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

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