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 Mat. Sb. (N.S.), 1977, Volume 102(144), Number 1, Pages 124–150 (Mi msb2641)

Boundary values of solutions of some classes of differential equations

V. I. Gorbachuk, M. L. Gorbachuk

Abstract: Differential equations of the following forms are considered:
$$y'+Ay=0\quadand\quad-y"+A^2y=0,$$
where $A$ is a positive selfadjoint operator in a Hilbert space $H$. The question of whether the solutions of such equations have boundary values at the end points of the interval $(a,b)$ on which they are considered is investigated, as well as the problem of recovering a solution from its boundary values. A characterization of the boundary values is given in terms of the behavior of the solution near the end points $a$ and $b$. A number of examples are cited in which $A$ is realized as a differential operator in various function spaces. When applied to these concrete situations, the abstract theorems yield the existence and characteristics of the boundary values for certain classes of elliptic and parabolic equations; in particular, the well-known results of F. Riesz, Köthe and Komatsu are obtained and sharpened in this way. The approach is based on the spectral theory of selfadjoint operators.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 31:1, 109–133

Bibliographic databases:

UDC: 517.947.5.37
MSC: Primary 34G05; Secondary 35J05, 35K05

Citation: V. I. Gorbachuk, M. L. Gorbachuk, “Boundary values of solutions of some classes of differential equations”, Mat. Sb. (N.S.), 102(144):1 (1977), 124–150; Math. USSR-Sb., 31:1 (1977), 109–133

Citation in format AMSBIB
\Bibitem{GorGor77} \by V.~I.~Gorbachuk, M.~L.~Gorbachuk \paper Boundary values of solutions of some classes of differential equations \jour Mat. Sb. (N.S.) \yr 1977 \vol 102(144) \issue 1 \pages 124--150 \mathnet{http://mi.mathnet.ru/msb2641} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=486970} \zmath{https://zbmath.org/?q=an:0356.34066|0392.34040} \transl \jour Math. USSR-Sb. \yr 1977 \vol 31 \issue 1 \pages 109--133 \crossref{https://doi.org/10.1070/SM1977v031n01ABEH002293} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977FV43600007} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Suleimanov N., “On the Exactness of Wiman-Valiron-Type Estimates for the Solutions of Evolutionary Equations”, 253, no. 3, 1980, 541–544
2. Kashpirovsky A., “Analytic Representation of Generalized-Functions of S'-Type”, no. 4, 1980, 12–14
3. N. M. Suleimanov, “Theorems of Wiman–Valiron type for solutions of parabolic equations”, Math. USSR-Sb., 43:1 (1982), 63–84
4. Gorbachuk V., Gorbachuk M., “Trigonometric Series and Generalized Periodic-Functions”, 257, no. 4, 1981, 799–804
5. Gorbachuk M., Dudnikov P., “On Initial Data of the Cauchy-Problem for Parabolic Equations for Which the Solution Is Infinitely Differentiable”, no. 4, 1981, 9–11
6. Suleimanov N., “One of the Refinements of Wiman-Valiron-Type Estimates for Solutions of Evolutional Equations”, 265, no. 3, 1982, 545–546
7. Fedorova L., “Boundary-Values of Solutions for Inhomogeneous Differentially-Operator Equations”, no. 7, 1983, 21–24
8. Knyazyuk A., “Boundary-Values of Solutions of Differential-Equations in the Banach-Space”, no. 9, 1984, 12–13
9. Gorbachuk M., Pivtorak N., “Solutions of Evolution Parabolic Equations with Degeneration”, Differ. Equ., 21:8 (1985), 892–897
10. Bondarenko N., “Self-Conjugate Expansions of the Minimal Operator Generated by the Degenerating Differential-Operator Sturm-Liouville Equation”, no. 8, 1987, 3–6
11. V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111
12. Litovchenko V.A., “The Cauchy Problem for a Class of Evolution Systems of the Parabolic Type with Unbounded Coefficients”, Differ. Equ., 44:6 (2008), 835–854
13. Gorbachuk M.L., Gorbachuk V.I., “On Behavior of Weak Solutions of Operator Differential Equations on (0, Infinity)”, Modern Analysis and Applications: Mark Krein Centenary Conference, Vol 2, Operator Theory Advances and Applications, 191, eds. Adamyan V., Berezansky Y., Gohberg I., Gorbachuk M., Gorbachuk V., Kochubei A., Langer H., Popov G., Birkhauser Verlag Ag, 2009, 115–126
14. Horbachuk V.M., Horbachuk M.L., “Spaces of Smooth and Generalized Vectors of the Generator of An Analytic Semigroup and Their Applications”, Ukr. Math. J., 69:4 (2017), 561–597
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