Skip to main content
Log in

To Favard’s theory for functional equations

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain some conditions for existence of almost periodic solutions for almost periodic functional equations in a Banach space which do not use the H-classes of these equations.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Levitan B. M., Almost-Periodic Functions [Russian], Gostekhizdat, Moscow (1953).

    MATH  Google Scholar 

  2. Demidovich B. P., Lectures on the Mathematical Theory of Stability [Russian], Nauka, Moscow (1967).

    MATH  Google Scholar 

  3. Mukhamadiev E., “On the inversion of functional operators in a space of functions bounded on the axes,” Math. Notes, vol. 11, no. 3, 169–172 (1972).

    Article  MATH  Google Scholar 

  4. Mukhamadiev E., “Investigations in the theory of bounded and periodic solutions of differential equations,” Math. Notes, vol. 30, no. 3, 713–722 (1981).

    Article  MATH  Google Scholar 

  5. Slyusarchuk V. Yu., “Almost periodic solutions of nonlinear discrete systems that are not necessarily almost periodic in Bochner’s sense,” Nonlinear Oscillations, vol. 17, no. 3, 407–418 (2014).

    Google Scholar 

  6. Amerio L., “Soluzioni quasiperiodiche, o limital, di sistemi differenziali non lineari quasi-periodici, o limitati,” Ann. Mat. Pura Appl., vol. 39, 97–119 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  7. Kolmogorov A. N. and Fomin S. V., Elements of the Theory of Functions and Functional Analysis, Dover Publications, Mineola (1999).

    MATH  Google Scholar 

  8. Soukhomlinoff G., “über Fortsetzung von linearen Funktionalen in linearen komplexen Räumen und linearen Quaternionr äumen,” Mat. Sb., vol. 3, no. 2, 353–358 (1938).

    MATH  Google Scholar 

  9. Slyusarchuk V. Yu., “Almost periodic solutions of nonlinear equations that are not necessarily almost periodic in Bochner’s sense,” Ukrainian Math. J., vol. 67, no. 2, 267–282 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. Slyusarchuk V. Yu., “Conditions for almost periodic bounded solutions of nonlinear difference equations with continuous argument,” Nonlinear Oscillations, vol. 16, no. 1, 118–124 (2013).

    Google Scholar 

  11. Slyusarchuk V. Yu., “Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces,” Ukrainian Math. J., vol. 65, no. 2, 341–347 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  12. Slyusarchuk V. Yu., “Conditions for almost periodicity of bounded solutions of nonlinear difference equations with discrete argument,” Nonlinear Oscillations, vol. 16, no. 3, 416–425 (2013).

    Google Scholar 

  13. Slyusarchuk V. Yu., “Conditions for almost periodicity of bounded solutions of nonlinear differential equations unsolved with respect to the derivative,” Ukrainian Math. J., vol. 66, no. 3, 432–442 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. Slyusarchuk V. Yu., “Almost periodic solutions of difference equations with discrete argument on metric space,” Miskolc Math. Notes, vol. 15, no. 1, 211–215 (2014).

    MathSciNet  MATH  Google Scholar 

  15. Slyusarchuk V. E., “The study of nonlinear almost periodic differential equations without recourse to the H-classes of these equations,” Sb. Math., vol. 205, no. 6, 892–911 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. Slyusarchuk V. E., “Conditions for almost periodicity of bounded solutions of non-linear differential-difference equations,” Izv. Math., vol. 78, no. 6, 1232–1243 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  17. Favard J., “Sur les équations différentielles `a coefficients presquepériodiques,” Acta Math., vol. 51, 31–81 (1927).

    Article  MATH  Google Scholar 

  18. Slyusarchuk V. E., “Invertibility of almost periodic-continuous functional operators,” Math. USSR-Sb., vol. 44, no. 4, 431–446 (1983).

    Article  Google Scholar 

  19. Slyusarchuk V. E., “Invertibility of nonautonomous functional-differential operators,” Math. USSR-Sb., vol. 58, no. 1, 83–100 (1987).

    Article  MATH  Google Scholar 

  20. Slyusarchuk V. E., “Necessary and sufficient conditions for invertibility of nonautonomous functional-differential,” Math. Notes, vol. 42, no. 2, 648–651 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  21. Amerio L., “Sull equazioni differenziali quasi-periodiche astratte,” Ric. Mat., vol. 30, 288–301 (1960).

    MathSciNet  MATH  Google Scholar 

  22. Zhikov V. V., “Proof of the Favard theorem on the existence of almost-periodic solution for an arbitrary Banach space,” Math. Notes, vol. 23, no. 1, 66–69 (1978).

    Article  MathSciNet  Google Scholar 

  23. Slyusarchuk V. E., “Necessary and sufficient conditions for existence and uniqueness of bounded and almost-periodic solutions of nonlinear differential equations,” Acta Appl. Math., vol. 65, no. 1–3, 333–341 (2001).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. E. Slyusarchuk.

Additional information

Rivne. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 1, pp. 206–218, January–February, 2017; DOI: 10.17377/smzh.2017.58.120.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Slyusarchuk, V.E. To Favard’s theory for functional equations. Sib Math J 58, 159–168 (2017). https://doi.org/10.1134/S0037446617010207

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446617010207

Keywords

Navigation