Obobshchennye resheniya pervoy kraevoy zadachi dlya differentsial'no-raznostnogo uravneniya v divergentnom vide na intervale konechnoy dliny

Abstract

We consider the Dirichlet problem for a second-order differential–difference equation in divergence form with variable coefficients on a finite interval Q = (0, d). Conditions on the right-hand side of the equation ensuring the smoothness of the generalized solution on the entire interval are studied. It is proved that the generalized solution of the problem belongs to the Sobolev space W22(Q) if the right-hand side is orthogonal in the space L2(Q) to finitely many linearly independent functions