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EXISTENCE RESULTS FOR A CLASS OF ABSTRACT IMPULSIVE DIFFERENTIAL EQUATIONS

Part of: Groups and semigroups of linear operators, their generalizations and applications Functional-differential and differential-difference equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  18 March 2013

EDUARDO HERNÁNDEZ  and
DONAL O’REGAN
EDUARDO HERNÁNDEZ*
Affiliation:
Departamento de Computação e Matemática, Faculdade de Filosofia Ciêcias e Letras de Ribeirão Preto, Universidade de São Paulo, CEP14040-901 Ribeirão Preto, SP, Brazil
DONAL O’REGAN
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland email donal.oregan@nuigalway.ie
*
✉lalohm@ffclrp.usp.br
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Abstract

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We study the existence of solutions for a class of abstract impulsive differential equations. Our technical framework allows us to study partial differential equations with impulsive conditions involving partial derivatives and nonlinear expressions of the solution. Some applications to impulsive partial differential equations are presented.

Keywords

impulsive differential equationsnonlinear impulseanalytic semigroupnonlinear equations

MSC classification

Secondary: 34K45: Equations with impulses 34K30: Equations in abstract spaces 35R12: Impulsive partial differential equations 47D06: One-parameter semigroups and linear evolution equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 3 , June 2013 , pp. 366 - 385
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Abada, N., Benchohra, M. and Hammouche, H., ‘Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions’, J. Differential Equations 246 (10) (2009), 3834–3863.CrossRefGoogle Scholar
Anguraj, A. and Karthikeyan, K., ‘Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions’, Nonlinear Anal. 70 (7) (2009), 2717–2721.CrossRefGoogle Scholar
Barreira, L. and Valls, C., ‘Lyapunov regularity of impulsive differential equations’, J. Differential Equations 249 (2010), 1596–1619.CrossRefGoogle Scholar
Benchohra, M., Henderson, J. and Ntouyas, S., Impulsive Differential Equations and Inclusions, Contemporary Mathematics and its Applications, 2 (Hindawi, New York, 2006).CrossRefGoogle Scholar
Chang, Y.-K., Anguraj, A. and Karthikeyan, K., ‘Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators’, Nonlinear Anal. 71 (10) (2009), 4377–4386.CrossRefGoogle Scholar
Chang, Y.-K., Kavitha, V. and Arjunan, M. M., ‘Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators’, Nonlinear Anal. Hybrid Syst. 4 (1) (2010), 32–43.CrossRefGoogle Scholar
Chu, J. and Nieto, J. J., ‘Impulsive periodic solutions of first-order singular differential equations’, Bull. Lond. Math. Soc. 40 (1) (2008), 143–150.CrossRefGoogle Scholar
Fan, Z. and Li, G., ‘Existence results for semilinear differential equations with nonlocal and impulsive conditions’, J. Funct. Anal. 258 (5) (2010), 1709–1727.CrossRefGoogle Scholar
Fu, X. and Cao, Y., ‘Existence for neutral impulsive differential inclusions with nonlocal conditions’, Nonlinear Anal. 68 (12) (2008), 3707–3718.CrossRefGoogle Scholar
Franco, D., Liz, E., Nieto, J. J. and Rogovchenko, Y. V., ‘A contribution to the study of functional differential equations with impulses’, Math. Nachr. 218 (2000), 49–60.3.0.CO;2-6>CrossRefGoogle Scholar
Frigon, M. and O’Regan, D., ‘Existence results for first-order impulsive differential equations’, J. Math. Anal. Appl. 193 (1) (1995), 96–113.CrossRefGoogle Scholar
Frigon, M. and O’Regan, D., ‘First order impulsive initial and periodic problems with variable moments’, J. Math. Anal. Appl. 233 (2) (1999), 730–739.CrossRefGoogle Scholar
Hernández, E. and O’Regan, D., ‘On a new class of abstract impulsive differential equations’, Proc. Amer. Math. Soc. 141 (5) (2013), 1641–1649.CrossRefGoogle Scholar
Kou, C., Zhang, S. and Wu, S., ‘Stability analysis in terms of two measures for impulsive differential equations’, J. London Math. Soc. (2) 66 (1) (2002), 142–152.CrossRefGoogle Scholar
Lin, A. and Hu, L., ‘Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions’, Comput. Math. Appl. 59 (1) (2010), 64–73.CrossRefGoogle Scholar
Liu, J. H., ‘Nonlinear impulsive evolution equations’, Dynam. Contin. Discrete Impuls. Systems 6 (1) (1999), 77–85.Google Scholar
Nieto, J. J. and O’Regan, D., ‘Variational approach to impulsive differential equations’, Nonlinear Anal. Real World Appl. 10 (2) (2009), 680–690.CrossRefGoogle Scholar
Rogovchenko, Y. V., ‘Impulsive evolution systems: main results and new trends’, Dynam. Contin. Discrete Impuls. Systems 3 (1) (1997), 57–88.Google Scholar
Rogovchenko, Y. V., ‘Nonlinear impulse evolution systems and applications to population models’, J. Math. Anal. Appl. 207 (2) (1997), 300–315.CrossRefGoogle Scholar
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44 (Springer, New York, 1983).CrossRefGoogle Scholar
Yan, Z., ‘Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators’, J. Comput. Appl. Math. 235 (8) (2011), 2252–2262.CrossRefGoogle Scholar
Yu, J. S. and Tang, X. H., ‘Global attractivity in a delay population model under impulsive perturbations’, Bull. London Math. Soc. 34 (3) (2002), 319–328.CrossRefGoogle Scholar