Factorization method for solving nonlocal boundary value problems in Banach space

Abstract

This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B₁ u = A u - S Φ( u )- G Ψ(A₀ u) = f, u ∈ D (B₁) , where A, A₀ are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B₁ under certain conditions can be factorized into a product of two simpler lower order operators as B ₁ = BB₀. Then the solvability and the unique solution of the equation B₁ u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B₀ u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations.