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On a new class of abstract impulsive differential equations

Authors: Eduardo Hernández and Donal O’Regan
Journal: Proc. Amer. Math. Soc. 141 (2013), 1641-1649
MSC (2010): Primary 34K30, 34K45, 35R12, 47D06
Published electronically: October 25, 2012
MathSciNet review: 3020851
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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.

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  • 1. Abada, Nadjet.; Benchohra, Mouffak.; Hammouche, Hadda. Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differential Equations 246 (2009), no. 10, 3834-3863. MR 2514728 (2010b:34134)
  • 2. Benchohra, M.; Henderson, J.; Ntouyas, S. Impulsive differential equations and inclusions. Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. MR 2322133 (2008f:34001)
  • 3. Chu, Jifeng.; Nieto, Juan J. Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 40 (2008), no. 1, 143-150. MR 2409187 (2009c:34060)
  • 4. Fan, Zhenbin.; Li, Gang. Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258 (2010), no. 5, 1709-1727. MR 2566317 (2010k:34074)
  • 5. Franco, Daniel.; Liz, Eduardo.; Nieto, Juan J.; Rogovchenko, Yuri V. A contribution to the study of functional differential equations with impulses. Math. Nachr. 218 (2000), 49-60. MR 1784637 (2002h:34169)
  • 6. Frigon, M.; O'Regan, D. First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl. 233 (1999), no. 2, 730-739. MR 1689642 (2000c:34016)
  • 7. Frigon, Marlène.; O'Regan, Donal. Existence results for first-order impulsive differential equations. J. Math. Anal. Appl. 193 (1995), no. 1, 96-113. MR 1338502 (96e:34021)
  • 8. Hernández, Eduardo.; Henríquez, Hernán.; Rabello, Marco. Existence of solutions for a class of impulsive partial neutral functional differential equations. J. Math. Anal. Appl. 331 (2007) 2, 1135-1158. MR 2313705 (2008m:35356)
  • 9. Kou, Chunhai.; Zhang, Shunian.; Wu, Shujin. Stability analysis in terms of two measures for impulsive differential equations. J. London Math. Soc. (2) 66 (2002), no. 1, 142-152. MR 1911226 (2003h:34109)
  • 10. Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S. Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1082551 (91m:34013)
  • 11. Liu, James H. Nonlinear impulsive evolution equations. Dynam. Contin. Discrete Impuls. Systems 6 (1999), no. 1, 77-85. MR 1679758 (99m:34138)
  • 12. Nieto, Juan J.; O'Regan, Donal. Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 680-690. MR 2474254 (2009m:34070)
  • 13. Rogovchenko, Yuri V. Impulsive evolution systems: main results and new trends. Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 1, 57-88. MR 1435816 (98g:34027)
  • 14. Rogovchenko, Yuri V. Nonlinear impulse evolution systems and applications to population models. J. Math. Anal. Appl. 207 (1997), no. 2, 300-315. MR 1438916 (98g:34028)
  • 15. Yu, J. S.; Tang, X. H. Global attractivity in a delay population model under impulsive perturbations. Bull. London Math. Soc. 34 (2002), no. 3, 319-328. MR 1887704 (2002k:34149)
  • 16. Samoilenko, A. M.; Perestyuk, N. A. Impulsive differential equations. With a preface by Yu. A. Mitropol'skiĭ and a supplement by S. I. Trofimchuk. Translated from the Russian by Y. Chapovsky. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1355787 (97i:34002)
  • 17. Martin, R. H. Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publ. Co., Florida, 1987. MR 887947 (88i:34125)
  • 18. Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin, 1983. MR 710486 (85g:47061)

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Additional Information

Eduardo Hernández
Affiliation: Departamento de Computação e Matemática, Faculdade de Filosofia, Ciencias e Letras de Ribeirão Preto, Universidade de São Paulo, CEP 14040-901 Ribeirão Preto, SP, Brazil

Donal O’Regan
Affiliation: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Keywords: Impulsive differential equation, mild solution, classical solution, $C_{0}$-semigroup of linear operators.
Received by editor(s): September 2, 2011
Published electronically: October 25, 2012
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society

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