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

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 $f-\varkappa T_1 f=g_0.$