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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.2 Antofagasta Apr. 2020

http://dx.doi.org/10.22199/issn.0717-6279-2020-02-0024 

Artículos

On stability functional differential equation with delay variable by using fixed point-theory

1University of Duhok, College of Basic Education, Dept.of Mathematics, Zakho, Iraq. e-mail: sizar@uod.ac

Abstract

I will explain how to use the Banach fixed point theory in the asymptotic stability of nonlinear differential equations; I will obtain appropriate generalizations and strong forms of some of the results in [2, 3, 5, 6, 10, 11, 12,13].

Specifically, in the above-mentioned paper, asymptotic stability is achieved, while I will discuss how to achieve a asymptotic stability as well as stability by making a simple observation, also circulate the previous asymptotic stability results to the Functional Differential Equations systems, not only on the scalar Functional Differential Equations as is the case in the mentioned paper.

This raises the question of how much this particular method can afford us, and what are the limitations of this technique. I will refer to the important limitation of the fixed point theory on the uniqueness of solutions only within the complete metric space area where they are not specified. If the metric space onto which the contraction mapping principle is applied is very small, i do not get a satisfactory result. I will discuss this in detail below.

Keywords: Nonlinear; Asymptotical stability; Banach fixed point theorem; Delayed functional differential equation

Texto completo disponible sólo en PDF

Full text available only in PDF format.

References

[1] L. C. Becker and T. A. Burton, “Stability, fixed points and inverses of delays”, Proceedings of the Royal Society of Edinburgh: section a mathematics, vol. 136, no. 2, pp. 245-275, Apr. 2006, doi: 10.1017/S0308210500004546 [ Links ]

[2] T. A. Burton , “Fixed points and stability of a nonconvolution equation”,Proceedings of the American Mathematical Society, vol. 132, no. 12, pp. 3679-3688, Dec. 2004, doi: 10.1090/S0002-9939-04-07497-0 [ Links ]

[3] T. A. Burton , “Fixed points, stability, and exact linearization”,Nonlinear analysis: theory, methods & applications, vol. 61, no. 5, pp. 857-870, May 2005, doi: 10.1016/j.na.2005.01.079 [ Links ]

[4] T. A. Burton , “The case for stability by fixed point theory”, Dynamics of continuous, discrete and impulsive systems series a: mathematical analysis, vol. 13B, suppl., pp. 253-263, 2006. [On line]. Available: https://bit.ly/351iN5eLinks ]

[5] T. A. Burton ,Stability by fixed point theory for functional differential equations. Mineola : Dover Publ. Inc., 2006. [ Links ]

[6] T. A. Burton and T. Furumochi, “Fixed points and problems in stability theory for ordinary and functional differential equations”. Dynamic systems and applications, vol. 10, no. 1, pp. 89-116, 2001. [On line]. Available: https://bit.ly/2KzJVP9Links ]

[7] T. A. Burton , “Stability by fixed point theory or liapunov theory: a comparison”, Fixed point theory, vol. 4, no. 1, pp. 15-32, 2003. [On line]. Available: https://bit.ly/2KxPFci Links ]

[8] T. A. Burton , “Stability by fixed point methods for highly nonlinear delay equations”, Fixed point theory , vol. 5, no. 1, pp. 3-20, 2004. [On line]. Available: https://bit.ly/2W1IfDnLinks ]

[9] T. A. Burton andT. Furumochi , “Fixed points and problems in stability theory for ordinary and functional differential equations”, Dynamic systems and applications , vol. 10, pp. 89-116, 2001. [On line]. Available: https://bit.ly/2xc0e1NLinks ]

[10] T. A. Burton and I. Purnaras, “Positive kernels, fixed points, and integral equations”,Electronic journal of qualitative theory of differential equations, no. 44, pp. 1-21, Jun. 2018. doi: 10.14232/ejqtde.2018.1.44 [ Links ]

[11] E. Zeidler,Nonlinear functional analysis and its applications, vol. 1. New York, NY: Springer, 1986. [ Links ]

[12] B. Zhang. “Contraction mapping and stability in a delay-differential equation”, Proceedings of Dynamic systems and applications , vol. 4, pp. 183-190, 2004. [On line]. Available: https://bit.ly/2xa93JhLinks ]

[13] B. Zhang, “Fixed points and stability in differential equations with variable delays”,Nonlinear analysis: theory, methods & applications , vol. 63, no. 5-7, pp. 233-242, Nov. 2005, doi: 10.1016/j.na.2005.02.081 [ Links ]

Received: October 30, 2018; Accepted: January 30, 2020

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