Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a Banach space

The solvability condition and the unique exact solution by the universal factorization (decomposition) method for a class of the abstract operator equations of the type
$$ B_1u=\mathcal{A}u-S\Phi(A_0u)-GF(\mathcal{A}u)=f ,\quad u\in D(B_1), $$
where $\mathcal{A}, A_0$ are linear abstract operators, $G, S$ are linear vectors and $\Phi, F$ are linear functional vectors is investigagted. This class is useful for solving Boundary Value Problems (BVPs) with Integro-Differential Equations (IDEs), where $\mathcal{A}, A_0$ are differential operators and $F(\mathcal{A}u), \Phi(A_0u)$ are Fredholm integrals. It was shown that the operators of the type $B_1$ can be factorized in the some cases in the product of two more simple operators $B_G$, $B_{G_0}$ of special form, which are derived analytically. Further the solvability condition and the unique exact solution for $B_1u=f$ easily follow from the solvability condition and the unique exact solutions for the equations $B_G v=f$ and $B_{G_0}u=v$.