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

This article is cited in 14 scientific papers (total in 14 papers)

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
Received: 10.06.1976

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
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    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  mathscinet  isi
    2. Kashpirovsky A., “Analytic Representation of Generalized-Functions of S'-Type”, no. 4, 1980, 12–14  mathscinet  isi
    3. N. M. Suleimanov, “Theorems of Wiman–Valiron type for solutions of parabolic equations”, Math. USSR-Sb., 43:1 (1982), 63–84  mathnet  crossref  mathscinet  zmath
    4. Gorbachuk V., Gorbachuk M., “Trigonometric Series and Generalized Periodic-Functions”, 257, no. 4, 1981, 799–804  mathscinet  isi
    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  mathscinet  zmath  isi
    6. Suleimanov N., “One of the Refinements of Wiman-Valiron-Type Estimates for Solutions of Evolutional Equations”, 265, no. 3, 1982, 545–546  mathscinet  isi
    7. Fedorova L., “Boundary-Values of Solutions for Inhomogeneous Differentially-Operator Equations”, no. 7, 1983, 21–24  mathscinet  isi
    8. Knyazyuk A., “Boundary-Values of Solutions of Differential-Equations in the Banach-Space”, no. 9, 1984, 12–13  mathscinet  isi
    9. Gorbachuk M., Pivtorak N., “Solutions of Evolution Parabolic Equations with Degeneration”, Differ. Equ., 21:8 (1985), 892–897  mathscinet  zmath  isi
    10. Bondarenko N., “Self-Conjugate Expansions of the Minimal Operator Generated by the Degenerating Differential-Operator Sturm-Liouville Equation”, no. 8, 1987, 3–6  mathscinet  zmath  isi
    11. V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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