## 1 Introduction
The linear operator \(B:X\to X\) is a self-composition of the operator \(B_{1}:X\to X\),  
\[B_{1}u=Au-G\Psi(Au),\quad D(B_{1})=\{u\in D(A):\Theta(u)=N\mathrm{Y}(u)\},\]  
where \(A:X\to X\) is a maximal \(n\)-th order linear ordinary differential operator, \(G=(g_{1},\ldots,g_{m})\in X^{m}\), \(\Psi=\mathrm{col}(\Psi_{1},\ldots,\Psi_{m})\) is a vector of \(m\) linear bounded functionals on \(X,\Theta=\mathrm{col}(\Theta_{1},\ldots,\Theta_{n}),\;\mathrm{Y}=\mathrm{col}( Y_{1},\ldots,Y_{m})\) are vectors of bounded linear boundary functionals, \(N\) is a \(n\times m\) constant matrix, \(f\in X\), and \(u\in D(B)\) is unknown. The paper [37] deals with the decomposition and the exact solution of the boundary value problem  
\[Bu=Au-S\Phi(u)-G\Psi(A_{1}u)=f,\quad D(B)=D(A),\]  
where \(B:X\to X\) is a linear operator, \(A,\,A_{1}:X\to X\) are linear differential operators of order \(n\) and \(n_{1}\), respectively, \(S=(s_{1},\ldots,s_{m})\), \(G=(g_{1},\ldots,g_{m})\in X^{m}\), \(\Phi=\mathrm{col}(\Phi_{1},\ldots,\Phi_{m})\), \(\Psi=\mathrm{col}(\Psi_{1},\ldots,\Psi_{m})\) are vectors of linear bounded functionals on \(X,f\in X\), and \(u\in D(B)\). The strict assumptions that were made and the requirement that the operators \(A_{1},\,A_{2}\) must be known in advance in [35] are restrictive in implementing the factorization technique in solving practical boundary value problems. On the other hand, the formulation in Banach space in [36; 37] for solving the corresponding two problems above has been proved more convenient. Here, we generalize the results in [35] in Banach space and at the same time loosen the requirements. Further, we provide a step by step procedure for the factorization and solution of the boundary value problem. In particular, in a Banach space \(X\) of complex valued functions of \(x\) defined on the region \(\Omega\subseteq\mathbb{R}^{k},\;k\in\mathbb{N},\;k\geq 1\), we consider the following boundary value problem  
\[Bu = Au-\sum_{j=1}^{m}\int_{\Omega}K_{\mathrm{ij}}(x,t)A_{1}u(t)dt- \sum_{j=1}^{m}\int_{\Omega}K_{j}(x,t)Au(t)dt=f,\quad x\in\Omega,\] \[D(B) = D(A)\subset X, \tag{1}\]  
where \(B:X\to X\) is a linear operator with its domain \(D(B)\) specified by the prescribed multipoint and integral conditions, \(A:X\to X\) is a bijective linear differential operator of order \(n\) defined on the same domain as the operator \(B\), \(A_{1}:X\to X\) is a bijective linear differential operator of order \(n_{1}<n\) whose domain \(D(A_{1})\) is determined by some multipoint and integral conditions, \(K_{\mathrm{ij}}(x,t)\), \(K_{\mathrm{j}}(x,t)\in X(\Omega\times\Omega)\) are known kernel functions, \(f(x)\in X\) is an input function and \(u(x)\in D(B)\) is the unknown function. We first examine under which conditions this problem can be decomposed into two simpler boundary value problems and then by solving these lower-order problems we derive its unique solution in closed form. In addition, we investigate the factorization and solution of the boundary value problem  
\[Bu = A^{2}u-\sum_{j=1}^{m}\int_{\Omega}K_{1j}(x,t)Au(t)dt-\sum_{j=1}^{m} \int_{\Omega}K_{j}(x,t)A^{2}u(t)dt=f,\quad x\in\Omega,\] \[D(B) = D(A^{2})\subset X, \tag{2}\]  
which is a special case of the boundary value problem (1). The paper is organized as follows. Some preliminaries are given in Section 2. Next, the main results are presented in Section 3. In Section 4, two examples problems, one for ordinary and one for partial integro-differential equations, are solved to elucidate the implementation and highlight the efficiency of the procedure. Finally, in Section 5 some conclusions are given.