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In this paper, the numerical-analytic method has been used to study the existence and approximation of the periodic solutions for the vector T-system of new nonlinear multiple integro-differential equations of mixed (Volterra-Fredholm) and (Fredholm-Volterra) types. Our main task provided sufficient conditions for investigating the family continuation theorems (numerical-analytic method and Banach fixed point theorem) in compact spaces for the existence of periodic solutions for the vector T-system of nonlinear multiple integrodifferential equations. All functions satisfies a Hölder condition (Hölder inequality) of orders α, β and γ where 0<α, β, γ<1.

Periodic solutions for nonlinear systems of multiple integro-integral differential equations of (V F) and (F V) type with isolated singular kernels