A problem with an integral condition of the first kind for an equation of the fourth order

The article deals with a non-local problem with an integral condition for fourth-order pseudo-hyperbolic equation. The equation contains both a mixed derivative and a fourth order derivative in the spatial variable. The integral condition is a condition of the first kind, which leads to difficulties in the study of solvability of a problem. One of the successful methods of overcoming the difficulties of such a plan is the transition from the conditions of the first kind to the conditions of the second kind. The article proves the equivalence of the conditions of the first kind to the conditions of the second kind for this problem. The conditions on the coefficients of the equation and the input data are obtained and they guarantee the existence of a single problem solving. In the literature, such an equation is called the Rayleigh–Bishop equation.