Volterra integral operators in spaces of functions of two variables

Abstract

The solvability in the spaces C and Lp, 1⩽p⩽ ∞, of the linear equation $$\lambda x(t,s) = Ax(t,s) + f(t,s),$$ , where $$Ax(t,s) = c(t,s)x(t,s) + \int\limits_0^t {m(t,\tau ,s)x(\tau ,s)d\tau + } \int\limits_0^s n (t,s,\sigma )x(t,\sigma )d\sigma + \int\limits_0^t {\int\limits_0^s {k(t,\tau ,s,\sigma )x(\tau ,\sigma )d\tau d\sigma .} } $$ is investigated.