Global Solvability of the Kernel Identification Problem for the Integro-Differential Equation of Beam Vibrations

We investigate the inverse problem of identifying the kernel that characterizes the memory effects of the medium in the integro-differential equation governing the forced vibrations of a beam. The direct problem is formulated as a Cauchy problem for this equation, which is subsequently reduced to a system of second-kind integral equations of the Volterra type involving the solution of the direct problem and the unknown kernel of the inverse problem. To address this system, we apply the method of compressive mappings within the space of continuous functions equipped with an exponential weight norm. Our analysis establishes the global solvability of the proposed inverse problem.