Generalized Solutions of the First Boundary Value Problem for a Differential–Difference Equation in Divergence Form on a Finite Interval

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 \(W_2^2(Q) \) if the right-hand side is orthogonal in the space \(L_2(Q) \) to finitely many linearly independent functions.