# American Institute of Mathematical Sciences

September  2006, 6(5): 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

## Existence and uniqueness of nonlinear impulsive integro-differential equations

 1 Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore-641 004, India 2 Department of Mathematics, PSG of Arts and Sciences, Coimbatore, India

Received  October 2005 Revised  February 2006 Published  June 2006

In this paper we study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive integral evolution equation

$u'(t)= Au(t)+f(t,u(t),\int_0^tk(t,s)u(s)ds), t>0, t\ne t_i,$
$u(0)= u_0,$
$\Delta u(t_i)= I_i(u(t_i)). i= 1,2,....,p.$

in a Banach space X, where A is the infinitesimal generator of a strongly continuous semigroup,$\Delta u(t_i)=u(t^+_i)-u(t^-_i)$ and $I's$ are some operator. We apply the semigroup theory to study the existence and uniqueness of the mild solutions, and then show that the mild solution give rise to classical solution if $f$ is continuously differentiable.

Citation: Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
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