Since 1892, the Lyapunov’s direct method has been used for the study of stability properties of ordinary, functional, differential and partial differential equations. Nevertheless, the applications of this method to problem of stability in differential equations with delay have encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms (see    ). Recently, investigators, such as Burton and Furumochi have noticed that some of these difficulties vanish or might be overcome by means of fixed point theory (see  -  ). The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov’s direct method. Sadovskii’s fixed point theorem (see  ) and techniques of the theory of the measure of noncompactness are used to prove the existence and stability of the solution of the problem investigated in this paper.
Let E denote a Banach space. We consider the nonlinear differential equation with variable delay
with the initial condition
where , , .
Here denotes the set of all continuous and bounded functions with the supremum norm defined . Throughout this paper, we assume that , with as , . We also assume that the function is uniformly continuous. Moreover, we assume that the function G is bounded, so there exists a constant , such that and .
Special cases of the equation (1) have been investigated by many authors. For example, Burton in  and Zhang in  have studied boundedness and stability of the linear equation:
In  , Burton and Furumochi using the fixed point theorem of Krasnosielski obtained boundedness and asymptotic stability for the equation:
where is constant; and is a quotient with odd positive integer denominator and .
Next, Jin and Luo (see  ), proved the boundedness and stability of solutions of the equation
and generalized the results claimed in    . Ardjouni and Djoudi in  considered the more general neutral nonlinear differential equation
and obtained the boundedness and stability results.
We mentioned here that the neutral delay differential equations appear in modelling of the networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, theory of automatic control and in neuromechanical systems in which inertia plays an important role, we refer the reader to the papers by Boe and Chang  , Brayton and Willoughby  and to the books by Driver  , Hale  and Popov  and reference cited therein.
The fundamental tool in this paper is the Kuratowski measure of noncompactness (see  ).
For any bounded subset A of E we denote by the Kuratowski measure of noncompactness of A, i.e. the infimum of all such that there exists a finite covering of A by sets of diameters smaller than . The properties of the measure of noncompactness are:
1) if then ;
2) , where denotes the closure of A;
3) if and only if A is relatively compact;
5) , ;
7) , where denotes the convex hull of A,
8) , where .
Lemma 1.  . Let be a family of strongly equicontinuous functions. Let , for and . Then
where denotes the measure of noncompactness in and the function is continuous.
Let us denote by the set of all nonnegative real sequences. For , , we write if (i.e. , for ) and .
Let X be a closed convex subset of and let be a function which assigns to each nonempty subset Z of X a sequence such that
if (the zero sequence), then is compact. (4)
In the proof of the main theorem, we will apply the following fixed point theorem
Theorem 1  If is continuous mapping satisfying for arbitrary nonempty subset Z of X with , then F has a fixed point in X.
2. Main Results
A solution of the problem (1) is a continuous function such that x satisfies (1) on and on . Stability definitions may found in  , for example.
In this paper, we extend stability theorem proved in  by giving a necessary and sufficient conditions for asymptotic stability of the zero solution of the Equation (1). By using some conditions expressed in terms of the measure of noncompactness which G satisfies, we define a continuous, bounded operator over the Banach space , whose fixed points are solutions of (1). The fixed point theorem of Sadovskii is used to prove the existence of a fixed point of the operator . To construct our mapping, we begin transforming (1) to a more tractable, but equivalent equation, which we invert to obtain an equivalent integral equation for which we derive a fixed point mapping. We need the following lemma in our proof of the main theorem.
Lemma 2 Let be an arbitrary continuous function and suppose that
Then x is a solution of (1) if and only if
Now we will present our main results. We set
Let be a fixed. Put
and , this mean that
Let us recall that for continuous function , the first order modulus of smoothness for f is the function defined for any by
Now, we use (6) to define the operator by if and for we let
Theorem 2 Assume that the function is uniformly continuous, satisfies the condition and there exists a constant such that
for each and for each , where denotes the Kuratowski measure of noncompactness. Moreover, we assume that there exists a constant satisfies, for :
and , as .
