Skip to main content
Log in

A Criterion for the Existence of Almost Periodic Solutions of Nonlinear Differential Equations with Impulsive Perturbation

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations with impulsive perturbation 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); S. Bochner, “Beitrage zur Theorie der fastperiodischen. II Teil. Funktionen mehrerer Variablen,” Math. Ann., 96, 383–409 (1927).

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

    Google Scholar 

  3. J. Favard, “Sur les èquations différentielles à coefficients presquepèriodiques,” Acta Math., 51, 31–81 (1927).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Yu. Slyusarchuk, “Conditions of almost periodicity for bounded solutions of nonlinear difference equations with continuous argument,” Nelin. Kolyv., 16, No. 1, 118–124 (2013); English translation: J. Math. Sci., 197, No. 1, 122–128 (2013).

  7. V. Yu. Slyusarchuk, “Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces,” Ukr. Mat. Zh., 65, No. 2, 307–312 (2013); English translation: Ukr. Math. J., 65, No. 2, 341–347 (2013).

  8. V. Yu. Slyusarchuk, “Conditions for the existence of almost periodic solutions of nonlinear difference equations with discrete argument,” Nelin. Kolyv., 16, No. 3, 416–425 (2013); English translation: J. Math. Sci., 201, No. 3, 391–399 (2013).

  9. V. Yu. Slyusarchuk, “Conditions for almost periodicity of bounded solutions of nonlinear differential equations unsolvable with respect to the derivative,” Ukr. Mat. Zh., 66, No. 3, 384–393 (2014); English translation: Ukr. Math. J., 66, No. 3, 432–442 (2014).

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

    MathSciNet  MATH  Google Scholar 

  11. V. E. Slyusarchuk, “Investigation of nonlinear almost periodic differential equations without using the \( \mathcal{H} \)-classes of these equations,” Mat. Sb., 205, No. 6, 139–160 (2014).

    Article  MathSciNet  Google Scholar 

  12. V. E. Slyusarchuk, “Conditions for almost periodicity of bounded solutions of nonlinear differential-difference equations,” Izv. Ros. Akad. Nauk, Ser. Mat., 78, No. 6, 179–192 (2014).

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

    MathSciNet  Google Scholar 

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

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

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

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 6, pp. 838–848, June, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Slyusarchuk, V.Y. A Criterion for the Existence of Almost Periodic Solutions of Nonlinear Differential Equations with Impulsive Perturbation. Ukr Math J 67, 948–959 (2015). https://doi.org/10.1007/s11253-015-1125-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-015-1125-x

Keywords

Navigation