Invertibility of linear relations generated by integral equation with operator measures

Abstract

We investigate linear relations generated by an integral equation with operator measures on a segment in the infinite-dimensional case. In terms of boundary values, we obtain necessary and sufficient conditions under which these relations possess the properties: is a closed relation; is an invertible relation; the kernel of is finite-dimensional; the range of is closed; is a continuously invertible relation and others. The results are applied to a system of integral equations becoming a quasi-differential equation whenever the operator measures are absolutely continuous as well as to an integral equation with multi-valued impulse action.