A linear inverse problem with semi-nonlocal boundary conditions for a three-dimensional equation of mixed type of the second kind of the fourth order in a parallelepiped

TL;DR

This paper proves the existence and uniqueness of solutions for a three-dimensional fourth-order mixed type inverse problem with semi-nonlocal boundary conditions in a parallelepiped, using advanced mathematical methods.

Contribution

It establishes the first rigorous existence and uniqueness theorems for this specific inverse problem with semi-nonlocal boundary conditions.

Findings

Existence of a generalized solution is proven.

Uniqueness of the solution is established.

The solution is shown to be in a certain class of integrable functions.

Abstract

In this article the correctness of al inear inverse problem with semi-nonlocal boundary conditions for a three-dimensional equation in a parallelepiped is considered. The equation itself is a fourth order mixed type equation of the second kind. The existence and uniqueness theorems for a generalized solution of the inverse problem in a certain class of integrable functions are~proved using the methods of Fourier, "$\varepsilon $-regularization", a priori estimates, approximating sequences and contracting mappings.