On class of integral equations with partial integrals and its applications

Abstract

We prove the existence and uniqueness of the solution to one class of systems of integral equations with partial integrals. Equations with partial integrals are equations containing an unknown function in the integrands of integrals of different dimension. The feature of the considered class of integral equations is that the equations involve integrals with both variables and constant upper integration limits. We first prove the unique solvability theorem for integral equations in the three-dimensional space. A similar statement is proved for equations with arbitrary many independent variables. Some applications of the obtained result are provided. For a hyperbolic system with dominant derivatives of the second order with three independent variables, we prove the existence and uniqueness of the solution of the main characteristic problem. The main characteristic problem for the system of equations with higher derivatives of the second order can be considered as an analogue of the Goursat problem for a hyperbolic system with no multiple characteristics. The solution of this problem is constructed explicitly in terms of the Riemann matrix. The Riemann matrix is defined as the solution of a system of Volterra integral equations. The problem with boundary conditions on five sides of the characteristic parallelepiped for this system of equations with higher derivatives of the second order is formulated. By reducing the problem to a system of equations with partial integrals and basing on our results, we prove the existence and uniqueness of the solution to this problem.