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 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; * 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 "Topological degree methods in non-linear boundary value problems, Conf.Board Math.Sci. Regional Conf. Series in Math. (Providence: AMS, 1979) by J. Mawhin. 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, 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 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. 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. 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 T1. Periodic and almost periodic solutions. Objectives: To study the existence and stability of periodic and almost periodic solutions of various classes of functional differential equations and impulsive functional differential equations. Inputs: The suggested study will use, in particular, some methods of A. M. Samoilenko (Elements of the Mathematical Theory of Multifrequency Oscillations, Kluwer, Dordrecht, 1991), Th. Bartsch and J. Mawhin ("The Leray-Schauder degree of $S\sp1$-equivariant operators associated to autonomous neutral equations in spaces of periodic functions," J. Differential Equations 92 (1991), No. 1, 90--99), and K. Gopalsamy and S. Trofimchuk ("Almost periodic solution of Lasota-Wazewska type delay differential equation," J. Math. Anal. Appl., 237 (1999), 106-127). Outputs: It is planned: * To obtain conditions sufficient for the existence and stability of periodic and almost periodic solutions of differential functional equations (including equations with pulse effect), for which purpose it is supposed to study the differential properties of solutions and integral sets of functional differential equations and impulsive functional differential equations; to construct asymptotic expansions of solutions of regularly perturbed non-linear functional differential equations; * To establish conditions of existence, uniqueness and stability of almost periodic solutions of the delay differential equations with piecewise constant delays. Schedule: months ... Methodologies: Topological degree arguments and some results about almost periodicity in semiflows will be used. Success criteria: ... Teams: CO, CR1, CR3, and CR4, co-ordinated by CR4. T2. Functional differential equations as biological models. Objectives: To study the stability and global attractivity in delay differential equations of population dynamics. Inputs: The following selected results obtained recently by the applicants will contribute to the successful completion of the task: conditions for persistence and global asymptotic stability in Lasota-Wazewska and Nicholson delay differential and difference equations of population dynamics (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); global attractivity results for Wright type delay differential equations (E. Liz, M. Pinto, G. Robledo, V. Tkachenko, and S. Trofimchuk, "Wright type delay differential equations with negative Schwarzian" (submitted)); analysis of oscillatory behaviour in Nicholson's blowflies, and the linearized oscillation theorem for a wide class of functional differential equations (A. Ivanov, E. Liz, and S. Trofimchuk, "Halanay inequality, Yorke 3/2 stability criterion, and differential equations with `maxima,'" University of Ballarat, Australia, Research Report No. 40/99, 1999). Outputs: The following problems related to the mathematical models of population dynamics will be studied. * 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 which changes almost periodically or recurrently; * It is supposed to give an analytical proof of the conjecture on the equivalence between local and global attractivity properties of positive equilibrium of 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 the parameters which are almost periodic (or recurrent) in time. Schedule: months ... Methodologies: Some notions and facts from the theory of one-dimensional dynamical systems will be used. Success criteria: ... Teams: CO, CR2, and CR3, co-ordinated by CR3. T3. Differential equations with `maxima'. Objectives: To carry out the investigation of a number of qualitative questions in the theory of differential equations with `maxima.' Inputs: 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 Studies, 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. Outputs: The following aspects of the theory of differential equations with `maxima' will be studied: * 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. Schedule: months ... Methodologies: Various continuation theorems, Lyapunov-Schmidt type methods, and positivity techniques will be used. Success criteria: ... Teams: CO and CR3, co-ordinated by CO. T4. Stability of differential functional equations with pulse effect. Objectives: To study the stability conditions for various differential functional equations with impulses at non-fixed moments of time. Inputs: The Lyapunov functionals and Lyapunov-Razumikhin conditions (see, e.g., M. Akhmetov and A. Zafer, "Stability of the zero solution of impulsive differential equations by the Lyapunov second method," J. Math. Anal. Appl., 248 (2000), 68-82) serve as a powerful tool in studies of stability of solutions of differential functional equations. Output: * New theorems on stability of solutions for functional differential equations and impulsive functional