Asymptotic stability in delay nonlinear fractional differential equations
DOI:
https://doi.org/10.4067/S0716-09172016000300004Keywords:
Delay fractional differential equations, fixed point theory, asymptotic stability, ecuaciones diferenciales fraccionarias, teoría de punto fijo, estabilidad asintóticaAbstract
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 α (1 < α < 2)
. By using the Banach’s contraction mapping principle in a weighted Banach space, we establish new results on the asymptotic stability of the zero solution provided that g (t, 0) = f (t, 0, 0) = 0, which include and improve some related results in the literature.
References
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[2] R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional functional differential equations, Computers and Mathematics with Applications 59, pp. 1095-1100, (2010).
[3] T. A. Burton, B. Zhang, Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems, Nonlinear Anal.
75, pp. 6485—6495, (2012).
[4] F. Chen, J. J. Nieto, Y. Zhou, Global attractivity for nonlinear fractional differential equations, Nonlinear Analysis: Real Word Applications 13, pp. 287-298, (2012).
[5] F. Ge, C. Kou, Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations, Applied Mathematics and Computation 257, pp. 308-316, (2015).
[6] F. Ge, C. Kou, Asymptotic stability of solutions of nonlinear fractional differential equations of order 1 < α < 2, Journal of Shanghai Normal University, Vol. 44, No. 3, pp. 284-290, (2015).
[7] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, (2006).
[8] C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal. 74, pp. 5975—5986, (2011).
[9] Y. Li, Y. Chen, I. Podlunby, Mittag—Leffler stability of fractional order nonlinear dynamic systems, Automatica 45, pp. 1965—1969, (2009).
[10] Y. Li, Y. Chen, I. Podlunby, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag—Leffler stability, Comput. Math. Appl. 59, pp. 1810—1821, (2010).
[11] C. Li, F. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Special Topics. 193, pp. 27—47, (2011).
[12] I. Ndoye, M. Zasadzinski, M. Darouach, N. E. Radhy, Observerbased control for fractional-order continuous-time systems, 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).
[13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, (1999).
[14] D. R. Smart, Fixed point theorems, Cambridge Uni. Press., Cambridge, (1980).
[15] J. Wang, Y. Zhou, M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64, pp. 3389—3405, (2012)
Published
2017-03-23
How to Cite
[1]
A. Ardjouni, H. Boulares, and A. Djoudi, “Asymptotic stability in delay nonlinear fractional differential equations”, Proyecciones (Antofagasta, On line), vol. 35, no. 3, pp. 263-275, Mar. 2017.
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