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On Solutions of Zero Exponential Type for Some Inhomogeneous Differential-Difference Equations in a Banach Space

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Progress and Challenges in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 54))

Abstract

Let A be a closed linear operator on a Banach space having a bounded inverse operator and f be an entire function of zero exponential type. The problem on well-possedeness of the differential-difference equation w′(z) = Aw(z − h) + f(z) in the space of entire functions of zero exponential type is considered. Moreover, an explicit formula for the zero exponential type entire solution is found.

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References

  1. Balser, W., Duval, A., Malek, S.: Summability of formal solutions for abstract Cauchy problems and related convolution equations. Ulmer Seminare über Funktionalanalysis und Differetialgleichungen. 11, 29–44 (2007)

    Google Scholar 

  2. Bellman, R., Cooke, K.L.: Differential-Difference Equations. Mathematics in Science and Engineering. The RAND Corporation, Santa Monica/New York Academic Press, London (1963)

    MATH  Google Scholar 

  3. Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 11, 129–136 (1980)

    Article  MATH  Google Scholar 

  4. Da Prato, G., Sinestrati, E.: Differential operators with non dense domain. Annali della scuola normale superiore. Di Pisa. 14, 285–344 (1987)

    MATH  Google Scholar 

  5. Dalec’kii, Ju., Kreǐn, M.: Stability of differential equations in Banach space. American Mathematical Society, Providence (1974)

    Google Scholar 

  6. Datko, R.: Linear autonomous neutral differential equations in a banach space. J. Differ. Equ. 25, 258–274 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay equations. Springer, New York (1995)

    Book  MATH  Google Scholar 

  8. Engel, K.I., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    MATH  Google Scholar 

  9. Favini, A., Vlasenko L.: Degenerate non-stationary differential equations with delay in Banach spaces. J. Differ. Equ. 192(1), 93–110 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gefter, S., Stulova, T.: On Holomorphic Solutions of Some Implicit Linear Differential Equation in a Banach Space. Operator Theory: Advances and Applications, vol. 191, pp. 323–332. Birkhauser Verlag, Basel (2009)

    Google Scholar 

  11. Gefter, S., Stulova, T.: On entire solutions of some inhomogeneous linear differential equations in a banach space. In: Proceedings of the 3rd Nordic EWM Summer School for PhD Students in Mathematics. TUCS General Publication, vol. 53, pp. 211–214. Turku Centre for Computer Science, Turku (2009). www.doria.fi/bitstream/handle/10024/66221/TUCSGeneral53.pdf

  12. Gorbachuk, M.: An operator approach to the Cauchy-Kovalevskay theorem. J. Math. Sci. 5, 1527–1532 (2000)

    Article  MathSciNet  Google Scholar 

  13. Gorbachuk, M.: On analytic solutions of operator-differential equations. Ukr. Math. J. 52(5), 680–693 (2000)

    Article  MathSciNet  Google Scholar 

  14. Gorbachuk, M., Gorbachuk, V.: On the well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space. Methods Funct. Anal. Topol. 11(2), 113–125 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  MATH  Google Scholar 

  16. Hille, E.: Ordinary Differential Equations in the Complex Domain. Wiley-InterScience, New York/London (1976)

    MATH  Google Scholar 

  17. Kreǐn, S.: Linear Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 29. American Mathematical Society, Providence (1971)

    Google Scholar 

  18. Sil’chenko, Yu.T.: Differential equations with non-densely defined operator coefficients, generating semigroups with singularities. Nonlinear Anal. A Theory Methods 36(3), 345–352 (1999)

    Article  MathSciNet  Google Scholar 

  19. Sil’chenko, Yu.T., Sobolevskii, P.E.: Solvability of the Cauchy problem for an evolution equation in a Banach space with a non-densely given operator coefficient which generates a semigroup with a singularity (Russian). Sib. Math. J. 27(4), 544–553 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vlasenko, L.A.: On a class of neutral functional differential equations. Funct. Differ. Equ. 13(2), 305–321(2006)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Kharkiv National University, 4 sq. Svoboda, Kharkiv, 61022, Ukraine

    Sergey Gefter & Tetyana Stulova

Authors
  1. Sergey Gefter
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  2. Tetyana Stulova
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Corresponding author

Correspondence to Sergey Gefter .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Santiago Ibáñez

  2. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Jesús S. Pérez del Río

  3. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Antonio Pumariño

  4. Department of Mathematics, University of Oviedo, Oviedo, Spain

    J. Ángel Rodríguez

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Gefter, S., Stulova, T. (2013). On Solutions of Zero Exponential Type for Some Inhomogeneous Differential-Difference Equations in a Banach Space. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_15

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