Asymptotically almost periodic solutions of fractional evolution equations
DOI:
https://doi.org/10.21638/spbu01.2021.309Abstract
We study the asymptotic behavior of solutions of nonlinear fractional evolution equations in Banach spaces. Asymptotically almost periodic solutions on half line are obtained by establishing a sharp estimate on the resolvent operator family of evolution equations. In particular, when the semigroup generated by A is exponentially stable then the conditions of the existence asymptotically almost periodic solutions is satisfied. An application to a fractional partial differential equation with initial boundary condition is also considered.Keywords:
fractional evolution equations, almost periodic solutions, resolvent operator family
Downloads
Download data is not yet available.
References
Литература/References
1. Feckan M. Note on Periodic and Asymptotically Periodic Solutions of Fractional Differential Equations. In: Dutta H., Peters J., (ed.) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol. 177. Cham, Springer (2020). https://doi.org/10.1007/978-3-319-99918-0_6
2. Xiao Ma, Xiao-Bao Shu, Jianzhong Mao. Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay. Stochastics and Dynamics 20 (01), 2050003 (2020). https://doi.org/10.1142/S0219493720500033
3. Stamov G.Tr., Stamova I.M. Almost periodic solutions for impulsive fractional differential equations. Dynamical Systems 29 (1), 119–132 (2014). https://doi.org/10.1080/14689367.2013.854737
4. Hino Y., Naito T., Nguyen V.M., Shin J. S. Almost periodic solutions of differential equations in Banach spaces. In: Stability and Control: Theory, Methods and Applications, vol. 15. London, Taylor & Francis (2002).
5. Arendt W., Batty C. J.K. Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line. Bull. London Math. Soc. 31 (3), 291–304 (1999). https://doi.org/10.1112/S0024609398005657
6. Arendt W., Pruss J. Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations. SIAM J. Math. Anal. 23 (2), 412–448 (1992). https://doi.org/10.1137/0523021
7. Batty C. J.K., Hutter W., Rabiger F. Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems. J. Differential Equations 156, 309–327 (1999). https://doi.org/10.1006/jdeq.1998.3610
8. Cheban D. Asymptotically almost periodic solutions of differential equations. New York, Hindawi Publ. Corp. (2009).
9. Fink A.M. Almost Periodic Differential Equations. In: Lecture Notes in Math., vol. 377. Berlin, New York, Springer (1974).
10. Luong V.T., Minh N.V. Almost periodic solutions of periodic linear partial functional differential equations. Funkcialaj Ekvacioj 62 (2), 209–226 (2020). https://doi.org/10.1619/fesi.62.209
11. Araya D., Lizama C. Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69 (11), 3692–3705 (2008). https://doi.org/10.1016/j.na.2007.10.004
12. Agarwal R. P., de Andrade B., Cuevas C. On Type of Periodicity and Ergodicity to a Class of Fractional Order Differential Equations. Advances in Difference Equations 2010, 179750 (2010). https://doi.org/10.1155/2010/179750
13. Cuesta E. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete and Continuous Dynamical Systems (Supplement), 277–285 (2007). https://doi.org/10.3934/proc.2007.2007.277
14. Ponce R. Bounded mild solutions to fractional integro differential equations in Banach spaces. Semigroup Forum 87 (2), 377–392 (2013). https://doi.org/10.1007/s00233-013-9474-y
15. Keyantuo V., Lizama C., Warma M. Asymptotic behavior of fractional order semilinear evolution equations. Differential Integral Equations 26 (7/8), 757–780 (2013).
16. Levitan B.M., Zhikov V.V. Almost Periodic Functions and Differential Equations. Cambridge, Cambridge Univ. Press (1982). [Russ. ed.: Pochti periodicheskie funktsii i differentsial’nye uravneniia. Moscow, Moscow Univ. Press (1978)].
17. Bazhlekova E. Fractional Evolution Equations in Banach Spaces. PhD thesis. Eindhoven (2001).
18. Gu H., Trujillo J.J. Existence of integral solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015). https://doi.org/10.1016/j.amc.2014.10.083
19. Zhou Y., Jiao F. Existence of mild solutions for fractional neutral evolution equations. Comp. Math. Appl. 59 (3), 1063–4475 (2010). https://doi.org/10.1016/j.camwa.2009.06.026
20. Wei Z., Dong W., Che J. Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Analysis: Theory, Methods & Applications 73 (10), 3232–3238 (2010). https://doi.org/10.1016/j.na.2010.07.003
21. Pruss J. Evolutionary integral equations and applications. In: Monographs in Mathematics, vol. 87. Basel, Birkhauser, Verlag (1993).
22. Wang R.-N., Chen D.-H., Xiao T.-J. Abstract fractional Cauchy problems with almost sectorial operators. Journal of Differential Equations 252 (1), 202–35 (2012). https://doi.org/10.1016/j.jde.2011.08.048
23. Engel K. J., Nagel R. One-Parameter Semigroups for Linear Evolution Equations. New York, Springer-Verlag, Inc. (2000).
Downloads
Published
2021-09-26
How to Cite
Nguyen, D. H., & Vu, T. L. (2021). Asymptotically almost periodic solutions of fractional evolution equations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 475–483. https://doi.org/10.21638/spbu01.2021.309
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
