Non-local boundary value problem with integral conditions for a second order hyperbolic equation

Abstract

In this paper, the classic solution of one-dimensional boundary value problem for a hyperbolic equation with non-classic boundary conditions is investigated. For that, the stated problem is reduced to the not-self-adjoint boundary value problem with equivalent boundary condition. Then, using the method of separation of variables, by means of the known spectral problem the given not self-adjoint boundary value problem is reduced to an integral equation. The existence and uniqueness of the integral equation are proved by means of the contraction mappings principle and it is shown that this solution is unique for a not-adjoint boundary value problem. Finally, using the equivalence, the theorem on the existence and uniqueness of a non-local boundary value problem with integral condition is proved.