Abstract:
We consider integral equations
with operator measures on a segment in the infinite-dimensional case.
These measures are defined on Borel sets of the segment and take values in the set of linear bounded operators acting in a separable Hilbert space.
We prove that these equations have unique solutions
and we construct a family of evolution operators. We apply the obtained results to the study of linear relations generated by an integral equation and boundary conditions.
In terms of boundary values,
we obtain necessary and sufficient conditions under which these relations $T$ possess the properties: $T$ is a closed relation; $T$ is an invertible relation; the kernel of $T$ is finite-dimensional;
the range of $T$ is closed; $T$ is a continuously invertible relation and others. We give examples to illustrate the obtained results.
Keywords:Hilbert space, integral equation, boundary value problem, operator measure, linear relation.
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