Abstract
In this paper, we give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of delay nonlinear fractional differential equations of order \(\alpha \) \((1<\alpha <2\)). By using the Krasnoselskii’s fixed point theorem in a weighted Banach space, we establish new results on the asymptotic stability of the zero solution provided that \(g\left( t,0\right) =f\left( t,0,0\right) \), which include and improve some related results in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbas, S.: Existence of solutions to fractional order ordinary and delay differential equations and applications. Electron. J. Differ. Equ. 2011(09), 1–11 (2011)
Agarwal, R.P., Zhou, Y., He, Y.: Existence of fractional functional differential equations. Computers Math. Appl. 59, 1095–1100 (2010)
Burton, T.A., Zhang, B.: Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems. Nonlinear Anal. 75, 6485–6495 (2012)
Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. Real Word Appl. 13, 287–298 (2012)
Ge, F., Kou, C.: Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations. Appl. Math. Comput. 257, 308–316 (2015)
Ge, F., Kou, C.: Asymptotic stability of solutions of nonlinear fractional differential equations of order \(1 \le \alpha \le 2\). J. Shanghai Normal Univ. 44(3), 284–290
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kou, C., Zhou, H., Yan, Y.: Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal. 74, 5975–5986 (2011)
Li, Y., Chen, Y., Podlunby, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)
Li, Y., Chen, Y., Podlunby, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)
Li, C., Zhang, F.: A survey on the stability of fractional differential equations. Eur. Phys. J. Special Topics. 193, 27–47 (2011)
Ndoye, I., Zasadzinski, M., Darouach, M., Radhy, N. E.: Observer based control for fractional-order continuous-time systems. In: Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, WeBIn5.3, pp. 1932–1937, December 16–18 (2009)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Smart, D.R.: Fixed point theorems. Cambridge Uni. Press, Cambridge (1980)
Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boulares, H., Ardjouni, A. & Laskri, Y. Stability in delay nonlinear fractional differential equations. Rend. Circ. Mat. Palermo, II. Ser 65, 243–253 (2016). https://doi.org/10.1007/s12215-016-0230-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-016-0230-5