Opuscula Math. 33, no. 2 (), 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

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.
Ishak Derrardjia, Abdelouaheb Ardjouni, Ahcene Djoudi, Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations, Opuscula Math. 33, no. 2 (2013), 255-272, http://dx.doi.org/10.7494/OpMath.2013.33.2.255

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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
Contact: opuscula@agh.edu.pl