Existence of solutions of nonlinear functional integro-differential equations in Banach spaces

Abstract

We consider the existence of generalized solutions of a nonlinear functional integro-differential equation of type $$ x'(t)+A(t, x_t)x(t)\ni G(t,x_t,\int^t_0k(t,s,x_s)ds),\quad t\in [0,T),\quad x_0=\phi $$ in general Banach spaces. Our study is performed by means of the concept of the Method of Lines and well known Banach fixed point theorems. We extend the results of Kartsatos and Parrott, Tanaka to an abstract nonlinear integro-differential equation.