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3.1 Title Non-linear functional differential equations: qualitative theory and applications. 3.1.2 OBJECTIVES The aim of the project is to obtain fundamental results for functional differential equations, including impulsive and stochastic functional differential equations, and to develop new approaches of their application to mathematical biology. The following aspects of the qualitative and analytical theory of functional differential equations having both theoretical and practical interest will be considered: * Investigation of qualitative properties of solutions for functional differential equations with a special attention to the stability, global attractivity, oscillation, boundedness, periodicity and almost periodicity, and bifurcation analysis; * Development of the theory of linear and non-linear mixed functional differential equations and stochastic functional differential equations; * Theory and applications of delay differential equations with `maxima'; * Non-linear boundary value problems for functional differential and difference systems; * Applications to mathematical biology. 3.1.3. BACKGROUND Ordinary and partial differential equations have been playing an important role in the theory of dynamic systems for a long time. Without any doubt, such equations will continue to serve as an indispensable tool in the future investigations. However, they are generally the first approximations of real systems. More realistic models should include some of the past states of these systems; that is a real system should be modelled by differential equations with time delays. Time delays occur so often, in almost every situation, that to neglect them is to ignore reality. Many complex processes in nature and technologies are described by functional differential or functional difference equations, because the functional components in equations allow one to consider aftereffect or pre-history influence. It was Volterra who, in his works of 1928 and 1931, first called attention to the importance of such systems. Since that time, the topic has been elaborated in many directions, a considerable advance in the field being due to the efforts of the authors of the project suggested. One can consult, e.g., the books "Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients" (Kluwer, 1993) by Mitropolsky, Samoilenko, and Martinyuk and "Introduction to the Theory and Applications of Functional Differential Equations" (Kluwer, 1999) by Kolmanovskii and Myshkis. Various classes of functional differential equations are of fundamental importance in many problems arising in immunology, epidemiology, theory of neural networks, theory of automatic control, etc. Delay differential equations have shown their high efficiency in the study of the behaviour of real populations (see, e.g., the works of the Australian entomologist Nicholson followed by numerous papers regarding the mathematical justification of periodicity and chaos which have been observed in the population of blowflies). The main functional differential equations appearing in these theories, as a rule, are essentially non-linear, and each of them has its own specific features. Because of this, it is often necessary to seek for an individual approach to every concrete equation. The modern computer software is sometimes applied to investigate these equations and to plot the corresponding bifurcation diagrams; there are also a few papers concerning the general properties possessed not by single equations but rather certain their classes. However, a rigorous mathematical justification of the obtained numerical results have not been given yet for a number of systems of mathematical biology. The theory of processes described by the functional differential equations of retarded and neutral type, integral and integro-differential equations of the Volterra type, mixed functional differential equations (i.e., those involving the translation operator and differentiation with respect to different arguments), stochastic functional differential equations, and functional differential equations with impulses is being now intensively developed. The investigation of the qualitative properties and stability of these processes, as well as construction of control and estimation algorithms for stochastic systems with aftereffect, are of great interest from both theoretical and practical viewpoints. In particular, despite the large number of results available in stability theory for the functional differential equations, the theory mentioned has not yet reached the level comparable to the that of stability theory for ordinary differential equations. A significant contribution to stability theory for functional differential equations, initiated primarily by Krasovskii and Razumikhin, is due to the authors of this