differential equations (especially, of the Volterra type) with non-fixed moments of impulses will be proved. Schedule: months ... Methodologies: Methods of Lyapunov-Krasovskii functionals and Razumikhin type methods will be used. Success criteria: ... Teams: CR1, CR2, and CR4, co-ordinated by CR2. T5. Boundary value problems for functional differential equations. T5.1. General existence and uniqueness results. Objectives: New criteria of the solvability of boundary value problems (with Fredholm linear part) for new classes of functional differential and difference systems will be established. Inputs: The well-elaborate methods involving the generalised inverse operators are quite often and efficiently used in boundary value problems (A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalised Inverse Operators and Noether Boundary Value Problems [in Russian], Kiev, 1995; R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, 1977). The theory allows one to study also a number of other problems, including the problem on bounded solutions for non-linear differential equations. Outputs: For the problems of the specified kind, the following results are expected: * For weakly non-linear, in a proper sense, problems, various perturbation type results will be obtained; * New conditions for the existence of solutions of non-linear boundary value problems in the critical case (when the equations for generating amplitudes have multiple roots) will be given; * New conditions will be established, under which there arise solutions of the perturbed systems, while the unperturbed system may not have any solution; * By using a similar scheme, conditions for the existence of solutions bounded on the entire real axis will be obtained for non-linear functional differential and difference systems on the assumption that the operator defined by the corresponding linear homogeneous system is a Fredholm operator (in particular, when the homogeneous system has exponential dichotomy on both half-lines or/and exponential trichotomy on whole line); * New conditions for the existence of bounded solutions will be obtained in the cases when the corresponding equation for generating amplitudes has either simple or multiple roots. Schedule: months ... Methodologies: Topological degree techniques, generalised inversion of operators, Schmidt lemma, and the Nikolskii characterisation theorem for Fredholm operators with zero index will be used. Success criteria: Obtaining stronger statements for Fredholm problems and their extension to new classes of equations. In particular, boundary value problems for differential functional equations may generate poorly studied topologically Fredholm operators, which fact justifies the detailed study of problems of that kind. The general results should significantly contribute also to more traditional problems: e.g., the result concerning the existence of bounded solutions, in the case of multiple roots, should be new even for ordinary differential equations (A. A. Boichuk, "Solutions of weakly nonlinear differential equations bounded on the whole line," Nonlinear Oscillations, 2 (1999), No. 1, 3-10). Teams: CR1, CR3, and CR4, co-ordinated by CR1. T5.2. Constructive methods of analysis of boundary value problems. Objectives: 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 (including the error estimates). Inputs: At present, there exist powerful methods (I. T. Kiguradze, "Boundary value problems for systems of ordinary differential equations," Current problems in mathematics. Newest results, Vol. 30, 3--103, 204, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987) based upon the use of differential inequalities and one-sided estimates. Outputs: * It is expected to establish optimal, in a sense, conditions sufficient for the existence of a unique solution for some classes of affine boundary value problems for non-linear functional differential equations; * Certain classes of functional differential systems with non-linearities satisfying certain one-sided growth restrictions (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 (with the emphasis to conditions for the existence of multiple periodic solutions with the same period); * Following Yorke and Lasota, new lower estimates for the periods of periodic solutions of autonomous Lipschitzian functional differential and difference equations will be established. Similar techniques should also lead one to the necessary conditions for the solvability of more general two- and multipoint boundary value problems. Schedule: months ... Methodologies: Theory of positive operators on partially ordered Banach spaces, (versions of) the successive approximation method, and the Lyapunov-Schmidt type methods will be used. Success criteria: The combination of Lyapunov-Schmidt type reductions and positivity techniques should lead one to sharp estimates for perturbation terms in existence results for weakly non-linear systems, which should prove very useful in the cases when as precise as possible a priori information on the `unperturbed' problems is needed (e.g., for establishing new continuation theorems). Teams: CO, CR1, and CR3, co-ordinated by CR3. T5.3. Controllability of boundary value problems. Objectives: To study the controllability of non-linear boundary value problems. Inputs: The problem on controllability of boundary value