Existence and uniqueness of solutions to the Goursat–Darboux system with integral boundary conditions

Currently, local boundary value problems for hyperbolic-type differential equations have been studied in considerable details. However, mathematical modeling of various real-world processes leads to nonlocal boundary value problems for nonlinear hyperbolic differential equations, which remain insufficiently investigated. This paper is devoted to a general integral boundary value problem in a characteristic rectangle for hyperbolic equations. Under natural conditions on the input data, we construct the Green's function and establish uniqueness criteria for the solution. The proofs of the main results demonstrate the essential nature of the imposed conditions: their violation makes it impossible to construct the Green's function and leads to the loss of required solvability properties. For a special case, by using Banach's contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of the boundary value problem solution. A specific example is provided to illustrate the obtained results.