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