Stability for Nonlinear Neutral Integro-Differential Equations with Variable Delay

In this paper we use the contraction mapping principle to obtain asymptotic stability results of a nonlinear neutral integrodifferential equation with variable delay. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous results due to Burton [7], Becker and Burton [6] and Jin and Luo [17]. In the end we provide an example to illustrate our claim.


Introduction
Incontestably, Lyapunov's direct method has been, for more than 100 years, the main tool for investigating the stability properties of a wide variety of ordinary, functional, partial differential and integro-differential equations. Nevertheless, the application of this method to problems of stability in differential and integro-differential equations with delays has encountered serious obstacles if the delays are unbounded or if the equation has unbounded terms [8]- [10]. In recent years, several investigators have tried stability by using a new technique. Particularly, Burton, Furumochi, Becker and others began a study in which they noticed that some of these difficulties vanish or might be overcome by means of fixed point theory (see [1]- [21], [23]). The fixed point theory does not only solve the problem on stability but has other significant advantage over Lyapunov's. The conditions of the former are often averages but those of the latter are usually pointwise (see [8]).
Here C (S 1 , S 2 ) denotes the set of all continuous functions ϕ : S 1 → S 2 with the supremum norm · . Throughout this paper we assume that a ∈ C(R + × [m (0) , ∞) , R) and τ ∈ C (R + , R + ) with t − τ (t) → ∞ as t → ∞. The function Q (t, x) is globally Lipschitz continuous in x. That is, there is positive constant L such that (2) |Q (t, x) − Q (t, y)| ≤ L x − y .
We also assume that Special cases of equation (1) have been investigated by many authors. For example, Burton in [7], Becker and Burton in [6], Jin and Luo in [17] have studied the equation (4) x a (t, s) x (s) d s, and have respectively proved the following theorems.
Theorem 1.1 (Burton [7]). Suppose that τ (t) = r and there exists a constant α < 1 such that Then the zero solution of (4) is asymptotically stable.

Stability for Integro-Differential Equations
In a recent work, we have studied the linear neutral equation (14) x and have have obtained the following result.
Theorem 1.4 (Ardjouni, Djoudi and Soualhia [5]). Suppose that τ is twice continuously differentiable with τ (t) = 1 for all t ∈ R + , c is continuously differentiable on R + , and there exist continuous function h : [m (0) , ∞) → R and a constant α ∈ (0, 1) such that for t ≥ 0 where B is given by (12) and Then the zero solution of (14) is asymptotically stable if and only if Note that in our consideration the neutral term d d t Q (t, x (t − τ (t))) of (1) produces nonlinearity in the derivative term d d t x (t − τ (t)). The neutral term d d t x (t − τ (t)) in [5] enters linearly. So, the analysis made here is different form that in [5].
Our objective here is to improve Theorem 1.3 and extend it to investigate a wide class of nonlinear integro-differential equation with variable delay of neutral type presented in (1). To do this we define a suitable continuous function h (see Theorem 2.1 below) and find conditions for h, with no need of further assumptions on the inverse of the delay t − τ (t), so that for a given continuous initial function ψ a mapping P for (1) is constructed in such a way to map a complete metric space S ψ in itself and in which P possesses a fixed point. This procedure will enable us to establish and prove an asymptotic stability theorem for the zero solution of (1) with a necessary and sufficient condition and with less restrictive conditions. The results obtained in this paper improve and generalize the main results in [6,7,17]. We provide an example to illustrate our claim.

Main results
Stability definitions may be found in [8], for example.
Our purpose here is to extend Theorem 1.3 by giving a necessary and sufficient condition for asymptotic stability of the zero solution of equation (1). But, to reach this end, one crucial step in the investigation of the stability of an equation using fixed point technic involves the construction of a suitable fixed point mapping. This can, in so many cases, be an arduous task. So, to construct our mapping, we begin by transforming (1) to a more tractable, but equivalent, equation, which we then invert to obtain an equivalent integral equation from which we derive a fixed point mapping. After then, we define prudently a suitable complete space, depending on the initial condition, so that the mapping is a contraction. Using Banach's contraction mapping principle, we obtain a solution for this mapping, and hence a solution for (1), which is asymptotically stable.
First, we have to transform (1) into an equivalent equation that possesses the same basic structure and properties to which we apply the variation of parameters to define a fixed point mapping. where Proof. Differentiating the integral term in (19), we obtain Substituting this into (19), it follows that (19) is equivalent to (1) provided B satisfies the following conditions for some function φ and B (t, s) must satisfy Consequently, Substituting this into (21), we obtain This definition of B satisfies (20). Consequently, (1) is equivalent to (19).  Proof. Use Lemma 2.1 to rewrite (1) in the following equivalent form Multiplying both sides of (25) by e t 0 h(u) d u and integrating with respect to s from 0 to t, we obtain

Stability for Integro-Differential Equations
Performing an integration by parts, we get Then the zero solution of (1) is asymptotically stable if and only if Proof. First, suppose that (28) holds. We set The set C ([m (0) , ∞) , R) of real valued bounded functions on [m (0) , ∞) is a Banach space when it is endowed with the supremum norm · ; that is, Otherwise speaking, we carry out our investigations in the complete metric Let ψ ∈ C ([m (0) , 0] , R) be fixed and define for t ∈ [m (0) , 0] and ϕ (t) → 0 as t → ∞ .
For the special case Q (t, x) = 0, we can get It is easy to see that all the conditions of Theorem 2.1 hold for α = 0.112 + 0.238 + 0.366 + 0.238 = 0.954 < 1. Thus, Theorem 2.1 implies that the zero solution of (35) is asymptotically stable.