# Article

Ufa Mathematical Journal
Volume 6, Number 4, pp. 48-59

# Invertibility of linear relations generated by integral equation with operator measures

Bruk V.M.

DOI:10.13108/2014-6-4-48

We investigate linear relations generated by an integral equ\-a\-tion 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 $S$ possess the properties: $S$ is a closed relation; $S$ is an invertible relation; the kernel of $S$ is finite-dimen\-si\-o\-nal; the range of $S$ is closed; $S$ 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.