Georgian Mathematical Journal Volume 13 (2006), Number 1, 25–34 FIXED POINT TECHNIQUES AND STABILITY FOR NEUTRAL NONLINEAR DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAYS AHCENE DJOUDI AND RABAH KHEMIS Abstract. We use the contraction mapping theorem to obtain stability re- sults of the scalar nonlinear neutral differential equation with functional delay x ′ (t)= −ax(t)+ b(t)x 2 (t − r(t)) + c(t)x(t − r(t))x ′ (t − r(t)). 2000 Mathematics Subject Classification: 34K20, 47H10. Key words and phrases: Contraction mapping, stability, nonlinear neutral differential equation, integral equation. 1. Introduction Liapunov’s method has been very effective in establishing stability results (see [2], [7] and [9]) and the existence of periodic solutions (see [2] and [8]) for a wide variety of differential equations. Nevertheless, in the applications of Li- apunov’s direct method to problems of stability in delay differential equations, serious difficulties occur if the functions in the equations are unbounded with time (see [7]), the delay is unbounded or the derivative of the delay is not small (Seifert [9]). In [4], when studying stability theory for ordinary and functional differential equations by means of fixed point theory, T. A. Burton and T. Fu- rumuchi noticed that these difficulties vanish when we apply fixed point theory. In particular, in the papers [1], [3] and [5], the authors examined particular problems which have offered great difficulties for Liapunov’s theory and pre- sented solutions by means of various fixed point theories. Following these ideas, we present here a study of stability in neutral nonlinear differential equations with functional delays by means of the contraction mapping theorem. Let us begin with a classical and simple example to show the difficulties encountered in using the Liapunov’s direct method. In [6], the problem x ′ (t)= −a(t)x(t)+ b(t)x(t − r) (1.1) is considered, where a and b are bounded continuous functions such that a(t) ≥ δ> 0, |b(t)|≤ θδ, θ< 1. Consider the Liapunov functional V (t, x t )= 1 2 x 2 (t)+ δ t  t−r x 2 (s)ds. ISSN 1072-947X / $8.00 / c  Heldermann Verlag www.heldermann.de