for functional dif- ferential impulse equation.

Functional Differential Equations and Applications 2012