On two problems with displacement at three points of boundary characteristics for a degenerate hyperbolic equation

Abstract

This paper is devoted to proving the well-posedness of the boundary characteristic problem with three displacements for the equation $-(-y)^mu_{xx}+u_{yy}-\frac{m}{2y}u_y=0$ in the characteristic triangle. The proof of the existence of a solution to the problem, based on integral representations of solutions to the modified Cauchy problem, is reduced to solving a functional equation with two shifts p(x) and q(x). A combined method of successive approximations and iterations is used to solve the functional equation. A counterexample is given indicating the importance of the class of functions where the solution to the functional equation is sought.