Abstract
In this paper, we prove that Caputo type linear fractional evolution equations do not have nonconstant periodic solutions. Then, we study asymptotically periodic solutions of semilinear fractional evolution equations and establish existence and uniqueness results by using theory of semigroup and fixed point theorems. Finally, two examples are given to illustrate the theoretical results.
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R.P. Agarwal, S. Hristova, D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, No. 2 (2016), 290–318; DOI: 10.1515/fca-2016-0017https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.
E. Bajlekova, Fractional Evolution Equations in Banach Spaces. Ph.D. Thesis, Eindhoven University of Technology, (2001).
T.A. Burton, B. Zhang, Asymptotically periodic solutions of fractional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A 20 (2013), 1–21.
C. Cuevas, H.R. Henríquez, H. Soto, Asymptotically periodic solutions of fractional differential equations. Appl. Math. Comput. 236 (2014), 524–545.
C. Cuevas, M. Pierri, A. Sepulveda, WeightedS-asymptotically ω-periodic solutions of a class of fractional differential equations. Adv. Difference Equ. 2011 (2011), Art. ID 584874, 1–13.
C. Cuevas, J. Souza, S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22 (2009), 865–870.
C. Cuevas, J. Souza, Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal.:TMA 72 (2010), 1683–1689.
K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, Springer, New York (2010).
M. Fečkan, J. Wang, M. Pospíčil, [tiFractional-Order Equations and Inclusions.} De Gruyter (2017).
H. Henríquez, M. Pierri, P. Táboas, On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343 (2008), 1119–1130.
T. Ke, N. Loi, V. Obukhovskii, Decay solutions for a class of reactional differential variational inequalities. Fract. Calc. Appl. Anal. 18, No. 3 (2015), 531–553; DOI: 10.1515/fca-2015-0033https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.
A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science BV (2006).
J. Mu, Y. Zhou, L. Peng, Periodic solutions and asymptotically periodic solutions to fractional evolution equations. Discrete Dyn. Nature Society 2017 (2017), Art. ID 1364532, 1–12.
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York (1983).
M. Pierri, On S-asymptotically ω-periodic functions and applications. Nonlinear Anal. 75 (2012), 651–661.
A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, 2nd Edn. John Willey & Sons, New York (1980).
J. Wang, M. Fečkan, Y. Zhou, Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 18 (2013), 246–256.
J. Wang, M. Fečkan, Y. Zhou, A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, No. 4 (2016), 806–831; DOI: 10.1515/fca-2016-0044https://www.degruyter.com/view/j/fca.2016.19.issue-4/issue-files/fca.2016.19.issue-4.xml.
H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328 (2007), 1075–1081.
Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comp. Math. Appl. 59 (2010), 1063–1077.
Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations. Nonl. Anal. Real World Appl. 11 (2010), 4465–4475.
Y. Zhou, J. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, 2nd Edn. World Scientifc, Singapore (2016).
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Ren, L., Wang, J. & Fečkan, M. Asymptotically Periodic Solutions for Caputo Type Fractional Evolution Equations. FCAA 21, 1294–1312 (2018). https://doi.org/10.1515/fca-2018-0068
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DOI: https://doi.org/10.1515/fca-2018-0068