On Volterra partial integral equations in the space of continuous functions of three variables

It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial integral equations under examination. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.