Boundary value problems with interior point boundary conditions

Abstract

Rather than approach the problem via the Green's function, this article considers the problem as that of a differential operator in a Hubert space, derives the adjoint operator, whose domain specifies the adjoint boundary conditions, and then produces necessary and sufficient conditions for selfad jointness. To do this we employ a variation of the fundamental lemma of the calculus of variations in the Hubert space setting, and we note our method is applicable even when the Green's function fails to exist. For convenience we only consider a first order vector equation, although our results are easily extended to n-ϊh order vector systems. Finally, our method is extendable to systems whose boundary conditions are applied at an infinite set of points. We hope to pursue this line in a future paper. I* The problem and its adjoint. Let us consider an interval [α, b] which is subdivided into m subintervals by Gi, α2, , am^(a = α0 < a, < < am_, < am = b) . We denote by H the Hubert space of n x 1 vectors X = (x19 x2, , xnγ , Y = (ylf y2, , yn)* , defined on [α, b] whose components are in L 2(α, 6) and whose inner product is given by