Decomposition of abstract linear operators in Banach spaces

In this paper we investigate a class of integro-differential equations οn Banach space with nonlocal and initial boundary conditions in terms an abstract operator equation
 B1 x=Ax-S0 F(A x)-G0 Φ(Ax)=f ,x∈D(B1) (1) 
 where A, A are linear operators, S0, G0 vectors Φ, F the functional vectors. The under study has decomposition form B1=B0 B B0 being different operators special forms. Methods. Extensions on spaces used. Decomposition Method proposed here is essentially from other Methods relevant literature. Results. Our main research result existence uniqueness solution its representation closed form. necessary sufficient for correctness intermediate, secondary results. Overall, direct method analytically solving problem proposed, algorithmic procedure that reproducible any program symbolic calculations. stages illustrated by three examples. Computer algebra system Mathematica employed to demonstrate outcomes.