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Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations

Derrardjia I., Ardjouni A., Djoudi A.

Opuscula Mathematica

2013

Vol. 33, no. 2

255--272

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)x3(t) + c(t)x'(t-r(t)) + b(t)x3(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

Fixed points and stability in neutral nonlinear differential equations with variable delays

Ardjouni A., Djoudi A.

Opuscula Mathematica

2012

Vol. 32, no. 1

5-19

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.