TY  - JOUR
ID  - Derrardjia2013
LB  - Derrardjia2013
AU  - Derrardjia, Ishak
AU  - Ardjouni, Abdelouaheb
AU  - Djoudi, Ahcene
TI  - Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations
ST  - Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2013
VL  - 33
IS  - 2
SP  - 255
EP  - 272
DO  - http://dx.doi.org/10.7494/OpMath.2013.33.2.255
UR  - http://dx.doi.org/10.7494/OpMath.2013.33.2.255
AB  - 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.
KW  - fixed point
KW  - stability
KW  - nonlinear neutral equation
KW  - Krasnoselskii-Burton theorem
SN  - 1232-9274
ER  -