Solvability of a nonlocal inverse problem for a fourth-order equation

In this paper, we consider a nonlocal inverse problem of finding an unknown right-hand side with one variable for a fourth-order partial differential equation in a rectangular domain. The eigenfunctions and functions of the corresponding spectral problem and its biorthogonal functions are complete and form a Riesz basis in the space $L_2(0,1)$. Criteria for the uniqueness and existence of a solution to the considered nonlocal inverse problem for a fourth-order equation are established. The uniqueness of the solution of the nonlocal inverse problem follows from the completeness of the system of biorthogonal functions. The solution of the problem is constructed as the sum of a series in terms of eigenfunctions and associated functions of the corresponding spectral problem. Sufficient conditions are established for the boundary functions that guarantee existence and stability theorems for the solution of the problem under consideration. In a closed domain, absolute and uniform convergence of the found solution of the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to $t$ and $x$ two and four times, respectively, depending on the smoothness of the function given the initial conditions. In this case, small denominators arise, which hinder the convergence of these series. It is proved that, depending on the size of the domain, the set of non-zero solutions of the expression in the denominator is not empty. And also, it is shown that if this denominator is equal to zero, then this problem will have a non-trivial solution under homogeneous conditions. It is also proved that the solution of the inverse problem is stable in the norms of the spaces $L_2$, $W_2^n$ and $C(\Omega_\pm)$ with respect to changes in the input data.