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On Behavior of Weak Solutions of Operator Differential Equations on (0, ∞)

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Modern Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 191))

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Abstract

The aim of this work is to describe the weak solutions of a first-order differential equation on the interval (0, ∞) in a Banach space and their behavior when approaching to the ends of this interval.

This work was completed with the support of NASU Research Fund (Program 0107U002333).

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References

  1. S.G. Krein, Linear Differential Equations in Banach Space. Nauka, Moscow, 1963.

    Google Scholar 

  2. W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 1999.

    Google Scholar 

  3. P. Koosis, Lectures on H p Spaces. Cambridge University Press, London, 1980.

    Google Scholar 

  4. H. Komatsu, Ultradistributions. J. of the Facul. of Sci. Univ. Tokyo 20 (1973), 25–105.

    MATH  MathSciNet  Google Scholar 

  5. M.G. Krein, Lectures on the Theory of Stability of solutions of Differential Equations in a Banach Space. Inst. of Math. Akad. Nauk Ukrain. SSR, Kiev, 1964.

    Google Scholar 

  6. M.G. Krein, Introduction to Geometry of Indefinite J-Spaces and the Operator Theory in this Spaces. The Second Summer Math School (Katsiveli, June–July 1964). Part 1. Inst. of Math. Akad. Nauk Ukrain. SSR, Kiev, 1965, 15–92.

    Google Scholar 

  7. Yu.L. Daletsky and M.G. Krein, Stability of Solutions of Differential Equations in a Banach Space. Nauka, Moscow, 1970.

    Google Scholar 

  8. E. Nelson, Analytic vectors. Ann. Math. 70 (1959), 572–615.

    Article  Google Scholar 

  9. R. Goodman, Analytic and entire vectors for representations of Lie groups. Trans. Amer. Math. Soc. 143 (1969), 55–76.

    Article  MATH  MathSciNet  Google Scholar 

  10. V.I. Gorbachuk and M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht, 1991.

    Google Scholar 

  11. Ya.V. Radyno, The space of vectors of exponential type. Dokl. Akad. Nauk BSSR 27 (1983), 791–793.

    MATH  MathSciNet  Google Scholar 

  12. A.I. Kashpirovski i, Analytic representation of generalized functions of S-type. Dokl. Akad. Nauk Ukrain. SSR. Ser. A. 4 (1980), 12–14.

    MathSciNet  Google Scholar 

  13. I.M. Gel’fand, On one-parameter groups of operators in a normed space. Dokl. Akad. Nauk SSSR 25 (1939), 713–718.

    Google Scholar 

  14. A. Nussbaum, Quasi-analytic vectors. Ark. Mat. 6 (1965), 179–192.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Beals, Semigroups and abstract Gevrey spaces. J. Funct. Anal. 10 (1972), 300–308.

    Article  MATH  MathSciNet  Google Scholar 

  16. M.L. Gorbachuk, Yu.G. Mokrousov, On density of some sets of infinitely differentiable vectors of a closed operator on a Banach space. Methods Funct. Anal. Topology 8 (2002), 23–29.

    MATH  MathSciNet  Google Scholar 

  17. M.L. Gorbachuk, V.I. Gorbachuk, On approximation of smooth vectors of a closed operator by entire vectors of exponential type. Ukrain. Mat. Zh. 47 (1995), 616–628.

    MATH  MathSciNet  Google Scholar 

  18. J.M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 63 (1977), 370–373.

    Article  MATH  MathSciNet  Google Scholar 

  19. M.L. Gorbachuk, V.I. Gorbachuk, On a generalization of the Berezanskii evolution criterion of self-adjointness of an operator. Ukrain. Mat. Zh. 52 (2000), 608–615.

    MATH  MathSciNet  Google Scholar 

  20. K. Hoffman, Banach Spaces of Analytic Functions. Prentice-Hall, INC. Englewood Cliffs, N. J., 1962.

    MATH  Google Scholar 

  21. V.I. Gorbachuk, A.V. Knyazyuk, Boundary values of solutions of operator differential equations. Uspekhi Mat. Nauk 44 (1989), 55–91.

    MathSciNet  Google Scholar 

  22. V.I. Gorbachuk, M.L. Gorbachuk, Boundary values of solutions of some classes of differential equations. Mat. Sbornik 102 (1977), 124–150.

    Google Scholar 

  23. M.G. Krein, A remark on a theorem in the paper of V.A. Yakubovich titled “A frequency theorem for the case where ...”. Sibirsk. Mat. Zh. 18 (1977), 1411–1413.

    MATH  MathSciNet  Google Scholar 

  24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.

    MATH  Google Scholar 

  25. M.L. Gorbachuk, I.T. Matsishin, The behavior at infinity of solutions of a first-order parabolic differential equations in Banach space. Dokl. Akad Nauk SSSR 312 (1990), 521–524.

    MathSciNet  Google Scholar 

  26. R. deLaubenfels, Inverses of generators. Proc. Amer. Math. Soc. 104 (1988), 443–448.

    Article  MATH  MathSciNet  Google Scholar 

  27. V.I. Gorbachuk, M.L. Gorbachuk, On behavior at infinity of orbits of uniformly stable semigroups. Ukrain. Mat. Zh. 58 (2006), 148–159.

    MATH  Google Scholar 

  28. M.G. Krein, On a generalization of Stieltjes’ investigations. Dokl. Akad. Nauk SSSR 87 (1952), 881–884.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska St., 01601, Kyiv-4, Ukraine

    M. L. Gorbachuk & V. I. Gorbachuk

Authors
  1. M. L. Gorbachuk
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  2. V. I. Gorbachuk
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Vadim M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    Israel Gohberg

  3. Insitute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska st., 01601, Kyiv-4, Ukraine

    Anatoly Kochubei , Yurij Berezansky , Myroslav Gorbachuk  & Valentyna Gorbachuk , ,  & 

  4. Department of Mathematical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Gennadiy Popov

  5. Institute of Analysis and Scientific Computing, Technical University of Vienna, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Heinz Langer

Additional information

To the memory of M. G. Krein

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Gorbachuk, M.L., Gorbachuk, V.I. (2009). On Behavior of Weak Solutions of Operator Differential Equations on (0, ∞). In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 191. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9921-4_7

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