project proposed. One may refer, e.g., to the monograph "Stability of Functional Differential Equations" (Academic Press, 1986) by Kolmanovskii, or to the books cited above. It is known that optimal algorithms for solving control and estimation problems in systems described by functional differential equations are quite complicated even in the linear-quadratic case; see, e.g., the book "Control of Systems with Aftereffect" (Amer. Math. Soc., 1996) by Kolmanovskii and Shaikhet. In particular, even for delay differential equations, these algorithms involve a procedure for solving a system of matrix functional partial differential equations. Solving the minimax estimation problems, which arise under a more realistic non-classical assumption about noises, and control problems with non-smooth cost functionals reduce to rather complicated mathematical problems. Multi-criteria control problems requiring specific mathematical methods are also of great interest. The construction of effective control and estimation algorithms for the systems described by functional differential and difference equations is thus a problem of fundamental importance. An interesting class of functional differential equations called differential equations with `maxima' appeared several decades ago in connection with the particular problem of modelling the automatic controller in certain electric circuits, and the extreme complexity of such equations was soon noticed, see Myshkis (Russ. Math. Surveys, 32, No. 2, 181-213 (1977)). Equations with `maxima' are characterised by the circumstance that, besides other terms, the unknown variable is usually involved in an expression which contains its maximal value taken over a certain set depending upon the current moment of time. This feature, greatly increasing the possible diversity in the structure of the solution set, simultaneously causes grave complications for their study. Later on, it turned out that the theory of such equations is fairly rich and that they may be of interest from many other points of view. As is well-known, it is quite often the case when the state of one or another evolution process is governed by a solution of some boundary value problem for a functional differential or difference system, especially that with Fredholm linear part. The peculiarity of such boundary value problems, which are the least studied as of now, consists mainly in the fact that the operator in the linear is non-invertible. This fact makes impossible the direct application the traditional methods for investigating boundary value problems based on various fixed point principles. The problems of such kind have Fredholm linear part (generally, with non-zero index); they include rather complicated and poorly studied underdetermined and overdetermined problems, which may be both non-critical and critical. 3.1.4. SCIENTIFIC/TECHNICAL DESCRIPTION 3.1.4.1. Research programme As is well-known, Lyapunov's direct method has been the theoretical basis for stability theory. For functional differential equations, this method is often referred to as the method of Lyapunov-Krasovskii functionals. It should be noted that the majority of theorems on the stability of functional differential equations are available at present for equations of retarded type. The so-called method of degenerate functionals suggested by the authors of this project proved to be more suitable for equations of neutral type. However, many general theoretical problems related with it, unfortunately, still remain open. The problem of stability or instability of solutions of Volterra type equations, stability of stochastic functional differential and difference equations are poorly investigated at present. New conditions of stability of stochastic difference equations will be obtained within this project (Task #1.1, Teams 4 and 6, co-ordinated by Team 4, months 1-6). For practical purposes, the construction of special functionals which permit to obtain coefficient stability conditions, are of great importance. In the project, it is planned to obtain new general results on stability theory for stochastic functional differential equations (Task #1.2, Team 4, months 7-18). In particular, a two-stage method for the construction of special Lyapunov functionals will be developed. Recently, a number of fruitful results in stability theory have been obtained using the concept of logarithmic matrix norm. This concept is particularly suitable for multi-dimensional systems. By choosing different logarithmic norms, the stability of various types of functional differential equations can be studied. It should be noted that the stability conditions for the systems of functional differential equations of order more than two were almost not known earlier. In this project, the concept of logarithmic norm will also be an important tool for obtaining