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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.34 no.1 Antofagasta mar. 2015

http://dx.doi.org/10.4067/S0716-09172015000100003 

Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory

Abdelouaheb Ardjouni

University Souk Ahras, Algeria 

Ahcene Djoudi

University Annaba, Algeria 


ABSTRACT

The totally nonlinear neutral differential equation

(d/ dt) (x(t))=−a(t)g(x(t−τ (t))) + (d/ dt)( G(t,x(t−τ (t)))),

with variable delay τ(t) ≥ 0 is investigated. We find suitable conditions for t, a, g and G so that for a given continuous initial function 0 a mapping P for the above equation can be defined on a carefully chosen complete metric space S0ψ ; and in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Becker and Burton [6]. An example is given to illustrate our main result.

Subjclass[2010] : 34K20, 34K30, 34K40.

Keywords : Fixed points, Stability, Neutral differential equations, Variable delays.


REFERENCES

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Abdelouaheb Ardjouni

Faculty of Sciences and Technology, Department of Mathematics and Informatics, University Souk Ahras,

P. O. Box 1553,

Souk Ahras, 41000,

Algeria

e-mail : abd_ardjouni@yahoo.fr

Ahcene Djoudi

Applied Mathematics Lab,

Faculty of Sciences,

Department of Mathematics,

University Annaba,

P. O. Box 12, Annaba 23000,

Algeria

e-mail : adjoudi@yahoo.com

Received : September 2014. Accepted : December 2014

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