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Abstrakty
In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: [formula] where J : = [0; b] and 0 = t0 < t1 < ... < tm < tm+1 = b (m ∈ N*), F is a set-valued map and g is single map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1, ... m). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuous dependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.
Czasopismo
Rocznik
Tom
Strony
249--283
Opis fizyczny
Bibliogr. 60 poz.
Twórcy
autor
autor
- Sidi-Bel-Abbès University Department of Mathematics P.B. 89, 22000. Sidi-Bel-Abbès, Algeria, mhelal_abbes@yahoo.fr
Bibliografia
- [1] Z. Agur, L. Cojocaru, G. Mazaur, R.M. Anderson, Y.L. Danon, Pulse mass measles vaccination across age cohorts, Proc. Nat. Acad. Sci. USA, 90 (1993), 11 698–11 702.
- [2] N.U. Ahmed, Optimal control for impulsive systems in Banach spaces, Int. J. Differ. Equ. Appl. 1 (2000) 1, 37–52.
- [3] J. Andres, L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003.
- [4] J.P. Aubin, Impulse Differential Inclusions and Hybrid Systems: a Viability Approach, Lecture Notes, Université Paris-Dauphine, 2002.
- [5] J.P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg-NewYork, 1984.
- [6] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
- [7] E.P. Avgerinos, N.S. Papageorgiou, Topological properties of the solution set of integrodifferential inclusions, Comment. Math. Univ. Carolinae 36 (1995) 3, 429–442.
- [8] S.A. Aysagaliev, K.O. Onaybar, T.G. Mazakov, The controllability of nonlinear systems, Izv. Akad. Nauk. Kazakh-SSR.-Ser. Fiz-Mat 1 (1985), 307–314.
- [9] S. Barnet, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, 1975.
- [10] D.D. Bainov, P.S. Simeonov, Systems with Impulse Effect, Ellis Horwood Ltd., Chichester, 1989.
- [11] M. Benchohra, L. Górniewicz, S.K. Ntouyas, A. Ouahab, Controllability results for impulsive functional differential inclusions, Reports on Mathematical Physics 54 (2004), 211–227.
- [12] M. Benchohra, A. Ouahab, Impulsive neutral functional differential inclusions with variable times, Electr. J. Differ. Equ. 2003 (2003), 1–12.
- [13] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 70–85.
- [14] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 580, 1977.
- [15] H. Covitz, S.B. Nadler (Jr.), Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5–11.
- [16] K. Deimling, Multi-valued Differential Equations, De Gruyter, Berlin-New York, 1992.
- [17] S. Djebali, L. Górniewicz, A. Ouahab, First order periodic impulsive semilinear differential inclusions existence and structure of solution sets, Math. Comput. Modeling 52 (2010), 683–714.
- [18] S. Djebali, A. Ouahab, Existence results for -Laplacian Dirichlet BV Ps of differential inclusions with application to control theory, Discuss. Math. Differential Incl. 30 (2010), 23–49.
- [19] A.H. Filippov, On some problems of optimal control theory, Vestnik Moskowskogo Univesiteta, Math. 2 (1958), 25–32.
- [20] H. Frankowska, Set-Valued Analysis and Control Theory, Centre de Recherche de Mathématique de la Décision, Université Paris-Dauphine, 1992.
- [21] H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations 84 (1990), 100–128.
- [22] M. Frigon, A. Granas, Théorèmes d’existence pour des inclusions différentielles sans convexité, C. R. Acad. Sci. Paris, Ser. I 310 (1990), 819-822.
- [23] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
- [24] S. Gao, L. Chen, J.J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24 (2006) 6037–6045.
- [25] M. Guo, X. Xue, R. Li, Controllability of impulsive evolutions inclusions with nonlocal conditions, Z. Optim. Theory Appl. 120 (2004), 355–374.
- [26] L. Górniewicz, Topological Fixed Point Theory of Multi-valued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999.
- [27] H. Guo, L. Chen, The effects of impulsive harvest on a predator-prey system with distributed time delay, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), 2301–2309.
- [28] J.R. Graef, A. Ouahab, Impulsive differential inclusions with parameter, Submitted.
- [29] A. Halanay, D. Wexler, Teoria Calitativa a systeme cu Impulduri, Editura Republicii Socialiste Romania, Bucharest, 1968.
