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.
Similar content being viewed by others
References
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).
B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).
J. Favard, “Sur les èquations différentielles à coefficients presquepèriodiques,” Acta Math., 51, 31–81 (1927).
B. M. Levitan, Almost Periodic Functions [in Russian], Gostekhizdat, Moscow (1953).
L. Amerio, “Soluzioni quasiperiodiche, o limital, di sistemi differenziali non lineari quasi-periodici, o limitati,” Ann. Mat. Pura Appl., 39, 97–119 (1955).
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).
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).
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).
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).
V. Yu. Slyusarchuk, “Almost periodic solutions of difference equations with discrete argument on metric space,” Miskolc Math. Notes, 15, No. 1, 211–215 (2014).
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).
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).
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).
É. Mukhamadiev, “Investigations into the theory of periodic and bounded solutions of differential equations,” Mat. Zametki, 30, No. 3, 443–460 (1981).
V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116(158), No. 4(12), 483–501 (1981).
V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators,” Mat. Sb., 130(172), No. 1(5), 86–104 (1986).
V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki, 42, No. 2, 262–267 (1987).
L. Amerio, “Sull equazioni differenziali quasi-periodiche astratte,” Ric. Mat., 30, 288–301 (1960).
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).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 6, pp. 838–848, June, 2015.
Rights and permissions
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1125-x