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

  • Sergey GefterEmail author
  • Tetyana Stulova
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)


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(zh) + 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.


  1. 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. 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)zbMATHGoogle Scholar
  3. 3.
    Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 11, 129–136 (1980)CrossRefzbMATHGoogle Scholar
  4. 4.
    Da Prato, G., Sinestrati, E.: Differential operators with non dense domain. Annali della scuola normale superiore. Di Pisa. 14, 285–344 (1987)zbMATHGoogle Scholar
  5. 5.
    Dalec’kii, Ju., Kreǐn, M.: Stability of differential equations in Banach space. American Mathematical Society, Providence (1974)Google Scholar
  6. 6.
    Datko, R.: Linear autonomous neutral differential equations in a banach space. J. Differ. Equ. 25, 258–274 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay equations. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    Engel, K.I., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  9. 9.
    Favini, A., Vlasenko L.: Degenerate non-stationary differential equations with delay in Banach spaces. J. Differ. Equ. 192(1), 93–110 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 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).
  12. 12.
    Gorbachuk, M.: An operator approach to the Cauchy-Kovalevskay theorem. J. Math. Sci. 5, 1527–1532 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gorbachuk, M.: On analytic solutions of operator-differential equations. Ukr. Math. J. 52(5), 680–693 (2000)MathSciNetCrossRefGoogle Scholar
  14. 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)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hille, E.: Ordinary Differential Equations in the Complex Domain. Wiley-InterScience, New York/London (1976)zbMATHGoogle Scholar
  17. 17.
    Kreǐn, S.: Linear Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 29. American Mathematical Society, Providence (1971)Google Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vlasenko, L.A.: On a class of neutral functional differential equations. Funct. Differ. Equ. 13(2), 305–321(2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsKharkiv National UniversityKharkivUkraine

Personalised recommendations