Solvability and Green’s Function of a Degenerate Boundary Value Problem on a Graph

Abstract

We study conditions for the solvability of boundary value problems for differential equations of arbitrary order on a geometric graph. The boundary conditions are given by func-tionals that are linear combinations of the one-sided limits of the solution and its derivatives calculated at all graph vertices. The dimensions of the linear spaces of solutions of homogeneous mutually adjoint boundary value problems are proved to be the same. Conditions for the solvability and unique solvability of a degenerate boundary value problem are established. A generalized Green’s function is constructed, its uniqueness is proved, and its properties are described. A theorem on the uniform convergence of the sequence of solutions of the degenerate boundary value problem under the condition of uniform convergence of its right-hand sides is proved.