Then there exists at least one solution of the problem (1), which tends to zero if
Proof. We divide our proof into parts. In the first part, we show that complete with is well defined. Next, using Sadovskii’s fixed point theorem and techniques of the theory of the measure of noncompactness, we prove, that there exists a fixed point of the operator , which is a solution of the problem (1). In the last part of our proof, we will show, that the solution of (1) tends to zero as if , as .
Part I. It is clear that is continuous. We will show the boundedness of this operator. Notice that , so there exists a constant such that , .
For , the boundedness of the operator is clear. For we have
Therefore, because satisfies , we have, by (9), that .
Now we will show that for each we obtain
Let’s note that
Because the exponential function is continuous and using assumptions about functions and properties of Bochner integral we have
By equicontinuity of function G and equicontinuity of family we have that
So we obtain that .
In the same way, we estimate the remaining ingredients and we get
Now we will show, that as . It is obvious that the first term tends to zero as , by the condition (13). Because and as and b is a bounded function so the second term tends to zero as . Because functions are bounded functions, so , . Analogously and tends to zero if . Moreover,
In conclusion as . Hence maps into .
Part II. Suppose that and denotes the Kuratowski measure of noncompactness. Let , . Then,
So, using (12) and properties of we have
Because , we get for each .
Define for any nonempty subset V of , where , . Evidently . By the properties of , the function satisfies conditions (2)-(4). From (10) , whenever . If then for each we have . Hence Arzela-Ascoli’s theorem proves that the set V is relatively compact. Consequently, by Theorem 1, the operator has a fixed point which is a solution of the problem (1) with on and as , if , for . The proof is complete.
Remark 1. It is clear that we can assume about the function G other types of continuity and other conditions on measures of noncompactness. When we investigate the existence of solutions of (1) with non-continuous right-hand side, it is natural to consider the so-called Carathéodory-type solutions.
 Becker, L.C. and Burton, T.A. (2006) Stability, Fixed Points and Inverse of Delay, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 136, 245-275.
 Ardjouni, A. and Djoudi, A. (2012) Fixed Points and Stability in Neutral Nonlinear Differential Equations with Variable Delays. Opuscula Mathematica, 32, 5-19.
 Jin, C.-H. and Luo, J.-W. (2008) Stability in Functional Differential Equations Established Using Fixed Point Theory. Nonlinear Analysis: Theory, Methods & Applications, 68, 3307-3315.
 Ardjouni, A. and Djoudi, A. (2011) Fixed Points and Stability in Linear Neutral Differential Equations with Variable Delays. Nonlinear Analysis: Theory, Methods & Applications, 74, 2062-2070.
 Ardjouni, A. and Djoudi, A. (2011) Stability in Nonlinear Neutral Differential Equations with Variable Delays Using Fixed Point Theory. Electronic Journal of Qualitative Theory of Differential Equations, No. 43, 1-11.
 Burton, T.A. and Furumochi, T. (2002) Krasnosielskii’s Fixed Point Theorem and Stability. Nonlinear Analysis: Theory, Methods & Applications, 49, 445-454.
 Jin, C.-H. and Luo, J.-W. (2008) Fixed Points and Stability in Neutral Differential Equations with Variable Delays. Proceedings of the American Mathematical Society, 136, 909-918.
 Raffoul, Y.N. (2004) Stability in Neutral Nonlinear Differential Equations with Functional Delays Using Fixed-Point Theory. Mathematical and Computer Modelling, 40, 691-700.
 Zhang, B. (2005) Fixed Points and Stability in Differential Equations with Variable Delays. Nonlinear Analysis: Theory, Methods & Applications, 63, 233-242.
 Brayton, R.K. and Willoughby, R.A. (1976) On the Numerical Integration of a Symmetric System of Difference-Differential Equations of Neutral Type. Journal of Mathematical Analysis and Applications, 18, 182-189.