Abstract
In this paper, we give sufficient conditions for the existence and uniqueness of asymptotically \(\omega \)-periodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space via asymptotically \(\omega \)-periodic functions in the Stepanov sense. This is done using the Banach fixed point Theorem.
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Cuevas, C., de Souza, J.: Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 72, 1683–1689 (2010)
Dimbour, W., Manou-Abi, S.: \(S\)-asymptotically \(\omega \)-periodic solution for a nonlinear differential equation with piecewise constant argument via \(S\)-asymptotically \(\omega \)-periodic functions in the Stepanov sense (2018) (to appear)
Dimbour, W., Valmorin, V.: Asymptotically antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space. Appl. Math. 7, 1726–1733 (2016)
Dimbour, W., Mophou, G., N’Guérékata, G.M.: \(S\) asymptotically \(\omega \)-periodic solution for partial differential equations with finite delay. Electron. J. Differ. Equ. 1–12, 2011 (2011)
Henríquez, H.R., Pierri, M., Táboas, P.: On \(S\) asymptotically \(\omega \)-periodic function on Banach spaces and applications. J. Math. Anal. Appl. 343, 1119–1130 (2008)
Henríquez, H.R., Pierri, M., Táboas, P.: Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for abstract neutral equations. Bull. Aust. Math. Soc. 78, 365–382 (2008)
N’Guérékata, G.M., Valmorin, V.: Antiperiodic solutions of semilinear integrodifferential equations in Banach spaces. Appl. Math. Comput. 218, 11118–111124 (2012)
Nicola, S., Pierri, M.: A note on \(S\)-asymptotically periodic functions. Nonlinear Anal. Real World Appl. 10, 3285–3297 (2009)
Pierri, M.: On \(S\)-asymptotically \(\omega \)-periodic functions and applications. Nonlinear Anal. 75, 651–661 (2012)
Rong-Hua, He: Stepanov-like pseudo-almost automorphic mild solutions for some abstract differential equations. Adv. Fixed Point Theory 2(3), 258–272 (2012)
Wiener, J.: A second-order delay differential equation with multiple periodic solutions. J. Math. Anal. Appl. 229, 6596–676 (1999)
Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1999)
Wiener, J., Debnath, L.: A survey of partial differential equations with piecewise continuous arguments. Int. J. Math. Math. Sci. 18(2), 209–228 (1995)
Wiener, J., Debnath, L.: Boundary value problems for the diffusion equation with piecewise continuous time delay. Int. J. Math. Math. Sci. 20, 187–195 (1997)
Wiener, J., Lakshmikantham, V.: Excitability of a second-order delay differential equation. Nonlinear Anal. 38, 1–11 (1999)
Xia, Z.: Asymptotically periodic of semilinear fractional integro-differential equations. Adv. Differ. Equ. 2014, 19 (2014)
Xia, Z.: Weighted pseudo asymptotically periodic mild solutions of evolutions equations. Acta Math. Sin. 31(8), 1215–1232 (2015)
Xie, R., Zhang, C.: Criteria of asymptotic \(\omega \)-periodicity and their applications in a class of fractional differential equations. Adv. Differ. Equ. 2015, 68 (2015)
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Dimbour, W., Manou-Abi, S.M. Asymptotically \(\omega \)-Periodic Functions in the Stepanov Sense and Its Application for an Advanced Differential Equation with Piecewise Constant Argument in a Banach Space. Mediterr. J. Math. 15, 25 (2018). https://doi.org/10.1007/s00009-018-1071-6
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DOI: https://doi.org/10.1007/s00009-018-1071-6