Two-dimensional inverse problem for an integro-differential equation of hyperbolic type

A multidimensional inverse problem of determining the kernel of the integral term of an integro-differential wave equation is considered. In the direct problem it is required to find the displacement function from the initial-boundary value problem. In the inverse problem it is required to determine the kernel of the integral term that depends on both the temporal and one spatial variable. Local unique solvability of the problem posed in the class of functions continuous in one of the variables and analytic in the other variable is proved with the use of the method of scales of Banach spaces of real analytic functions.