problems for differential functional equations has attracted much attention. At present, the linear case is studied in most detail (M. Akhmetov and R. Sejilova, "The control of the boundary value problem for linear impulsive integro-differential systems," J. Math. Anal. Appl., 236 (1999), 312-326). Outputs: It is supposed: * To obtain conditions sufficient for the controllability of boundary value problems (in both separated and non-separated cases) for differential functional equations with strong non-linearities. Schedule: months ... Methodologies: The use will be made of coincidence degree theory and numerical-analytic methods combined with the techniques of control theory for linear and quasilinear systems. Success criteria: ... Teams: CR2 and CR4, co-ordinated by CR4. T6. Oscillation criteria for functional differential equations. Objectives: ... Inputs: The conditions for oscillation and non-oscillation of 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). Oscillation of solutions of delay logistic equations, describing various models in mathematical biology (population dynamics, spread of diseases, etc.), was investigated by Gopalsamy and Zhang (Quart. Appl. Math., XLVI (1988), 267-273), who have gave sufficient oscillation conditions. Outputs: 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. In particular, the oscillation or non-oscillation of solutions of logistic equations with several delays and logistic equations with impulses will be investigated; * 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. Schedule: months 1-18 and 31-36 (?) Methodologies: ... Success criteria: ... Teams: CR2, CR3, and CR4, co-ordinated by CR2. 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 CO. 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 CO is Prof. Eduardo Liz. The members of Team CO 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 T1, T2, T3, and T5.2 are allocated to this team. CR1. University of Louvain-la-Neuve, Louvain-la-Neuve, Belgium. The team involves Prof. Dr. J. Mawhin The leader of CR1 is Prof. Jean Mawhin. Tasks T1, T4, T5.1, and T5.2 are allocated to this team. CR2. 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 CR2 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 T2, T4, T5.3, and T6 are allocated to this team. CR3. 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. Anatoly 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 T1, T2, T3, T4, T5.1, and T5.2 are allocated to Team 5. CR4. 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 Miss M. Tleubergenova (29 years old) from the Mathematical Department of Aktobe Higher Military Aviation Academy. The leader of CR4 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 T1, T4, T5.1, T5.3, and T6 are allocated to this team. 3.1.5.2. SCIENTIFIC REFERENCES CO. University of 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 1. University of 1. TEAM 2. 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 3. 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 and S. Trofimchuk, "Almost periodic solution of Lasota-Wazewska type delay differential equation," J. Math. Anal. Appl., 237 (1999), 106-127. 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 4. West Kazakhstan Financial and Economical Institute, Aktobe, Kazakhstan. 1. M. Akhmetov, A. Zafer, Stability of the zero solution of Impulsive differential equatios by the Lyapunov second method, JMAA, 248 (2000)68-82. 2. M. Akhmetov and R. Sejilova, On the control of boundary value problem for linear impulsive integro-differential system. Differential'nye Uravneniya. 36(2000), no. 9,1-8. 3. M. Akhmetov and R. Sejilova, Rank criteria of controllability for boundary value problem of impulsive integro-differential system. Ukrain. Math. J. 52 (2000), no. 6. 4. M. Akhmetov and R. Nagaev, Periodic solutions of nonlinear impulsive system in a neighborhood of quasiperiodic solution's family. Differentsial'nye Uravneniya. 36(2000), no.5, 1-8. 5. M. Akhmetov and R. Sejilova, The control of the Boundary Value Problem for Linear Impulsive Integro- Differential Systems, JMAA, (236)(1999) 312-326. 6. M. Akhmetov, N. Perestyuk, The comparison method for differential equations with impulse action. Differentsialnye Uravneniya, 26 (1990), no. 9,1475-1483. Translation: Differential Equations, 26 (1990), no.9, 1079-1086 (1991) 7. M. Akhmetov and N. Perestyuk, Periodic and almost periodic solutions of strongly nonlinear impulse systems. Prikladnaya Matem. i Memkhanika,56(1992),no.6, 926-934. English Translation: J. Appl. Math. Mechs, 56 (1992), no.6, 829-837. 8. M. U. Akhmetov, N. A. Perestyuk, and M. A. Tleubergenova, The control of linear impulsive systems, Ukrain. Math. Zh., 47(1995), 307-314. 3.1.6. Management General co-ordination of the project will be done by the co-ordinator, Dr. E. Liz. The co-ordination of activities of each of the research teams will be done by their leaders, Professors J. Mawhin, E. Braverman, 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. University de Vigo, Spain. Participant 2. University of Ioannina, Ioannina, Greece. Participant 3. Technion - Israel Institute of Technology, Haifa, Israel Participant 4. Institute of Mathematics of National Academy of Sciences of Ukraine, Kyiv, Ukraine. Participant 5. 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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