Existence of mild solutions of second order evolution integro-differential equations in the Fre'chet spaces

In this article, we shall establish  sufficient conditions for the existence of mild solutions for second order semilinear integro-differential evolution equations in Fre'chet spaces C(\mathbb{R}_+ , E), where E is an Banach space. Our approach is based on the concept of a measure of noncompactness and Tykhonoff fixed point theorem. For illustration we give an example.

Keywords:

Semilinear integro-differential equation , measure of noncompactness , mild solutions , evolution system , Tykhonoff fixed point theorem