On a solvability of the nonlinear inverse problem for the hyperbolic equation

We investigate the nonlinear inverse problem with an unknown time depended coefficient for a second-order hyperbolic equation. The problem is stated as follows: it is required to find a solution and an unknown time depended coefficient at the solution in the equation. In the article we prove the theorems of the existence and the uniqueness of the problem’s regular solution. To study solvability of the inverse problem, we realize a conversion from the initial inverse problem to a some direct nonlocal supplementary problem for a high-order hyperbolic equation. We prove the solvability of the supplementary problem. Then we realize a conversion to the first problem again and as a result we receive the solvability of the inverse problem.