EXISTENCE OF GLOBAL SOLUTIONS FOR IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

Volume 4, Issue 2, pp 102-114 http://dx.doi.org/10.22436/jnsa.004.02.02

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Authors

S. SIVASANKARAN - Department of Mathematics, University College, Sungkyunkwan University, Suwon 440-746, South Korea. M. MALLIKA ARJUNAN - Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore - 641 114, Tamil Nadu, India. V. VIJAYAKUMAR - Department of Mathematics, Info Institute of Engineering, Kovilpalayam, Coimbatore - 641 107, Tamil Nadu, India.

Abstract

In this paper, we study the existence of global solutions for a class of impulsive abstract functional differential equation with nonlocal conditions. The results are obtained by using the Leray-Schauder alternative fixed point theorem. An example is provided to illustrate the theory.

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Keywords

• Impulsive functional differential equations
• mild solutions
• global solutions
• semigroup theory.

MSC

•  34A37
•  34K30
•  35R10
•  35R12
•  47D06
•  49K25

References

• [1] H. Akca, A. Boucherif, V. Covachev, Impulsive functional differential equations with nonlocal conditions, Inter. J. Math. Math. Sci., 29:5 (2002), 251-256.


• [2] A. Anguraj, K. Karthikeyan , Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Anal., 70(7) (2009), 2717- 2721.


• [3] A. Anguraj, M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Electronic Journal of Differential Equations, 111 (2005), 1-8.


• [4] A. Anguraj, M. Mallika Arjunan, Existence results for an impulsive neutral integro- differential equations in Banach spaces, Nonlinear Studies, 16(1) (2009), 33-48.


• [5] D. D. Bainov, P. S. Simeonov, Impulsive Dierential Equations: Periodic Solutions and Applications, Longman Scientific and Technical Group, England (1993)


• [6] K. Balachandran, J. Y. Park, M. Chandrasekaran, Nonlocal Cauchy problem for delay integrodifferential equations of Sobolve type in Banach spaces, Appl. Math. Lett., 15(7) (2002), 845-854.


• [7] M. Benchohra, J. Henderson, S. K. Ntouyas, Existence results for impulsive multi- valued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263(2) (2001), 763-780.


• [8] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive neutral functional differential inclusions in Banach spaces, Appl. Math. Lett., 15(8) (2002), 917-924.


• [9] M. Benchohra, A. Ouahab, Impulsive neutral functional differential equations with variable times, Nonlinear Anal., 55(6) (2003), 679-693.


• [10] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York (2006)


• [11] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162(2) (1991), 494-505.


• [12] L. Byszewski , Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. Systems Appl., 5(4) (1996), 595-605.


• [13] L. Byszewski, H. Akca, Existence of solutions of a semilinear functional differential evolution nonlocal problem, Nonlinear Anal., 34(1) (1998), 65-72.


• [14] L. Byszewski, H. Akca, On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stochastic Anal., 10(3) (1997), 265-271.


• [15] Y. K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.


• [16] Y. K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal.: Hybrid Systems, 2(1) (2008), 209-218.


• [17] Y. K. Chang, V. Kavitha, M. Mallika Arjunan, Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators, Nonlinear Anal.: Hybrid Systems, 4(1) (2010), 32-43.


• [18] C. Cuevas, E. Hernández, M. Rabelo, The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58 (2009), 744-757.


• [19] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York (2003)


• [20] E. Hernández, Existence results for partial neutral integrodifferential equations with nonlocal conditions, Dynam. Systems Appl., 11(2) (2002), 241-252.


• [21] E. Hernández, S. M. Tanaka Aki, Global solutions for abstract functional differential equations with nonlocal conditions, Electronic Journal of Qualitative Theory of Differential Equations, 50 (2009), 1-8.


• [22] E. Hernández, M. McKibben, H. Henríquez, Existence results for abstract impulsive second order neutral functional differential equations, Nonlinear Anal., 70 (2009), 2736-2751.


• [23] E. Hernández, M. Pierri, G. Goncalves, Existence results for an impulsive abstract partial differential equations with state-dependent delay, Comput. Math. Appl. , 52 (2006), 411-420.


• [24] E. Hernández, M. Mckibben, Some comments on: ''Existence of solutions of abstarct nonlinear second-order neutral functional integrodifferential equations'' , Comput. Math. Appl., 50(5-6) (2005), 655-669.


• [25] E. Hernández, M. Rabello, H. R. Henriquez , Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl., 331 (2007), 1135-1158.


• [26] E. Hernández, H. R. Henriquez, Impulsive partial neutral differential equations, Appl. Math. Lett., 19 (2006), 215-222.


• [27] Y. Hino, S. Murakami, T. Naito, In Functional-differential equations with infinite delay, Lecture notes in Mathematics, 1473, Springer-Verlog, Berlin (1991)


• [28] G. L. Karakostas, P. Ch. Tsamatos, Suficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl. Math. Lett., 15(4) (2002), 401-407.


• [29] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)


• [30] J. H. Liu, Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Sys., 6 (1999), 77-85.


• [31] J. Liang, J. H. Liu, Ti-Jun Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Model., 49 (2009), 798-804.


• [32] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publ. Co., Florida (1987)


• [33] S. K. Ntouyas, P. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210 (1997), 679-687.


• [34] W. E. Olmstead, C. A. Roberts, The one-dimensional heat equation with a non- local initial condition, Appl. Math. Lett., 10(3) (1997), 84-94.


• [35] J. Y. Park, K. Balachandran, N. Annapoorani, Existence results for impulsive neutral functional integrodifferential equations with infinite delay, Nonlinear Analysis, 71 (2009), 3152-3162.


• [36] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York-Berlin (1983)


• [37] Y. V. Rogovchenko, Impulsive evolution systems: Main results and new trends, Dynam. Contin. Discrete Impuls. Syst., 3(1) (1997), 57-88.


• [38] Y. V. Rogovchenko, Nonlinear impulsive evolution systems and application to population models, J. Math. Anal. Appl., 207(2) (1997), 300-315.


• [39] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations , World Scientific, Singapore (1995)