sufficient stability conditions for functional differential equations (Task #2, Teams 4 and 5, co-ordinated by Team 4, months 19-36). A detailed investigation of possibilities to use the vector norm as a Lyapunov function and logarithmic norm as a measure of stability degree for equations of second and third order, non-linear equations and predator-pray systems, will be carried out. By using the Lyapunov functionals and the Lyapunov-Razumikhin conditions for functional differential equations (see, e.g., the paper "Razumikhin's method in the qualitative theory of processes with delay" by Myshkis (J. Appl. Math. Stochastic Anal., 8 (1995), 233-247)), new theorems on stability of solutions for linear and non-linear impulsive functional differential equations of the Volterra type with non-fixed moments of impulses will be proved (Task #3, Teams 1, 2, and 6, co-ordinated by Team 6, months 7-18). New efficient control and estimation algorithms for functional differential and non-linear stochastic systems, including multi-criteria optimization problems for various biological systems (population dynamics models, chemostat models, mathematical models in immunology), will be constructed (Task #4, Team 4, months 19-36). As an interesting application to biology, the Mackey-Glass type delay differential equations (see Murray, "Mathematical Biology", Springer, 1993) describing production of blood cells in the human body will be investigated. It is planned to obtain sharp global attractivity results for this model when considering it in the environment that changes almost periodically or recurrently. The following selected results obtained recently by the applicants will facilitate the successful completion of this Task: sharp conditions for persistence and global asymptotic stability in Lasota-Wazhewska and Nicholson delay differential and difference equations of population dynamics; global attractivity results for some classes of functional differential equations including equations with infinite delay; analysis of oscillatory behavior in Nicholson's blowflies, and linearized oscillation theorem for a wide class of functional differential equations. In the framework of the project, the following problems related to the mathematical models of population dynamics will be studied (Task #5, Teams 1, 2, 3, and 5, co-ordinated by Team 1, months 1-18): To give an analytical proof of the conjecture on the equivalence between local and global attractivity properties of positive equilibrium to Mackey-Glass delay equations in the simplest case of two-dimensional delay discrete equations, and to establish conditions for the existence of a global attractor of the Nicholson type systems with parameters which are almost periodic (or recurrent) in time. Conditions of existence and stability of periodic and almost periodic solutions of various classes of functional differential equations and impulsive functional differential equations will be obtained (Task #6, Teams 2, 5, and 6, co-ordinated by Team 6, months 19-30). For this purpose, the differential properties of solutions and integral sets of functional differential equations and impulsive functional differential equations will be studied. Asymptotic expansions of solutions of regularly perturbed non-linear functional differential equations will also be constructed. As of now, most results concerning the analytical study of differential equations with `maxima' are available for the `linear' case, in which the term containing a maximum is involved linearly in the equation. The quotation marks are necessary here because the operator of calculating the `maximum' over a set different from a singleton is never linear; equations with `maxima' are thus always non-linear. It is natural that much less is known about the equations with `maxima' which are `essentially non-linear,' in the appropriate sense. At present, there are a number of papers concerning the equations with `maxima', including those (co-)authored by researchers involved in the project proposed, where various aspects of the qualitative theory of such equations have been studied; we should like to mention here the papers of Samoilenko, Trofimchuk, and Bantsur (Dopovidi Nats. Akad. Nauk Ukrainy, No. 1 (1998), 47-52), Xu and Liz (Nonlinear Stud., 3, (1996), 231-241), and the paper "Existence and stability of almost periodic solutions for quasilinear delay systems and Halanay inequality" of Liz and Trofimchuk (submitted to J. Math. Anal. Appl.). However, there is still much to be done, quite a number of important analytical questions remaining unanswered. Within the framework of the project proposed, the following aspects of the theory of differential equations with `maxima' will be studied in depth (Task #7, Teams 3 and 5, co-ordinated by Team 3, months 1-6 and 19-30): Obtaining conditions sufficient for the