# EXISTENCE OF SOLUTIONS FOR IMPULSIVE NONLINEAR DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

• Selvaraj, B. (DEPT. OF MATH., KARUNYA UNIV.) ;
• Arjunan, M. Mallika (DEPT. OF MATH., KARUNYA UNIV.) ;
• Kavitha, V. (DEPT. OF MATH., KARUNYA UNIV.)
• Accepted : 2009.08.12
• Published : 2009.09.25

#### Abstract

In this article, we study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions u'(t) = Au(t) + f(t, u(t); Tu(t); Su(t)), $0{\leq}t{\leq}T_0$, $t{\neq}t_i$, u(0) + g(u) = $u_0$, ${\Delta}u(t_i)=I_i(u(t_i))$, i = 1,2,${\ldots}$p, 0<$t_1$<$t_2$<$\cdots$<$t_p$<$T_0$, in a Banach space X, where A is the infinitesimal generator of a $C_0$ semigroup, g constitutes a nonlocal conditions, and ${\Delta}u(t_i)=u(t_i^+)-u(t_i^-)$ represents an impulsive conditions.

#### References

1. A. Anguraj and M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Elect. J. Diff. Eqns., Vol. 2005(2005), No.111, 1-8.
2. D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect, Ellis Horwood Ltd., Chichister, 1989.
3. M. Benchohra and S.K. Ntouyas, Existence of mild solutions of semilinear evolution inclusions with nonlocal conditions, Georgian Math. J. 7(2000), 221-230.
4. L. Byszewski, Theorems about the existence and uniqueness of a solution of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162(1991), 496-505.
5. Y.-K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with non-local conditions, J. Appl. Math. & Comp.,(2008), 28(1)(2008), 79-91. https://doi.org/10.1007/s12190-008-0078-8
6. K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179(1993), 630-637. https://doi.org/10.1006/jmaa.1993.1373
7. X. Fu and K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54(2004), 215-227.
8. X. Fu, On solutions of neutral nonlocal evolution equations with nondense domain, J. Math. Anal. Appl., 299(2004), 392-410. https://doi.org/10.1016/j.jmaa.2004.02.062
9. V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
10. J. Liang, J.H. Liu and Ti-Jun Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal. 37(2004), 183-189.
11. J. Liang, J.H. Liu and Ti-Jun Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comp. Model., 49(3-4)(2009), 794-804.
12. J.H. Liu, Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Sys., 6(1999), 77-85.
13. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
14. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.