Fixed point and stability of nonlinear differential equations with variable delays




Differential equations, Stability, Fixed point, Variable delays, Mackey--Glass equation


In this paper, we study the stability of a generalized nonlinear differential equation with variable delays via fixed point theory. An asymptotic stability theorem with sufficient conditions is proved, which improves and generalizes some previous results. Two examples are given to illustrate our results.

Author Biography

Abdelhafid Younsi, University of Djelfa.

Department of Mathematics.


L. Berezansky, E. Braverman, A note on stability of Mackey-Glass equations with two delays, J. Math. Anal. Appl., 450. 2, pp. 1208-1228, 2017.

L. Berezansky and E. Braverman, New stability conditions for linear differential equations with several delays. Abstract and Applied Analysis (Vol. 2011), 2011.

L. Berezansky and E. Braverman, Global linearized stability theory for delay differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71(7-8), pp. 2614-2624; 2009.

L. C. Becker and T. A. Burton, Stability, fixed points and inverses of delays. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 136(2), pp. 245-275, 2006.

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

T. A. Burton, Stability by fixed point theory or Liapunov's theory: A comparison, Fixed Point Theory, 4(1), pp. 15-32, 2003.

T. A. Burton and T. Furumochi, Dynamic systems and applications, 10(1), pp. 89-116, 2001.

T. A. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dynamic Systems and Applications, 11(4), pp. 499-520, 2002.

M. Fan, Z. Xia, and H. Zhu, Asymptotic stability of delay differential equations via fixed point theory and applications, Canadian applied mathematics quarterly, 18(4), pp. 361-380, 2010.

Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays. Math. Biosci. Eng, 17(5), pp. 4513-4526, 2020.

B. Zhang, Contraction mapping and stability in a delay-differential equation. Dynamical systems and appl, 4, pp. 183-190, 2004.

B. Zhang, Fixed points and stability in differential equations with variable delays. Nonlinear analysis: theory, methods & applications, 63(5-7), pp. e233-e242, 2005.



How to Cite

A. Younsi, “Fixed point and stability of nonlinear differential equations with variable delays ”, Proyecciones (Antofagasta, On line), vol. 43, no. 4, pp. 813-825, Jun. 2024.