existence, uniqueness, and stability of periodic solutions of `linear' equations with `maxima' with periodic forcing term; investigation of the cases when equations with `maxima' may have multiple periodic solutions, infinitely many periodic solutions, or solutions with `chaotic' behaviour; construction of a `quasi-linear' theory for the differential equations with `maxima' based upon the properties of additive self-mappings of strongly minihedral cones. The general theory of mixed functional differential equations (MFDEs for short) is not available as yet, only some particular classes of such equations have been considered (see, e.g., the paper "On mixed type functional-differential equations" (Nonlinear Analysis TMA, 30 (1997), 2577-2584) by Kamenskii and Myshkis). An approach to the construction of such theory has been suggested recently by the project participants on the base of the natural general form of these equations. In this project, the theory of MFDEs will be developed (Task #8.1, Teams 1 and 4, co-ordinated by Team 4, months 1-18): Theorems on the solvability of the initial-boundary problem for systems of MFDEs, which are difference ones with respect to some arguments and differential with respect to the others, will be established; new methods for qualitative analysis of properties of solutions for MFDEs will be developed; application of the Laplace transformation to the autonomous MFDEs will be studied; conditions of the Lyapunov stability for some classes of MFDEs will be found; L-two-stability criteria of linear space-homogeneous MFDEs will be obtained. The elaboration of impulsive differential equations began more than thirty years ago (see the paper "Systems with shocks at prescribed instants of time" (Mat. Sb., 74 (1967), 201-208) by Myshkis and Samoilenko). The instants of impulses were given in advance in the most of papers, which excludes the possibility to consider the autonomous systems. The theory of systems for which the instants of impulses are determined with the attainment of system critical state are still poorly studied. Recently, the asymptotic behaviour of solutions of different classes of autonomous impulsive differential systems has been considered by Myshkis. However, the qualitative properties of the corresponding trajectories in the phase space have been studied very little at present, and the conditions for stability of solutions are still unknown for some classes of such systems. It is supposed that these questions will be studied in this project (Task #8.2, Teams 1, 4, and 6, co-ordinated by Team 1, months 19-36). When modelling one or another evolution process, it is often the case that the boundary conditions imposed are given by a linear or weakly non-linear vector functional the number of components of which is not equal to the dimension of the system. Being essentially more complicated than the problems Fredholm of index zero, these require more powerful methods to be studied in detail. Within this project, by using the technique of generalised inverse operators and the method of Lyapunov-Schmidt, new criteria of solvability of boundary value problems (with Fredholm linear part) for wide classes of functional differential and difference systems will be established and iterative algorithms for constructing the solutions will be obtained (Task #9, Teams 3, 5, and 6, co-ordinated by Team 5, months 7-24). More precisely, efficient methods for constructing the generalised Green's operator of linear semi-homogeneous boundary value problems for functional differential and difference systems will be developed; properties of the constructed generalised Green's operator will be studied. For non-linear problems, various perturbation results will be obtained; new conditions will be established, under which there arise solutions of the perturbed systems, while the unperturbed system may not have any solution. Conditions for the existence of solutions bounded on the entire axis will be obtained for non-linear ordinary differential and difference systems under the assumption that the operator defined by the corresponding linear homogeneous system is a Fredholm operator and the corresponding inhomogeneous system has more than one linearly independent solutions bounded on the entire real axis. These conditions should be related in a natural manner with the well-known result of Palmer for ordinary differential equations (the case when the operator of the linear part is Fredholm with index zero and the number of linearly independent bounded solutions is one). It is supposed to obtain new conditions for the solvability of periodic and multi-point boundary value problems for functional differential equations with non-linear terms satisfying various two-sided and one-sided growth conditions, as well as to construct iteration algorithms for their finding (Task #10, Teams 3 and 5, co-ordinated by Team 3, months 25-36). Functional differential systems (in particular, systems having some symmetry properties close to those studied by Hale (McGraw-Hill, 1963) and, later, by Vanderbauwhede (Pitman, 1982)) will be studied from the viewpoint of the method of periodic successive approximations; Following Yorke and Lasota, new lower estimates for the periods of periodic solutions of autonomous Lipschitzian functional differential and difference equations will be established. For some classes of functional differential equations, conditions for the existence of a solution lying in a specified hyperplane will be obtained. As is well-known, the conditions for oscillation or non-oscillation of various functional differential equations have attracted much attention (see, e.g., the "Oscillation Theory for Functional Differential Equations" (Marcel Dekker, 1995) by Erbe, Kong, and Zhang). Within the project proposed, oscillatory properties of the solutions of delay difference equations, differential equations with deviating arguments in critical state, and neutral differential equations will be studied (Task #11, Teams 1 and 2, co-ordinated by Team 2, months 1-18 and 31-36). In particular, the oscillation or non-oscillation of solutions of logistic equations with several delays and logistic equations with impulses will be investigated. Delay logistic equations describe various models of mathematical biology (population dynamics, spread of diseases, etc.). The oscillation of solutions of delay logistic equations was investigated by Gopalsamy and Zhang (Quart. Appl. Math., XLVI (1988), 267-273), who have gave sufficient oscillation conditions. In this project, the connection of oscillation properties of delay logistic equations with oscillation of linear delay equations will be studied. The possibility of controlling the oscillatory (non-oscillatory) properties of delay logistic equations by introducing impulses will be investigated. 3.1.4.2. DELIVERABLES, EXPLOITATION & DISSEMINATION OF RESULTS The research results will be presented in the form of scientific publications in international editions and included in some monographs planned to be written. Reports will be sent to INTAS every 12 months. 3.1.5. Description of the Consortium In the project there are three research teams from different INTAS members and three from different NIS organisations. 3.1.5.1. RESEARCH TEAMS TEAM 1. Technion - Israel Institute of Technology, Haifa, Israel. The team involves Dr. E. Braverman (Technion - Israel Institute of Technology, Haifa, Israel) and Dr. L. Berezansky (Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Israel). The leader of Team 1 is Dr. Elena Braverman. The investigations carried out by the team concern stability conditions and oscillation or non-oscillation criteria for functional differential equations and functional differential equations with impulses. Tasks 3, 5, 8.2, and 11 are allocated to this team. TEAM 2. University of Ioannina, Ioannina, Greece. The team involves Prof. I.P. Stavroulakis and Prof. M.K. Grammatikopoulos (both from the University of Ioannina, Greece). The leader of Team 2 is Prof. Ioannis P. Stavroulakis. The team members have a more then 25 years experience in the field of qualitative theory for functional differential equations. They deal chiefly with the study of the asymptotic and oscillatory behaviour of solutions of various classes of delay differential and difference equations. Tasks 3, 5, 6, and 11 are allocated to this team. TEAM 3. University of Vigo, Vigo, Spain. The team involves Prof. E. Liz (University of Vigo, Spain) and Dr. J.J. Nieto (University of Santiago de Compostela). The leader of Team 3 is Prof. Eduardo Liz. The members of Team 3 work in the field of investigation of the qualitative and analytical properties of ordinary and functional differential equations. Their research interests cover periodic and more general boundary value problems for delay differential equations, integro-differential equations, impulse systems, and equations with `maxima'; the method of upper and lower solutions, monotone iterative techniques; applications to biomathematics. Tasks 5, 7, 9, and 10 are allocated to this team. TEAM 4. Moscow State University of Communications, Moscow, Russia. This team involves (i) Prof. A.D. Myshkis and Prof. A.M. Filimonov from Moscow State University of Communications, Russia; (ii) Prof. V.B. Kolmanovskii, Dr. N.I. Koroleva, Mr. S.V. Kuznetsov (post-graduate student, 23 years old), Mr. A.A. Kovalev (post-graduate student, 23 years old), Miss N.V. Kosareva (post-graduate student, 23 years old), and Miss E.V. Ivinskaja (post-graduate student, 23 years old) from Moscow State Institute of Electronics and Mathematics (Technical University). This team leader is Prof. Anatolii D. Myshkis. The research activities of the members of Team 4 concern a wide variety of problems in the theory of ordinary and functional differential equations. Their topics include qualitative analysis for functional differential, functional difference, and stochastic functional differential equations; the study of the asymptotic behaviour of solutions; investigation of various problems related to mixed functional differential equations and autonomous impulse systems. Professors Kolmanovskii and Myshkis are the authors of a number of well-known monographs on functional differential equations. Prof. Myshkis is known as the founder of the modern theory of functional differential equations. Tasks 1.1, 1.2, 2, 4, 8.1, and 8.2 are allocated to Team 4. TEAM 5. