Open AccessStability by Krasnoselskii’s theorem in totally nonlinear neutral differential equations
Author(s) -
Ishak Derrardjia,
Abdelouaheb Ardjouni,
Ahcéne Djoudi
Publication year - 2013
Publication title -
rocznik akademii górniczo-hutniczej im. stanisława staszica. opuscula mathematica/opuscula mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 16
eISSN - 2300-6919
pISSN - 1232-9274
DOI - 10.7494/opmath.2013.33.2.255
Subject(s) - mathematics , nonlinear system , stability (learning theory) , differential equation , fixed point theorem , mathematical analysis , physics , quantum mechanics , machine learning , computer science
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, andstability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181-190]) in which he takes \(c=0\) in the above equation