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 Ufimsk. Mat. Zh., 2014, Volume 6, Issue 4, Pages 50–62 (Mi ufa259)

Invertibility of linear relations generated by integral equation with operator measures

V. M. Bruk

Saratov State Technical University, Saratov, Russia

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.

Keywords: integral equation, operator measure, Hilbert space, linear relation, spectrum, quasiderivative, impulse action.

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English version:
Ufa Mathematical Journal, 2014, 6:4, 48–59 (PDF, 384 kB); https://doi.org/10.13108/2014-6-4-48

UDC: 517.98
MSC: 47A06, 47A10, 34B27

Citation: V. M. Bruk, “Invertibility of linear relations generated by integral equation with operator measures”, Ufimsk. Mat. Zh., 6:4 (2014), 50–62; Ufa Math. J., 6:4 (2014), 48–59

Citation in format AMSBIB
\Bibitem{Bru14} \by V.~M.~Bruk \paper Invertibility of linear relations generated by integral equation with operator measures \jour Ufimsk. Mat. Zh. \yr 2014 \vol 6 \issue 4 \pages 50--62 \mathnet{http://mi.mathnet.ru/ufa259} \transl \jour Ufa Math. J. \yr 2014 \vol 6 \issue 4 \pages 48--59 \crossref{https://doi.org/10.13108/2014-6-4-48} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84928262402}