On a new class of abstract impulsive differential equations

Hernández, Eduardo; O’Regan, Donal

doi:10.1090/S0002-9939-2012-11613-2

On a new class of abstract impulsive differential equations
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by Eduardo Hernández and Donal O’Regan PDF

Proc. Amer. Math. Soc. 141 (2013), 1641-1649 Request permission

Abstract:

In this note we introduce a new class of abstract impulsive differential equations for which the impulses are not instantaneous. We introduce the concepts of mild and classical solution and we establish some results on the existence of these types of solutions. An example involving a partial differential equation is presented.

References

Nadjet Abada, Mouffak Benchohra, and Hadda Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), no. 10, 3834–3863. MR 2514728, DOI 10.1016/j.jde.2009.03.004
M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publishing Corporation, New York, 2006. MR 2322133, DOI 10.1155/9789775945501
Jifeng Chu and Juan J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc. 40 (2008), no. 1, 143–150. MR 2409187, DOI 10.1112/blms/bdm110
Zhenbin Fan and Gang Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal. 258 (2010), no. 5, 1709–1727. MR 2566317, DOI 10.1016/j.jfa.2009.10.023
Daniel Franco, Eduardo Liz, Juan J. Nieto, and Yuri V. Rogovchenko, A contribution to the study of functional differential equations with impulses, Math. Nachr. 218 (2000), 49–60. MR 1784637, DOI 10.1002/1522-2616(200010)218:1<49::AID-MANA49>3.3.CO;2-Y
M. Frigon and D. O’Regan, First order impulsive initial and periodic problems with variable moments, J. Math. Anal. Appl. 233 (1999), no. 2, 730–739. MR 1689642, DOI 10.1006/jmaa.1999.6336
Marlène Frigon and Donal O’Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl. 193 (1995), no. 1, 96–113. MR 1338502, DOI 10.1006/jmaa.1995.1224
Eduardo Hernández M., Marco Rabello, and Hernán R. Henríquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 331 (2007), no. 2, 1135–1158. MR 2313705, DOI 10.1016/j.jmaa.2006.09.043
Chunhai Kou, Shunian Zhang, and Shujin Wu, Stability analysis in terms of two measures for impulsive differential equations, J. London Math. Soc. (2) 66 (2002), no. 1, 142–152. MR 1911226, DOI 10.1112/S0024610702003277
V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, vol. 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1082551, DOI 10.1142/0906
James H. Liu, Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Systems 6 (1999), no. 1, 77–85. MR 1679758
Juan J. Nieto and Donal O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 680–690. MR 2474254, DOI 10.1016/j.nonrwa.2007.10.022
Yuri V. Rogovchenko, Impulsive evolution systems: main results and new trends, Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 1, 57–88. MR 1435816
Yuri V. Rogovchenko, Nonlinear impulse evolution systems and applications to population models, J. Math. Anal. Appl. 207 (1997), no. 2, 300–315. MR 1438916, DOI 10.1006/jmaa.1997.5245
J. S. Yu and X. H. Tang, Global attractivity in a delay population model under impulsive perturbations, Bull. London Math. Soc. 34 (2002), no. 3, 319–328. MR 1887704, DOI 10.1112/S0024609301008943
A. M. Samoĭlenko and N. A. Perestyuk, Impulsive differential equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. With a preface by Yu. A. Mitropol′skiĭ and a supplement by S. I. Trofimchuk; Translated from the Russian by Y. Chapovsky. MR 1355787, DOI 10.1142/9789812798664
Robert H. Martin Jr., Nonlinear operators and differential equations in Banach spaces, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1987. Reprint of the 1976 original. MR 887947
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1