Nonlocal Inverse Problem for Determining the Unknown Coefficient in the Beam Vibration Equation

The paper is devoted to the study of the direct problem for the vibration of a homogeneous beam of finite length with nonlocal time conditions. Necessary and sufficient conditions for the existence of a solution of the direct problem are obtained. For the direct problem, we study the inverse problem of determining the time-dependent coefficient multiplying a lower-order derivative. Using the eigenvalues and eigenfunctions, the problem is reduced to a system of integral equations. The existence and uniqueness of the solution of inverse problems are shown using the Banach principle.