Skip to main content
Log in

Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the \( \mathcal{H} \)-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.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Bochner, “Beitrage zur Theorie der fastperiodischen. I Teil. Funktionen einer Variablen,” Math. Ann., 96, 119–147 (1927).

    Article  MathSciNet  Google Scholar 

  2. S. Bochner, “Beitrage zur Theorie der fastperiodischen. II Teil. Funktionen mehrerer Variablen,” Math. Ann., 96, 383–409 (1927).

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. V. E. Slyusarchuk, “Bounded and periodic solutions of nonlinear functional-differential equations,” Mat. Sb., 203, No. 5, 135–160 (2012).

    Article  MathSciNet  Google Scholar 

  6. V. Yu. Slyusarchuk, “Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations,” Ukr. Mat. Zh., 61, No. 11, 1541–1556 (2009); English translation: Ukr. Math. J., 61, No. 11, 1809–1829 (2009).

    Article  MathSciNet  Google Scholar 

  7. V. E. Slyusarchuk, “Method of local linear approximation in the theory of nonlinear functional-differential equations,” Mat. Sb., 201, No. 8, 103–126 (2010).

    Article  MathSciNet  Google Scholar 

  8. V. Yu. Slyusarchuk, “Method of local linear approximation of nonlinear differential operators by weakly regular operators,” Ukr. Mat. Zh., 63, No. 12, 1685–1698 (2011); English translation: Ukr. Math. J., 63, No. 12, 1916–1932 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Favard, “Sur les équations diff´erentielles à coefficients presquepériodiques,” Acta Math., 51, 31–81 (1927).

    Article  MATH  MathSciNet  Google Scholar 

  10. É. Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki, 11, No. 3, 269–274 (1972).

    MathSciNet  Google Scholar 

  11. É. Mukhamadiev, “Investigations in the theory of periodic and bounded solutions of differential equations,” Mat. Zametki, 30, No. 3, 443–460 (1981).

    MATH  MathSciNet  Google Scholar 

  12. V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116(158), No. 4(12), 483–501 (1981).

    Google Scholar 

  13. V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators,” Mat. Sb., 130(172), No. 1(5), 86–104 (1986).

  14. V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki, 42, No. 2, 262–267 (1987).

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. V. V. Zhikov, “Proof of the Favard theorem on the existence of almost periodic solutions in the case of an arbitrary Banach space,” Mat. Zametki, 23, No. 1, 121–126 (1978).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 2, pp. 307–312, February, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Slyusarchuk, V.Y. Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces. Ukr Math J 65, 341–347 (2013). https://doi.org/10.1007/s11253-013-0781-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0781-y

Keywords

Navigation