A problem with a non-local condition for a fourth-order equation with multiple characteristics

Abstract

In this article, we consider a non-local problem with an integral condition for a fourth-order equation. The unique solvability of the problem is proved. The proof of the uniqueness of a solution is based on the a priori estimates derived in the paper. To prove the existence of a solution, the problem is reduced to two Goursat problems for second-order equations, and the equivalence of the stated problem and the resulting system of Goursat problems is proved. One of the problems of the system is the classical Goursat problem. The second problem is a characteristic problem for an integro-differential equation with a non-local integral condition on one of the characteristics. It is impossible to apply the well-known methods of substantiating the solvability of problems with conditions on characteristics to the study of this problem. The introduction of a new unknown function made it possible to reduce the second problem to an equation with a completely continuous operator, to verify, on the basis of the uniqueness theorem, that it is solvable and, by virtue of the proven equivalence of the problems, that the problem posed is solvable.