Extension method for a class of loaded differential equations with nonlocal integral boundary conditions

In this paper we investigate a class of loaded ordinary differential equations with nonlocal integral boundary conditions in terms of an abstruse operator equation Bu = A 2 u - q Ψ( u ) = f, f ∈ Y, (1) D ( B ) = { u ∈ D ( A 2) : Φ( u ) = NF ( Au ) , Φ( Au ) = PF ( Au )} . A loaded part and nonlocal integral boundary conditions of these equations are described using functional vectors Ψ( u ) and F ( Au ) , respectively. Such equations follow from Extension Theory of linear operators. The necessary and sufficient solvability conditions of these equations are given by the determinant of some matrix. In the case when this determinant is nonzero, a direct method for exact solution of this class of loaded differential equations is proposed. If some problem can be reduced to the type of equation under consideration, then it can be easily solved using the extension method. This method, for q = 0 -> , also gives the necessary and sufficient solvability conditions and the exact solution of a class of ordinary differential equations with nonlocal integral boundary conditions in terms of an abstruse operator equation Bu = A 2 u = f, D ( B ) = { u ∈ D ( A 2) : Φ( u ) = NF ( Au ) , Φ( Au ) = PF ( Au )} , f ∈ Y.