Geodesic
Boundary value problems for integral equations with operator measures
Problemy analiza, Tome 6 (2017) no. 1, pp. 19-40 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {V. M. Bruk},
     title = {Boundary value problems for integral equations with operator measures},
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     url = {http://geodesic.mathdoc.fr/item/PA_2017_6_1_a2/}
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V. M. Bruk. Boundary value problems for integral equations with operator measures. Problemy analiza, Tome 6 (2017) no. 1, pp. 19-40. http://geodesic.mathdoc.fr/item/PA_2017_6_1_a2/

[1] Izv. RAN. Ser. Matem., 73:2 (2009), 3–68 | DOI | DOI | MR | Zbl

[2] Uspekhi Mat. Nauk, 68:1 (2013), 77–128 | DOI | DOI | MR | Zbl

[3] Behrndt Jussi, Hassi Seppo, Snoo Henk, Wietsma Rudi, “Squareintegrable Solutions and Weil Functions for Singular Canonical Systems”, Math. Nachr., 284:11–12 (2011), 1334–1384 | DOI | MR | Zbl

[4] Naukova Dumka, Kiev, 1965 | MR | Zbl

[5] Mat. Zametki, 24:4 (1978), 499–511 | DOI | MR | Zbl

[6] Bruk V. M., “On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression”, J. Math. Phys., Anal., Geom., 5:2 (2009), 123–144 | MR | Zbl

[7] Bruk V. M., “On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the Infinite-Dimensional Case”, J. Math. Phys., Anal., Geom., 10:2 (2014), 163–188 | DOI | MR | Zbl

[8] Cross R., Multivalued Linear Operators, Monogr. Textbooks Pure Appl. Math., 213, Dekker, New York, 1998 | MR | Zbl

[9] Orcutt B. C., Canonical Differential Equations, Dissertation, University of Virginia, 1969 | MR

[10] Dokl. Akad. Nauk SSSR, 184:5 (1969), 1034–1037 | MR | Zbl

[11] Vischa Schkola, Kiev, 1987 | DOI | MR | Zbl