Abstract
In this paper we discuss the pseudo almost automorphic solution of a fractional order neutral differential equation in a Banach space X. The results are established using the Krasnoselskii’s fixed point theorem.
Similar content being viewed by others
References
Acquistapace, P.: Evolution operators and strong solution of abstract linear parabolic equations. Diff. Integral Equ. 1, 433–457 (1988)
Acquistapace, P., Terreni, B.: Aunified approach to abstract linear parabolic equations. Rend. Semin. Math. Univ. Padova 78, 47–107 (1987)
El-Sayed, A.M.A.: On the fractional differential equations. Appl. Math. Comput. 49, 205–213 (1992)
El-Sayed, A.M.A.: Fractional order differential equation. Kyungpook Math. J. 28, 18–22 (1988)
El-Sayed, A.M.A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33(2), 181–186 (1998)
Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional next term semilinear previous evolution equations. Nonlinear Anal. TMA 71(10), 4471–4475 (2009)
Bochner, S.: Continuous mappings of almost automorphic and almost automorphic functions. Proc. Natl. Sci. USA 52, 907–910 (1964)
Yu, C., Gao, G.: Some results on a class of fractional functional differential equations. Commun. Appl. Nonlinear Anal. 11(3), 67–75 (2004)
Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(8), 134–147 (1971)
Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)
Ezzinbi, K., Fatajou, S., N’Guérékata, G.: Pseudo almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. Nonlinear Anal. TMA 70(4), 1641–1647 (2009)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)
Lakshmikanthama, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)
Liang, J., N’Guérékata, G.M., Xiao, T.-J., Zhang, J.: Some properties of pseudo almost automorphic functions and applications to abstract differential equations. Nonlinear Anal. TMA 70(7), 2731–2735 (2009)
Mophou, G.M., N’Guérékata, G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79(2), 315–322 (2009)
Mishura, S.Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. Springer, Berlin (2008)
N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)
N’Guérékata, G.M., Pankov, A.: Integral operators in spaces of bounded, almost periodic and almost automorphic functions. Diff. Integral Equ. 21(11–12), 1155–1176 (2008)
Daftardar-Gdjji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004)
Xiao, T.-J., Liang, J., Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach space. Semigroup Forum 76, 518–524 (2008)
Xiao, T.-J., Zhu, X.-X., Liang, J.: Pseudo almost automorphic mild solutions to nanautonomous differential equations and applications. Nonlinear Anal. TMA 70(11), 4079–4085 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein.
Rights and permissions
About this article
Cite this article
Abbas, S. Pseudo almost automorphic solutions of fractional order neutral differential equation. Semigroup Forum 81, 393–404 (2010). https://doi.org/10.1007/s00233-010-9227-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-010-9227-0