Stability for Functional Differential Equations with Delay in Banach Spaces

In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solutions of nonlinear differential equation with variable delay. Results presented in this paper extend some previous results due to Burton [1], Becker and Burton [2], Ardjouni and Djoudi [3], and Jin and Luo [4].


Introduction
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 [1] [5] [6]).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 [1]- [17]).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 [18]) 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 ( ) (

= , x t a t x t r t b t x t r t c t G x t x t r t
with the initial condition ( ) ( ) ( ) , for 0 , 0 , Here ( ) denotes the set of all continuous and bounded functions with the supremum norm ⋅ defined Throughout this paper, we assume that . We also assume that the function : continuous.Moreover, we assume that the function G is bounded, so there exists Special cases of the equation ( 1) have been investigated by many authors.For example, Burton in [6] and Zhang in [17] have studied boundedness and stability of the linear equation:

x t a t x t r t ′ = − −
In [13], Burton and Furumochi using the fixed point theorem of Krasnosielski obtained boundedness and asymptotic stability for the equation: γ ∈ and γ is a quotient with odd positive integer denominator and Next, Jin and Luo (see [4]), proved the boundedness and stability of solutions of the equation and generalized the results claimed in [6] [12] [17].Ardjouni and Djoudi in [3] 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 [19], Brayton and Willoughby [20] and to the books by Driver [21], Hale [14] and Popov [22] and reference cited therein.
The fundamental tool in this paper is the Kuratowski measure of noncompactness (see [23]).
For any bounded subset A of E we denote by ( ) A α the Kuratowski measure of noncompactness of A, i.e. the infimum of all 0 >  such that there exists a finite covering of A by sets of diameters smaller than  .The properties of the measure of noncompactness α are: 2) ( ) ( ) , where A denotes the closure of A; 3) ( ) be a family of strongly equicontinuous functions.Let ( ) ( ) where ( ) α denotes the measure of noncompactness in ( ) Let X be a closed convex subset of ( ) , C I E and let Φ be a function which assigns to each nonempty subset Z of X a sequence ( ) In the proof of the main theorem, we will apply the following fixed point theorem Theorem 1 [18] If for arbitrary nonempty subset Z of X with ( ) 0 Z Φ > , then F has a fixed point in X.

Main Results
A solution of the problem (1) is a continuous function Stability definitions may found in [1], for example.
In this paper, we extend stability theorem proved in [3] 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 P φ over the Banach space S ψ , 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 P φ .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.
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 where { } Let us recall that for continuous function : f R E → , the first order modulus of smoothness for f is the function Now, we use ( 6) to define the operator : P S S ψ ψ → by ( )( ) ( ) , Theorem 2 Assume that the function ( ) is uniformly continuous, satisfies the condition ( ) and there exists a constant { } for each 1 2 , V V S ψ ⊂ and for each ( ) , where α denotes the Kuratowski measure of noncompactness.Moreover, we assume that there exists a constant ( ) { } ,