Nonlinear integro-differential equation of pseudoparabolic type with nonlocal integral condition

The mathematical modeling of many processes occurring in the real world leads to the study of direct and inverse problems for equations of mathematical physics. Direct problems for partial differential and integro-differential equations by virtue of their importance in the application are one of the most important parts of the theory of differential equations. In the case, when the boundary of the flow of physical process is not applicable for measurements, as an additional information can be used on nonlocal conditions in the integral form.
We propose a method of studying the one-value solvability of the nonlocal problem for a nonlinear third-order integro-differential equation. Such type of differential equations models many natural phenomena and appears in many fields of sciences. For this reason, a great importance was given to this type of equations in the works of many researchers.
We use the Fourier method of separation of variables. The application of this method of separation of variables can improve the quality of formulation of the given problem and facilitate the processing procedure.
So in this article the author studies the questions of one-value solvability of nonlocal mixed-value problem for nonlinear pseudoparabolic type of integro-differential equation. By applying the Fourier method of separation of variables, the author obtained the countable system of nonlinear integral equations (CSNIE). The theorem of one-value solvability of CSNIE is proved using the method of successive approximations in combination with the method of compressing mapping. Further the author showed the convergence of Fourier series to unknown function—to the solution of the nonlocal mixed-value problem. It is also checked that the solution of the given is smooth. Every estimate was obtained with the help of the Holder inequality, Minkovski inequality and Bessel-type inequality. This paper advances the theory of nonlinear partial integro-differential equations.