In this study, we investigate the existence, uniqueness, and stability solutions of Volterra integrodifferential equations with retarded argument and symmetric matrices. The Picard approximation method and Banach fixed point theorem have been used in this study. Theorems on the existence and uniqueness of a solution are established under some necessary and sufficient conditions on closed and bounded domains (compact spaces).

We study the existence, uniqueness and stability solutions of new Volterra integrodifferential equations of the first order by using Picard approximation method, Banach fixed point theorem. Theorems on the existence, uniqueness and stability solutions are established under some necessary and sufficient conditions on closed and bounded domains. Furthermore the study of such nonlinear of Volterra integro-differential equations leads us to improve and extend the above methods and thus the non-linear Volterra integrodifferential equations that we have introduced in this study become more general and detailed than those introduced some results by Butris and Rafeq (2011).