Skip to main content
SpringerLink
Log in

Entire Solutions of One Linear Implicit Differential-Difference Equation in Banach Spaces

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish the existence and uniqueness conditions for the solution for the initial problem

$$ {Bu}^{\prime }(z)= Au\left(z+h\right)+f(z),\kern1em z\in \mathbb{C},\kern1em u(0)={u}_0 $$

in the classes of entire vector functions of exponential type. Closed linear operators A and B act on Banach spaces and can be degenerate. We present an example of application of abstract results to partial differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York–London (1963).

    MATH  Google Scholar 

  2. L. A. Vlasenko, Evolutionary Models with Implicit and Degenerate Differential Equations [in Russian], Sistemnye Tekhnologii, Dnepropetrovsk (2006).

  3. S. G. Krein and M. I. Khazan, “Differential equations in Banach spaces,” in: VINITI Series in Mathematical Analysis [in Russian], Vol. 21, VINITI, Moscow (1983), pp. 130–264.

  4. Yu. A. Mitropol’skii, A. M. Samoilenko, and D. I. Martynyuk, Systems of Evolutionary Equations with Periodic and Conditionally Periodic Coefficients [in Russian], Naukova Dumka, Kiev (1985).

  5. A. D. Myshkis, Linear Differential Equations with Retarded Argument [in Russian], Nauka, Moscow (1972).

  6. A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degenerations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).

  7. V. Yu. Slyusarchuk, Absolute Stability of Dynamical Systems with Aftereffects [in Ukrainian], Rivne State University of Water Management and Utilization of Natural Resources, Rivne (2003).

  8. V. I. Fodchuk, Ya. I. Bihun, I. I. Klevchuk, I. M. Cherevko, and I. V. Yakimov, Regularly and Singularly Perturbed Functional-Differential Equations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1996).

  9. W. Hahn, “Bericht ¨uber Differential – Dlfferenzengleichungen mit festen und veränderlichen Spannen,” Jahresbericht der Deutschen Mathematiker – Vereinigung, 57, No. 2, 55–84 (1954).

  10. J. Hale, Theory of Functional Differential Equations, Springer, New York (1977).

  11. S. L. Gefter, A. L. Piven, and A. S. Tanasichuk, “Operator Bruwier series and initial problem for a linear differential-difference equation in a Banach space,” Probl. Mat. Anal., 89, 3–10 (2017); English translation: J. Math. Sci., 226, Issue 4, 335–343 (2017).

  12. S. L. Gefter and T. E. Stulova, “On entire solutions of exponential type for some implicit linear differential-difference equation in a Banach space,” Zap. Nauchn. Sem. POMI, 416, 91–97(2013); English translation: J. Math. Sci., 202, Issue 4, 541–545 (2014).

  13. S. L. Gefter and T. E. Stulova, “On solutions of zero exponential type for some inhomogeneous differential-difference equations in a Banach space,” in: Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics and Statistics, 54 (2013), pp. 253–263.

  14. N. I. Radbel’, “On the initial manifold and dissipativity of the Cauchy problem for the equation Ax′(t) + Bx(t) = 0,” Differents. Uravn., 15, No. 6, 1142–1143 (1979).

  15. A. G. Rutkas, “Cauchy problem for the equation Ax′(t) + Bx(t) = f(t),” Differents. Uravn., 11, No. 11, 1996–2010 (1975).

  16. A. G. Rutkas, “Spectral methods of investigation of degenerate differential-operator equations,” Sovr. Mat. Prilozh., 35, 48–64 (2005).

  17. S. L. Campbell, Singular Systems of Differential Equations I, Pitman Publishing, San Francisco (1980).

  18. L. A. Vlasenko and A. G. Rutkas, “On one class of functional-differential equations with nonatomic difference operator,” Mat. Zametki, 95, Issue 1, 37–49 (2014).

  19. L. A. Vlasenko and A. G. Rutkas, “Optimal control by undamped Sobolev-type systems with delay,” Mat. Zametki, 102, Issue 3, 323–338 (2017).

  20. A. Favini and L. Vlasenko, “Degenerate nonstationary differential equations with delay in Banach spaces,” J. Different. Equat., 192, No. 1, 93–110 (2003).

    Article  Google Scholar 

  21. L. A. Vlasenko, “On a class of neutral functional differential equations,” Funct. Different. Equat., 13, No. 2, 305–321 (2006).

    MathSciNet  MATH  Google Scholar 

  22. V. M. Horbachuk and M. L. Horbachuk, “Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications,” Ukr. Mat. Zh., 69, No. 4, 478–509 (2017); English translation: 69, No. 4, 561–597 (2017).

  23. M. L. Gorbachuk and V. I. Gorbachuk, “On well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space,” Meth. Funct. Anal. Topology, 11, No. 2, 113–125 (2005).

    MathSciNet  MATH  Google Scholar 

  24. S. L. Hefter and T. E. Stulova, “On the well-posedness of a nonresonance operator-differential equation in the space of entire functions of exponential type,” Dop. Nats. Akad. Nauk Ukr., No. 9, 7–12 (2012).

  25. O. Perron, “Über Bruwiersche Reihen,” Math. Z., 45, 127–141 (1939).

    Article  MathSciNet  Google Scholar 

  26. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, AMS Colloq. Publ., Providence, Rhode Island (1957).

Download references

Author information

Authors and Affiliations

  1. Karazin Kharkiv National University, Kharkiv, Ukraine

    S. L. Hefter & O. L. Piven’

Authors
  1. S. L. Hefter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. L. Piven’
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 8, pp. 1044–1057, August, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hefter, S.L., Piven’, O.L. Entire Solutions of One Linear Implicit Differential-Difference Equation in Banach Spaces. Ukr Math J 70, 1205–1220 (2019). https://doi.org/10.1007/s11253-018-1563-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-018-1563-3

Search

Navigation