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 $f-\varkappa T_{1}f=g_0.$

9 pages