- [30] J. Henderson, A. Ouahab, Local and global existence and uniqueness results for second and higher order impulsive functional differential equations with infinite delay, Aust. J. Math. Anal. Appl. 4 (2007), 149–182.
- [31] Sh. Hu, N.S. Papageorgiou, Handbook of Multi-valued Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997.
- [32] Sh. Hu, N.S. Papageorgiou, Handbook of Multi-valued Analysis. Volume II: Applications, Kluwer, Dordrecht, 2000.
- [33] S. Hu, N.S. Papageorgiou, V. Lakshmikantham, On the properties of the solutions set of semilinear evolution inclusions, Nonlinear Anal. 24 (1995), 1683–1712.
- [34] J. Jiao, X. Meng, L. Chen, A new stage structured predator-prey Gomportz model with time delay and impulsive porturbations, Appl. Math. Comput. 196 (2008), 705–719.
- [35] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multi-valued Maps and Semilinear Differential Inclusions in Banach Spaces, Gruyter & Co., Berlin, 2001.
- [36] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, 1991.
- [37] E. Kruger-Thiemr, Formal theory of drug dosage regiments, J. Theo. Biol. 13 (1966), 212–235.
- [38] V.I. Korobov, Reduction of a controllability problem to a boundary value problem, Different. Uranen. 12 (1976), 1310–1312.
- [39] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
- [40] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786.
- [41] X. Meng, J. Jiao, L. Chen, The dynamics of an age structured predator-prey model with disturbing pulse and time delays, Nonlinear Anal. 9 (2008), 547–561.
- [42] V.D. Milman, A.A. Myshkis, On the stability of motion in the presence of impulses, Sib. Math. J. 1 (1960), 233–237 [in Russian].
- [43] J. Musielak, Introduction to Functional Analysis, PWN, Warszawa, 1976 [in Polish].282 Mohmed Helal and Abdelghani Ouahab
- [44] A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay, Nonlinear Anal. T.M.A. 67 (2006), 1027–1041.
- [45] A. Ouahab, Local and global existence and uniqueness results for impulsive functional differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006), 456–472.
- [46] S.G. Pandit, S.G. Deo, Differential Systems Involving Impulses, Lecture Notes in Mathematics,954, Springer-Verlag, 1982.
- [47] N.S. Papageorgiou, N. Yannakakis, Nonlinear parametric evolution inclusions, Math.Nachr. 233/234 (2002), 201–219.
- [48] Y. Pei, S. Liu, C. Li, L. Chen, The dynamics of an impulsive delay SI model with variable coefficients, Appl. Math. Modell. 33 (2009), 2766–1776.
- [49] B. Ricceri, Une propriété topologique de l’ensemble des points fixe d’une contraction moultivoque à valeurs convexes, Atti Accad. Naz. Linei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. (1987), 283–286.
- [50] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
- [51] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41, American Mathematical Society, Providence, 2002.
- [52] X. Song, H. Guo, Extinction and permanence of a kind of pest-predator models impulsive effect and infinite delay, J. Korean Math. Soc. 44 (2007), 327–342.
- [53] A.A. Tolstonogov, Differential Inclusions in Banach Spaces, Kluwer, Dordrecht, 2000.
- [54] A.A. Tolstonogov, Approximation of attainable sets of an evolution inclusion of subdifferential type, Sibirsk. Mat. Zh. 44 (2003), 883–904.
- [55] A.A. Tolstonogov, Properties of attainable sets of evolution inclusions and control systems of subdifferential type, Sibirsk. Mat. Zh. 45 (2004), 920–945.
- [56] D. Yujun, Periodic boundary value problems for functional differential equations with impulses, J. Math. Anal. Appl. 210 (1997), 170–181.
- [57] D. Yujun, Z. Erxin, An application of coincidence degree continuation theorem in existence of solutions of impulsive differential equations, J. Math. Anal. Appl. 197 (1996), 875–889.
- [58] D.Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859–903.
- [59] T. Wazewski, On an optimal control problems, Proc. Conference Differential Equations and Their Application, Prague, (1962), 222–242.
- [60] Q.J. Zhu, On the solution set of differential inclusions in Banach space, J. Diff. Eq. 93 (1991) 2, 213–237.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0007-0021