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Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative

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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 H-classes of these equations.

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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  6. V. Yu. Slyusarchuk, “Conditions of almost periodicity of bounded solutions of nonlinear difference equations with continuous argument,” Nelin. Kolyvannya, 16, No. 2, 118–124 (2013).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Favard, “Sur les équations différentielles 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. L. Amerio, “Sull equazioni differenziali quasi-periodiche astratte,” Ric. Mat., 30, 288–301 (1960).

    MATH  MathSciNet  Google Scholar 

  13. 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).

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. 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 

  16. 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).

  17. 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).

  18. 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 

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

    Article  MathSciNet  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 3, pp. 384–393, March, 2014.

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Slyusarchuk, V.Y. Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative. Ukr Math J 66, 432–442 (2014). https://doi.org/10.1007/s11253-014-0941-8

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  • DOI: https://doi.org/10.1007/s11253-014-0941-8

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