Opuscula Math. 33, no. 2 (2013), 255-272
http://dx.doi.org/10.7494/OpMath.2013.33.2.255

Opuscula Mathematica

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

Ishak Derrardjia
Abdelouaheb Ardjouni
Ahcene Djoudi

Abstract. 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.

Keywords: fixed point, stability, nonlinear neutral equation, Krasnoselskii-Burton theorem.

Mathematics Subject Classification: 47H10, 34K20, 34K30, 34K40.

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