Browsing by Author "Djoudi, Ahcene"

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Access status: Open Access ,
Fixed points and stability in neutral nonlinear differential equations with variable delays
(2012) Ardjouni, Abdelouaheb; Djoudi, Ahcene
By means of Krasnoselskii’s fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C.H. Jin and J.W. Luo [Nonlinear Anal. 68 (2008), 3307–3315], and also those of T.A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499–519] and B. Zhang [Nonlinear Anal. 63 (2005), e233–e242]. In the end we provide an example to illustrate our claim.
Access status: Open Access ,
Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations
(2013) Derrardjia, Ishak; Ardjouni, Abdelouaheb; Djoudi, Ahcene
In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns $x'(t)=a(t)x^3(t)+c(t)x'(t-r(t))+b(t)x^3(t-r(t)).$ The equation has proved very challenging in the theory of Liapunov's direct method. The stability results are obtained by means of Krasnoselskii-Burton's theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181-190]) in which he takes $c=0$ in the above equation.