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine. The team involves Prof. A.M. Samoilenko, Prof. A.A. Boichuk, Prof. V.I. Tkachenko, Prof. S.I. Trofimchuk, Dr. A.N. Ronto (26 years old) from the Institute of Mathematics of the National Academy of Sciences of Ukraine. The team leader is Prof. Anatolii M. Samoilenko. This team deals with the qualitative and asymptotic methods in the theory of ordinary differential equations, functional differential equations, and differential equations with impulses. The research interests of its members cover periodic and almost periodic solutions, stability and global stability, solvability analysis and approximation of solutions of boundary value problems, and various aspects of the theory of differential equations with `maxima.' Tasks 5, 6, 7, 9, and 10 are allocated to Team 5. TEAM 6. West Kazakhstan Financial and Economical Institute, Aktobe, Kazakhstan. The team involves Prof. M. Akhmetov (West Kazakhstan Financial and Economical Institute, Kazakhstan), Miss R.D. Sejilova (27 years old), and Mr. T.D. Amanov from the Mathematical Department of Aktobe Higher Military Aviation Academy. The leader of Team 6 is Prof. Marat Akhmetov [Marat Ubaydulla-uly Akhmet]. The team members work in the field of the theory of differential equations with impulses, differential systems with discontinuous right-hand side, and integro-differential equations. They study boundary value problems, periodic and almost periodic solutions, and stability problems. Tasks 1.1, 3, 6, 8.2, and 9 are allocated to this team. 3.1.5.2. SCIENTIFIC REFERENCES TEAM 1. Technion - Israel Institute of Technology, Haifa, Israel 1. L. Berezansky and E. Braverman, "Impulsive stabilization of linear delay differential equations," Dynamic Systems and Applications, 5 (1996), No. 2, 263-277. 2. H. Akca, L. Berezansky, and E. Braverman, "On linear differential equations with integral impulsive conditions," Zeitschrift Anal. Anwend., 15 (1996), No. 3, 709-727. 3. L. Berezansky and E. Braverman, "Exponential boundedness of solutions for impulsive delay differential equations," Appl. Math. Letters, 9 (1996), No. 6, 91-95. 4. L. Berezansky and E. Braverman, "Explicit conditions of exponential stability for a linear impulsive delay differential equation," J. Math. Anal. Appl., 214 (1997), 439-458. 5. L. Berezansky and E. Braverman, "On non-oscillation of a scalar delay differential equation," Dynamic Systems and Applications, 6 (1997), No 4, 567-580. 6. L. Berezansky and E. Braverman, "Some oscillation problems for a second order linear delay differential equation," J. Math. Anal. Appl., 220 (1998), 719-740. 7. L. Berezansky and E. Braverman, "On oscillation of a second order impulsive linear delay differential equation," J. Math. Anal. Appl., 233 (1999), 276-300. TEAM 2. University of Ioannina, Ioannina, Greece. 1. M. K. Grammatikopoulos, Y.G. Sficas, and I.P. Stavroulakis, "Necessary and sufficient conditions for oscillations of neutral equations with several coefficients," J. Differential Equations, 76 (1988), 294-311. 2. I.P. Stavroulakis, "Oscillations of mixed neutral equations," Hiroshima Math. J., 19 (1989), 441-456. 3. M.K. Grammatikopoulos and I.P. Stavroulakis, "Necessary and sufficient conditions for oscillation of neutral equations with deviating arguments," J. London Math. Soc., 41 (1990), 244-260. 4. M.K. Grammatikopoulos and I.P. Stavroulakis, "Oscillations of neutral differential equations," Radovi Matematicki, 7 (1991), 47-71. 5. J.R. Graef, M. K. Grammatikopoulos, and P.W. Spikes, "Asymptotic behavior of nonoscillatory solutions on neutral differential equations of arbitrary order," Nonlinear Analysis, 21 (1993), 23-42. 6. J. Jaros and I.P. Stavroulakis, "Necessary and sufficient conditions for oscillations of difference equations several delays," Utilitas Mathematica, 45 (1994), 187-195. 7. A. Elbert and I.P. Stavroulakis, "Oscillation and non-oscillation criteria for delay differential equations," Proc. Amer. Math. Soc., 123 (1995), 1503-1510. 8. I.P. Stavroulakis, "Oscillation of delay difference equations," Computers Math. Applic., 29 (1995), 83-88. 