Asymptotic stability in delay nonlinear fractional differential equations

  • Abdelouaheb Ardjouni University Souk Ahras.
  • Hamid Boulares UBMA.
  • Ahcene Djoudi UBMA.
Palabras clave: Delay fractional differential equations, fixed point theory, asymptotic stability, ecuaciones diferenciales fraccionarias, teoría de punto fijo, estabilidad asintótica

Resumen

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 . 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.

Biografía del autor

Abdelouaheb Ardjouni, University Souk Ahras.
Faculty of Sciences and Technology, Department of Mathematics and Informatics.
Hamid Boulares, UBMA.
Faculty of Sciences, Department of Mathematics.
Ahcene Djoudi, UBMA.
Faculty of Sciences, Department of Mathematics.

Citas

[1] S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Differential Equations, Vol. 2011, No. 09, pp. 1-11, (2011).

[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)
Publicado
2017-03-23
Cómo citar
Ardjouni, A., Boulares, H., & Djoudi, A. (2017). Asymptotic stability in delay nonlinear fractional differential equations. Proyecciones. Journal of Mathematics, 35(3), 263-275. https://doi.org/10.4067/S0716-09172016000300004
Sección
Artículos