Nonlocal inverse problem for determining the unknown coefficient in the beam vibration equation

The article is devoted to the study of the direct problem for the oscillation of a homogeneous beam of finite length with non-local time conditions. Necessary and sufficient conditions for the existence of a solution to the direct problem are obtained. For the direct problem, we study the inverse problem of determining the time-dependent coefficient at the lowest derivative. Using eigenvalues and eigenfunctions, the problem is reduced to a system of integral equations. With the help of the Banach principle, the existence and uniqueness of the solution of inverse problems are shown.