9. Y. Domshlak and I.P. Stavroulakis, "Oscillations of differential equations with deviating arguments in a critical state," Dynamic Systems and Applications, 7 (1998), 405-414. TEAM 3. University de Vigo, Vigo, Spain. 1. Eduardo Liz and Juan J. Nieto, "Periodic boundary value problems for a class of functional differential equations," J. Math. Anal. Appl., 200 (1996), 680-686 . 2. Eduardo Liz and Juan J. Nieto, "Boundary value problems for second order integro-differential equations of Fredholm type," J. Comput. Appl. Math., 73 (1996), 215-225 . 3. Hong-Kun Xu and Eduardo Liz, "Boundary value problems for differential equations with maxima," Nonlinear Studies, 3 (1996), 231-241. 4. Juan J. Nieto, "Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions," Proc. Amer. Math. Soc., 125 (1997), 2599-2604. 5. Hong-Kun Xu and Juan J. Nieto, "Extremal solutions of a class of nonlinear integro-differential equations in Banach spaces," Proc. Amer. Math. Soc., 125 (1997), 2605-2615. 6. Eduardo Liz and Rodrigo L. Pouso, "Upper and lower solutions with `Jumps'," J. Math. Anal. Appl., 222 (1998), 484-493. 7. Eduardo Liz and Rodrigo L. Pouso, "Approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions," J. Comput. Appl. Math., 95 (1998), 127-138. 8. Eduardo Liz and Juan J. Nieto, "Positive solutions of linear impulsive differential equations," Comm. Appl. Anal., 2 (1998), 565-572. 9. Juan J. Nieto, Yu Jiang, and Yan Jurang, "Comparison results and monotone iterative technique for impulsive delay differential equations," Acta Sci. Math. (Szeged), 65 (1999), 343-352. 10. Hong-Kun Xu and Eduardo Liz, "Boundary value problems for functional differential equations," Nonlinear Analysis TMA (to appear). TEAM 4. Moscow State University of Commucations, Russia. 1. V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Dordrecht (1999). 2. V.B. Kolmanovskii and A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer, Dordrecht (1992). 3. A.D. Myshkis, "On autonomous self-supporting impulsive-continuous system," Dynam. Systems Appls., 4 (1995), 541-548. 4. A. Myshkis, "Autonomous differential equations with impulsive self-support and infinite delay," Functional Differential Equations, 3 (1995), No. 1 & 2, 145-154. 5. A.D. Myshkis, "Vibrations of the string with energy dissipation and impulsive feedback support," Nonlinear Analysis TMA, 26 (1996), 1271-1278. 6. A.M. Filimonov, "Continuous approximations of difference operators," J. Difference Equations Appl., 2 (1996), 411-422. 7. V.B. Kolmanovskii, "Stability of some nonlinear functional differential equations," Nonlinear Differential Equations, No. 2 (1995), 185-198. 8. V.B. Kolmanovskii, N.I. Koroleva, and X. Mao, "The control of bilinear microbiological model with delay under random perturbation," Proceedings of CESA'98, vol. 1, pp. 839-843. 9. G.A. Kamenskii and A.D. Myshkis, "On mixed type functional-differential equations," Nonlinear Analysis TMA, 30 (1997), 2577-2584. 10. A.D Myshkis, "Razumikhin's method in the qualitative theory of processes with delay," J. Appl. Math. Stochastic Anal., 8 (1995), 233-247. TEAM 5. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine. 1. Yu.A. Mitropolsky, A.M. Samoilenko, and D.I. Martinyuk, Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients, Kluwer, Dodrecht (1993). 2. A.A. Boichuk, V.F. Zhuravlev, and A.M. Samoilenko, "Linear Noether boundary value problems for impulse differential systems with delay," Differential Equations, 30 (1994), 1677-1682. 3. A.A. Boichuk, V.F. Zhuravlev, and A.M. Samoilenko, Generalised Inverse Operators and Noether Boundary Value Problems, Kiev, Inst. Math. Nat. Acad. Sci. Ukraine (1995). 4. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations World Scientific, Singapore (1995). 5. A.M. Samoilenko, E.P. Trofimchuk, and N.R. Bantsur, "Periodic and almost periodic solutions of the system of differential equations with `maxima,'" Dopovidi Nats. Akad. Nauk Ukrainy, No. 1 (1998), 53-57. 6. I. Gyori and S.I. Trofimchuk, "Global attractivity in x'(t) = -q x(t) + p f(x(t-\tau))," Dynamic Systems and Applications, 8 (1999), 197-210. 7. M. Pinto and S. Trofimchuk, "Stability and existence of multiple periodic solutions for quasilinear differential equation with maxima," Proc. Royal Soc. Edinburg, Sect. A (to appear). 8. K. Gopalsamy, S. Trofimchuk, "Almost periodic solution of Lasota-Wazewska type delay differential equation," J. Math. Anal. Appl. (to appear). 9. V.I. Tkachenko, "On unitary almost periodic systems, In: Advances in Difference equations," Proceedings of the Second International Conference on Difference Equations and Applications, Gordon and Breach Science Publishers, 1997, pp. 545-552. 10. A. Ronto, "On some boundary value problems with non-local conditions," Nonlinear Oscillations (Kyiv), 2 (1999), No. 1, 92-108. TEAM 6. West Kazakhstan Financial and Economical Institute, Aktobe, Kazakhstan. 