Abstract

The paper investigates the solvability of an inverse boundaryvalue problem with an unknown coefficient and the righthand side, depending on the time variable, for the linearized BenneyLuke equation with nonselfadjoint boundary and additional integral conditions. The problem is considered in a rectangular domain. A definition of the classical solution of the problem is given. First, we consider an auxiliary inverse boundaryvalue problem and prove its equivalence (in a certain sense) to the original problem. To investigate the auxiliary inverse boundaryvalue problem, the method of separation of variables is used. By applying the formal scheme of the variable separation method, the solution of the direct boundary problem (for a given unknown function) is reduced to solving the problem with unknown coefficients. Then, the solution of the problem is reduced to solving a certain countable system of integrodifferential equations for the unknown coefficients. In turn, the latter system of relatively unknown coefficients is written as a single integrodifferential equation for the desired solution. Next, using the corresponding additional conditions of the inverse auxiliary boundaryvalue problem, to determine the unknown functions, we obtain a system of two nonlinear integral equations. Thus, the solution of an auxiliary inverse boundaryvalue problem is reduced to a system of three nonlinear integrodifferential equations with respect to unknown functions. A special type of Banach space is constructed. Further, in a ball from a constructed Banach space, with the help of contracted mappings, we prove the solvability of a system of nonlinear integrodifferential equations, which is also the unique solution to the auxiliary inverse boundaryvalue problem. Finally, using the equivalence of these problems the existence and uniqueness of the classical solution of the original problem are proved.

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