The work is devoted to the study of the solvability of an inverse boundary value problem with an unknown time-dependent coefficient for the Boussinesq-Love equation with Nonlocal Integral Condition. The goal of the paper consists of the determination of the unknown coefficient together with the solution. The problem is considered in a rectangular domain. The definition of the classical solution of the problem is given. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. Thus, the solution of an auxiliary inverse boundary value problem reduces to a system of three nonlinear integro-differential equations for unknown functions. A concrete Banach space is constructed. Further, in the ball from the constructed Banach space by the contraction mapping principle, the solvability of the system of nonlinear integro-differential equations is proved. This solution is also a unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.

In this paper, we study an inverse boundary value problem with unknown time depend coefficients for fourth-order pseudoparabolic equations with a nonlocal integral conditions of the second kind. The essence of the problem is that it is required, together with the solution, to determine the unknown coefficient. The problem is considered in a rectangular area. Solving the initial inverse boundary value problem, a transition is made from the original inverse problem to some auxiliary inverse problem. Using compressed mappings, the existence and uniqueness of the solution of the auxiliary problem is proved. In the conclusion we obtained the solvability of the original inverse problem.

In this paper it is constructed a new grand grand Sobolev-Morrey $S_{p),\varkappa ),a,\alpha }^{l}W(G)$ spaces with dominant mixed derivatives. With help integral representation of generalized mixed derivatives of functions, defined on $n$-dimensional domains satisfying flexible horn condition, an embedding theorem is proved. In other works, the embedding theorem is proved in these spaces and belonging of the generalized mixed derivatives of functions from these spaces to the Holder class, was studied.

In the present work, we consider an inverse boundary value problem for a fourth order pseudo parabolic equation with periodic and integral condition. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. By the contraction mapping principle is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis.

In the paper the solvability conditions of one class of fourth order elliptic type operator-differential equations are found. These conditions are expressed by some properties of coefficient sof the given operator-differential equation.