Inverse problem for an integro-differential equation of hyperbolic type with additional information of a special form in a bounded domain

A one-dimensional inverse problem of determining the kernel of the integral term of an integro-differential equation of hyperbolic type in a variable-bounded domain $x$ is considered. Firstly, the direct problem is investigated, for the regular part of which the Cauchy problem on the axis $x=0$ is obtained using the method of singularity extraction. Subsequently, an integral equation for the unknown function is derived by the d'Alembert formula.
For the direct problem, the inverse problem of determining the kernel entering the integral term of the equation is studied. To find it, an additional condition is specified in a special form. As a result, the inverse problem is reduced to an equivalent system of integral equations for unknown functions. The principle of contraction mappings in the space of continuous functions with weighted norms is applied to the obtained system.
For the given problem, a theorem of global unique solvability has been proven, which is the main result of the study.