Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 7 (2012), 1 -- 14

IMPULSIVE EVOLUTION INCLUSIONS WITH INFINITE DELAY AND MULTIVALUED JUMPS

Mouffak Benchohra and Mohamed Ziane

Abstract. In this paper we prove the existence of a mild solution for a class of impulsive semilinear evolution differential inclusions with infinite delay and multivalued jumps in a Banach space.

2010 Mathematics Subject Classification: 34A60; 34G25.
Keywords: Evolution system; Generalized Cauchy operator; Measure of noncompactness; Impulsive functional differential inclusions; Mild solutions.

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References

  1. N. Abada, R. Agarwal, M. Benchohra and H. Hammouche, Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay, Asian-Eur. J. Math. 1 (2008), no. 4, 449-468. MR2474181(2009i:34179). Zbl 1179.34070.

  2. N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear Anal. 69 (2008) 2892-2909. MR2452100(2009h:34088). Zbl 1160.34068.

  3. N. Abada, M. Benchohra, H. Hammouche, Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Anal. 4 (2010) 791-803. MR2680247(2011h:34132). Zbl 1207.34077.

  4. N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, in: Pitman Research Notes in Mathematics Series, vol. 246, Longman Scientific & Technical, Harlow, John Wiley & Sons, New York, 1991. MR1100706(92e:47069). Zbl 0727.47026.

  5. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing, Hackensack, NJ, 2006. MR2257896(2007m:93001). Zbl 1127.93001.

  6. N.U. Ahmed, Systems governed by impulsive differential inclusions on Hilbert spaces, Nonlinear Anal. 45 (2001) 693-706. MR1841203(2002e:34016). Zbl 0995.34053.

  7. D.D. Bainov, P.S. Simeonov, Systems with Impulsive Effect, Horwood, Chichester, 1989. MR1010418(90i:93082). Zbl 0683.34032.

  8. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, 2006. MR2322133(2008f:34001). Zbl 1130.34003.

  9. M. Benchohra, S.K. Ntouyas, Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces, J. Math. Anal. Appl. 258 (2001) 573-590. MR1835560(2002d:34130). Zbl 0982.45008.

  10. I. Benedetti, An existence result for impulsive functional differential inclusions in Banach spaces, Discuss. Math. Differ. Incl. Control Optim. 24 (2004), 13-30. MR2118212(2005j:34014). Zbl 1071.34087.

  11. I. Benedetti, P. Rubbioni, Existence of solutions on compact and non-compact intervals for semilinear impulsive differential inclusions with delay. Topol. Methods Nonlinear Anal. 32 (2008), 227-245. MR2494056(2010f:34114). Zbl 1189.34125.

  12. T. Cardinali, P. Rubbioni, On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl. 308 (2005), 620-635. MR2150113(2006a:34172). Zbl 1083.34046.

  13. A. Freidman, Partial Differential Equations. Holt, Rinehat and Winston, New York, 1969. MR445088(56 #3433). Zbl 0224.35002.

  14. L. Görniewicz, Topological Fixed Point Theory of Multivalued Mappings, Topological Fixed Point Theory and Its Applications Vol. 4, Kluwer Academic Publishers, Dordrecht, Second edition, 2006. MR2238622(2007f:58014). Zbl 1107.55001.

  15. C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal. 51 (2002), 765-782. MR1921375(2003f:34149). Zbl 1018.34076.

  16. J. Hale, J. Kato, Phase space for retarded equations with infnite delay, Funkcial. Ekvac. 21 (1978), 11-41. MR0492721(58 #11793). Zbl 0383.34055.

  17. J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, in: Applied Mathematical Sciences, vol. 99, Springer- Verlag, New York, 1993. MR1243878(94m:34169). Zbl 0787.34002.

  18. Y. Hino, S. Murakami and T. Naito, Functional-differential equations with infnite delay, Lecture Notes in Mathematics, Vol. 1473, Berlin, Springer- Verlag, 1991. MR1122588(92g:34088). Zbl 0732.34051.

  19. Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dordrecht, 1997. MR1485775(98k:47001). Zbl 0887.47001.

  20. M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2001. MR1831201(2002e:47066). Zbl 0988.34001.

  21. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. MR1135796(93c:49001). Zbl 0731.49001.

  22. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. MR1082551(91m:34013). Zbl 0719.34002.

  23. S. Migorski, A. Ochal, Nonlinear impulsive evolution inclusions of second order, Dynam. Systems Appl. 16 (2007), 155-173. MR2305434(2008a:34137). Zbl 1128.34038.

  24. V. Obukhovskii, J.-C. Yao, On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces. Nonlinear Anal. 73 (2010), 1715-1728. MR2661354(2011m:34184). Zbl 1214.34052.

  25. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. MR1355787(97i:34002). Zbl 0837.34003.

  26. J. Wu, Theory and Applications of Partial Functional Differential Equations, in: Applied Mathematical Sciences, vol. 119, Springer-Verlag, New York, 1996. MR1415838(98a:35135). Zbl 0870.35116.




Mouffak Benchohra Mohamed Ziane
Laboratoire de Mathématiques, Départemaent des Mathématiques,
Université de Sidi Bel-Abbès, IUniversité de Tiaret,
B.P. 89, 22000, Sidi Bel-Abbès, Algérie. B.P. 78, 14000, Tiaret, Algérie.
e-mail: benchohra@yahoo.com e-mail: ziane@mail.univ-tiaret.dz




http://www.utgjiu.ro/math/sma