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Systems of degenerate differential equations in Banach spaces

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Abstract

We consider systems of degenerate differential equations in Banach spaces of a special form. The main instrument of research is the technique of distributions in Banach spaces; namely, the construction of a fundamental operator function introduced by the first author. We translate the results obtained previously for a single equation to the systems of various types and illustrate them with examples.

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References

  1. Chen G. and Zhang H., “Initial boundary value problem for a system of generalized IMBq equations,” Math. Methods Appl. Sci., 27, No. 5, 497–518 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  2. Gantmakher F. R., The Theory of Matrices [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  3. Sidorov N., Loginov B., Sinitsyn A., and Falaleev M., Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Kluwer Acad. Publ., Dordrecht (2002).

    MATH  Google Scholar 

  4. Falaleev M. V., “Fundamental operator-functions of singular differential operators in Banach spaces,” Siberian Math. J., 41, No. 5, 960–973 (2000).

    Article  MathSciNet  Google Scholar 

  5. Falaleev M. V. and Grazhdantseva E. Yu., “Fundamental operator-functions of degenerate differential and difference-differential operators in Banach spaces which have a Noether operator in the principal part,” Siberian Math. J., 46, No. 6, 1123–1134 (2005).

    Article  MathSciNet  Google Scholar 

  6. Falaleev M. V. and Grazhdantseva E. Yu., “Fundamental operator-functions of singular differential operators under spectral boundedness conditions,” Differential Equations, 42, No. 6, 819–825 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  7. Falaleev M. V., “Fundamental operator-functions of singular differential operators under sectorial and radial conditions,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 68–75 (2006).

  8. Vladimirov V. S., Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  9. Vainberg M. M. and Trenogin V. A., The Branching Theory of Solutions to Nonlinear Equations [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  10. Sidorov N. A., Romanova O. A., and Blagodatskaya E. B., “Partial differential equations with the operator of finite index at the principal part,” Differentsial’nye Uravneniya, 23, No. 4, 726–728 (1987).

    MATH  Google Scholar 

  11. Nashed M. Z., Generalized Inverses and Applications, Academic Press, New York (1976).

    MATH  Google Scholar 

  12. Sviridyuk G. A., “On the general theory of operator semigroups,” Russian Math. Surveys, 49, No. 4, 45–74 (1994).

    Article  MathSciNet  Google Scholar 

  13. Sviridyuk G. A. and Fedorov V. E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht (2003).

    MATH  Google Scholar 

  14. Falaleev M. V., “The Cauchy problem for the heat equation with Fredholm operator of the time derivative in Banach spaces,” in: Proceedings of the All-Russian Scientific Conference “Mathematics. Mechanics. Informatics” (Chelyabinsk, 19–22 September, 2006), Chelyabinsk Univ., Chelyabinsk, 2007, pp. 201–210.

    Google Scholar 

  15. Boyarintsev Yu. E., Methods for Solving Degenerate Systems of Ordinary Differential Equations [in Russian], Nauka, Novosibirsk (1988).

    Google Scholar 

  16. Chistyakov V. F., Algebro-Differential Operators with Finite-Dimensional Kernel [in Russian], Nauka, Novosibirsk (1996).

    MATH  Google Scholar 

  17. Korobova O. V., “Singular systems of differential equations in Banach spaces,” Vestnik Irkutsk. Univ. Special Issue: Proceedings of the Annual Scientific-Theoretical Conference of Junior Scientists, Irkutsk Univ., Irkutsk, 2006, pp. 101–103.

    Google Scholar 

  18. Korobova O. V., “Singular systems of differential equations in Banach spaces,” in: Proceedings of the All-Russian Scientific Conference “Mathematics. Mechanics. Informatics” (Chelyabinsk, 19–22 September, 2006), Chelyabinsk Univ., Chelyabinsk, 2006, pp. 72.

    Google Scholar 

  19. Korobova O. V., “Singular systems of differential equations with Fredholm operator of the derivative in Banach spaces,” in: Proceedings of the 3rd Regional Conference “Mathematics and the Problems of Teaching Mathematics in Higher School” Dedicated to the Memory of Professor B. A. Bel’tyukov, Izdat. Irkutsk Ped. Inst., Irkutsk, 2007, pp. 51–53.

    Google Scholar 

  20. Falaleev M. V., “Generalized solutions of a linearized system of Boussinesq equations,” in: Proceedings of the 3rd Regional Conference “Mathematics and the Problems of Teaching Mathematics in Higher School” Dedicated to the Memory of Professor B. A. Bel’tyukov, Izdat. Irkutsk Ped. Inst., Irkutsk, 2007, pp. 77–79.

    Google Scholar 

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Authors and Affiliations

  1. Irkutsk State University, Irkutsk, Russia

    M. V. Falaleev & O. V. Korobova

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  1. M. V. Falaleev
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  2. O. V. Korobova
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Correspondence to M. V. Falaleev.

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Original Russian Text Copyright © 2008 Falaleev M. V. and Korobova O. V.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 916–927, July–August, 2008.

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Falaleev, M.V., Korobova, O.V. Systems of degenerate differential equations in Banach spaces. Sib Math J 49, 734–743 (2008). https://doi.org/10.1007/s11202-008-0070-4

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  • DOI: https://doi.org/10.1007/s11202-008-0070-4

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