1. M.U. Akhmetov and A. Zafer, "The controllability of boundary value problems for quasilinear impulsive systems," Nonlinear Analysis, 34 (1998) 1055-1065. 2. M.U. Akhmetov and A. Zafer, "Controllability of the Volle-Poussen value problem for Differential Impulsive systems," J. Optim. Theory and Appl., 102 (1999), no 2. (to appear). 3. M.U. Akhmetov and N.A. Perestyuk, "The comparison method for differential equations with impulse actions," Differential Equations, 26 (1990), 1079-1086. 4. M.U. Akhmetov and N.A. Perestyuk, "Periodic and almost periodic solutions of strongly nonlinear impulse systems," Prikladn. Mat. Mekh., 56 (1992), 926-934. 5. M.U. Akhmetov and R.D. Sejilova, "On boundary value problem for integro-differential equations with impulsive actions," Izvetiya AN Kazakhstana, Ser. Fiz. Mat., 5 (1998), 5-9. 6. T.D. Amanov, "On boundary value problems of general form for impulsive systems," Vestnik "Dunie", 3 (1999), 12-17. 7. M.U. Akhmetov and R.D. Sejilova, "The control of boundary value problem for linear impulsive integro-differential system," J. Math. Anal. Appl., 236 (1999) (to appear). 3.1.6. Management General co-ordination of the project will be done by the co-ordinator, Dr. E. Braverman. The co-ordination of activities of each of the research teams will be done by their leaders, Professors I. Stavroulakis, E. Liz, A. Myshkis, A. Samoilenko, and M. Akhmetov, The basic coordination of scientific and administrative activities during the project will be carried out by means of ordinary and electronic mail. It is also supposed that, for the purpose of participating in joint seminars and discussing obtained results, the researchers from the INTAS member teams will visit the corresponding institutions in Kyiv and Moscow, whereas those from the NIS teams are expected to visit their colleagues in Israel, Greece, and Spain. The latter should be particularly important in view of the economical difficulties experienced by the NIS countries; in particular, by virtue of the absence of many modern mathematical journals in the local libraries. The expenses needed for the meetings and exchange of scientists will be in accordance with the sums specified in the Cost Table. 3.1.6.1. PLANNING & TASKS ALLOCATION Tasks Participants Months Months Months Months Months Months 1-6 7-12 13-18 19-24 25-30 31-36 1.1 P4, P6 * 1.2 P4 * * 2 P4, P5 * * * 3 P1, P2, P6 * * 4 P4 * * * 5 P1, P2, P3, P5 * * * 6 P2, P5, P6 * * 7 P3, P5 * * * 8.1 P1, P4 * * * 8.2 P1, P4, P6 * * * 9 P3, P5, P6 * * * 10 P3, P5 * * 11 P1, P2 * * * * 3.1.6.2. COST TABLE MAIN COST TABLE INTAS MEMBER TEAMS ------------------------------------- TEAM STATUS COST CATEGORIES TOTAL Labour Overhears Travel and Equipment Consu- Other (Euro) NAME Costs subsistence mables cost ------------------------------------- 1. CO 0 0 6300 0 300 0 6600 2. CR 0 0 5300 0 300 0 5600 3. CR 0 0 5300 0 300 0 5600 -------------------------------------- SUBTOTAL 0 0 16900 0 900 0 17800 -------------------------------------- NIS TEAMS ------------------------------------- TEAM STATUS COST CATEGORIES TOTAL Labour Overhears Travel and Equipment Consu- Other (Euro) NAME Costs subsistence mables cost ------------------------------------- 4. CR 14000 0 14000 0 400 0 28400 5. CR 18000 0 8400 1800 200 0 28400 6. CR 13200 0 800 400 0 0 14400 -------------------------------------- SUBTOTAL 45200 0 22800 2600 600 0 71200 -------------------------------------- TOTAL 45200 0 39700 2600 1500 0 89000 -------------------------------------- Participant 1. Technion - Israel Institute of Technology, Haifa, Israel Participant 2. University of Ioannina, Ioannina, Greece. Participant 3. University de Vigo, Spain. Participant 4. Moscow State University of Commucations, Russia. Participant 5. Institute of Mathematics of National Academy of Sciences of Ukraine, Kyiv, Ukraine. Participant 6. West Kazakhstan Financial and Economical Institute, Aktobe, Kazakhstan. NIS Labour Cost Summary Table ------------------------------------------------------------------- TEAM Number of Cost/month Number of Total cost individual (Euro) Months (Euro) NAME grants ------------------------------------------------------------------- 4 1 200 22 4400 1 175 24 4200 1 150 36 5400 5 1 240 18 4320 4 190 18 13680 6 1 250 24 6000 2 150 24 7200 ------------------------------------------------------------------ TOTALS 11 45200 ----------------------------------------------------------------- The following equipment will be purchased for the team headed by Prof. A. Samoilenko: * a personal computer (900 Euro); * a laser printer (500 Euro); * a scanner (400 Euro). The equipment listed is necessary when preparing new papers (numerical computations), in particular, for producing electronic manuscripts (typesetting, inclusion of graphs, diagrams, etc.).
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