On solvability of a partial integral equation in the space ${L_2(\Omega\times\Omega)}$

TL;DR

This paper studies the solvability of a specific partial integral equation in the space of square-integrable functions over a multi-dimensional domain, introducing a determinant concept and providing explicit solutions for continuous kernels.

Contribution

It introduces a determinant for the partial integral equation and offers explicit solutions for continuous kernels, advancing understanding of solvability conditions.

Findings

Defined a determinant as a continuous function on the domain

Provided explicit solutions for equations with continuous kernels

Analyzed solvability conditions in the $L_2$ space

Abstract

In this paper we investigate solvability of a partial integral equation in the space $L_2(\Omega\times\Omega),$ where $\Omega=[a,b]^\nu.$ We define a determinant for the partial integral equation as a continuous function on $\Omega$ and for a continuous kernels of the partial integral equation we give explicit description of the solution.