Yu. Kh. Eshkabilov
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Methods Funct. Anal. Topology 15 (2009), no. 1, 67-73
Let $T_1: L_2(\Omega^2) \to L_2(\Omega^2)$ be a partial integral operator [4,7] with the kernel from $C(\Omega^3)$ where $\Omega=[a,b ]^ u$, $ u \in N$ is fixed. In this paper we investigate solvability of the partial integral equation $f-\varkappa T_1 f=g_0$ in the space $L_2(\Omega^2)$ in the case where $\varkappa$ is a cha ac eristic number. We prove a the theorem that gives a necessary and sufficient condition for solvability of the partial integral equation $f-\varkappa T_1 f=g_0.$
Methods Funct. Anal. Topology 14 (2008), no. 4, 323-329
In this paper we investigate solvability of a partial integral equation in the space $L_2(\Omega\times\Omega),$ where $\Omega=[a,b]^ u.$ 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.