ACTA MATHEMATICA
UNIVERSITATIS COMENIANAE



Vol. LXXXIII, 1 (2014)
p. 119 - 134
Stability in totally nonlinear neutral differential equations
with variable delay

A. Ardjouni, I. Derrardjia and A. Djoudi

Received: April 14, 2013;   Accepted: September 18, 2013


Abstract.   In this paper we use a fixed point technique and the concept of large contractions to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equation with functional delay. The study concerns the equation
x'(t) = -a(t)h(x(t)) + c(t)x'(t - r(t)) + b(t)G(x(t), x(t - r(t))),
which 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 and generalize the works of Burton, and Derrardjia, Ardjouni and Djoudi.

Keywords:  Fixed points; stability; nonlinear neutral differential equation; Krasnoselskii-Burton theorem; variable delay.  

AMS Subject classification: Primary:  47H10, 34K20, 34K30, 34K40  


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Acta Mathematica Universitatis Comenianae
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