APPROXIMATION OF MILD SOLUTIONS OF DELAY INTEGRO-DIFFERENTIAL EQUATIONS ON BANACH SPACES

. The aim of this work is to study an approximation of the mild solution of a delay semi linear integro-differential equation with an initial history condition & phi;. Using resolvent operators theory in the sense given by R. Grimmer, we can ensure an explicit form to the mild solution u & phi; of our considered equation. The approximation takes into account the mild solutions (u & phi;& sigma; )& sigma; >0 of the related family of integro-differential equations with piecewise constant arguments. Our main results is to show that u & phi;& sigma; converges to u & phi; as & sigma; & RARR; 0 uniformly on compact and unbounded intervals. For the error function, we receive an explicit exponential decay estimates by using the stability of the resolvent operator and the Halanay's Inequality. We also show that the approximation is stable and that the solution of the delayed integro-differential equation and its associated difference equation produced via piecewise constant arguments DEPCA method are asymptotically stable. In the end, some examples are given to illustrate our basic results.