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The purpose of this paper is to investigate the existence, uniqueness, and stability solution to a non-linear system of integro-differential equations by using both methods Picard approximation and Banach fixed point theorem. The existence, uniqueness, and stability theorems are established under necessary and sufficient conditions of closed and bounded domain.

This article presented some theorems on a novel non-linear multiple Integro-differential equations of boundary T-system. It has been studied the numerical-analytic method and Banach fixed point theorem for the existence and approximation of the solutions over considered boundary system in compact space. In this work, we demonstrate that the mention methods can be discussed and analyzed for the existence and uniqueness, of a solution for the vector system. The paper satisfies the Hölder condition.