The necessary and sufficient condition for solvability of a partial integral equation

TL;DR

This paper establishes the exact necessary and sufficient conditions for the solvability of a specific partial integral equation involving a partial integral operator with continuous kernel, focusing on cases where the scalar parameter is an eigenvalue.

Contribution

It provides a complete characterization of solvability conditions for a class of partial integral equations with continuous kernels, extending understanding of their spectral properties.

Findings

Derived the necessary and sufficient conditions for solvability.

Characterized the spectral properties related to the characteristic number.

Extended the theory of partial integral equations with continuous kernels.

Abstract

Let $T_1: L_2(\Omega^2) \to L_2(\Omega^2)$ be a partial integral operator with the kernel from $C(\Omega^3)$ where $\Omega=[a,b ]^\nu.$ In this paper we investigate solvability of a partial integral equation $f-\varkappa T_1 f=g_0$ in the space $L_2(\Omega^2)$ in the case when $\varkappa$ is a characteristic number. We proved the theorem describing the necessary and sufficient condition for solvability of the partial integral equation $f-\varkappa T_1 f=g_0.$