Boundedness and Stability in nonlinear delay difference equations employing fixed point theory

In this paper we study stability and boundedness of the nonlinear dierence equation x(t + 1) = a(t)x(t) + c(t) x(t g(t)) + q(x(t); x(t g(t)) : In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis. Liapunov's method is normally used to study the stability properties of the zero solution of dieren tial and dierence equations. Certain diculties arise when Liapunov's method is applied to equations with unbounded de- lay or equations containing unbounded terms (11), (20). It has been found that some of these diculties can be eliminated if xed point theory is used instead (3). In the present paper we study certain type of boundedness and stability properties of solutions of linear and nonlinear dierence equations with delay using xed point theory as the main mathematical tool. In particular we study equi-boundedness of solutions and stability of the zero solution of


Introduction
Liapunov's method is normally used to study the stability properties of the zero solution of differential and difference equations.Certain difficulties arise when Liapunov's method is applied to equations with unbounded delay or equations containing unbounded terms [11], [20].It has been found that some of these difficulties can be eliminated if fixed point theory is used instead [3].In the present paper we study certain type of boundedness and stability properties of solutions of linear and nonlinear difference equations with delay using fixed point theory as the main mathematical tool.
In particular we study equi-boundedness of solutions and stability of the zero solution of x(t + 1) = a(t)x(t) + c(t)∆x(t − g(t)) + q(x(t), x(t − g(t)) (1.1) where a, c : Z → R, q : R × R → R and g : Z → Z + .The operator ∆ is defined as ∆x(t) = x(t + 1) − x(t).We assume that a and c are bounded discrete functions whereas g can be unbounded.Continuous versions of (1.1) are generally known as neutral differential equations.Neutral differential equations have many applications.For example, they arise in the study of two or more simple oscillatory systems with some interconnections between them [4,22], and in modelling physical problems such as vibration of masses attached to an elastic bar [22].EJQTDE, 2005 No. 26, p. 1 In section 2, we consider a linear difference equation and obtain the asymptotic stability of the zero solution employing Liapunov's method.We then point out some difficulties and restrictions that arise when the method of Liapunov is used.In sections 3 and 4 we study equi-boundedness of solutions and stability of the zero solution of (1.1) employing the contraction mapping principle.
If g is bounded and the maximum of g is k, then for any integer t 0 ≥ 0, we define Z 0 to be the set of integers in [t 0 − k, t 0 ].If g is unbounded then Z 0 will be the set of integers in (−∞, t 0 ].Let ψ : Z 0 → R be an initial discrete bounded function. Definition 1.1.We say x(t) := x(t, t 0 , ψ) is a solution of (1.1) if x(t) = ψ(t) on Z 0 and satisfies (1.1) for t ≥ t 0 .Definition 1.2.The zero solution of (1.1) is Liapunov stable if for any > 0 and any integer t 0 ≥ 0 there exists a δ > 0 such that |ψ(t)| ≤ δ on Z 0 implies |x(t, t 0 , ψ)| ≤ for t ≥ t 0 .
Definition 1.3.The zero solution of (1.1) is asymptotically stable if it is Liapunov stable and if for any integer t 0 ≥ 0 there exists r(t 0 ) > 0 such that Definition 1.5.The solutions of (1.1) are said to be equi-bounded if for any t 0 and any B 1 > 0, there exists a We refer the readers to [16]  Our objective in this section is to illustrate the difficulties and restrictions that arise when Liaponuv's method is used to study the stability of the zero solution of difference equations with delay.To show this, we consider the linear difference equation where a, b and g are as defined above.
Also, suppose there is a δ > 0 such that fot all t ∈ Z and |b(t)| ≤ δ. (2.3) Then the zero solution of (2.1) is asymptotically stable.
Proof.Define the Liapunov functional V (t, x t ) by Then along solutions of (2.1) we have Remark.The first difficulty associated with the above method is the construction of an efficient Liapunov functional.Secondly, conditions (2.2) and (2.3) in Theorem 2.1 imply that which is a very restrictive condition on the functions a and b.

Boundedness and Stability of Linear Difference Equation Employing Fixed Point Theory
In this section we begin our study of boundedness and stability using the contraction mapping principle by considering the linear difference equation with delay where a, b, c, g and ∆ are defined above.
The use of the contraction mapping principle requires a map and a complete metric space.Thus we begin by inverting equation (3.1) to obtain the map.
In the process we will require the following: a) For any sequence x(k) b) where E is defined as Ex(t) = x(t + 1).
Lemma 3.1.Suppose that a(t) = 0 for all t ∈ Z .Then x is a solution of equation (3.1) if and only if where Proof.Note that equation (3.1) is equivalent to the equation Dividing both sides by gives where φ(r) = c(r) − c(r − 1)a(r).
This completes the proof of lemma 3.1.
Theorem 3.1.Suppose a(t) = 0 for t ≥ t 0 and a(t) satisfies for M > 0. Also, suppose that there is an α ∈ (0, 1) such that Then solutions of (3. We first show that P maps from S to S. By (3.6) Thus P maps from S into itself.We next show that P is a contraction under the supremum norm.Let ζ, η ∈ S. Then This shows that P is a contraction.Thus, by the contraction mapping principle, P has a unique fixed point in S which solves (3.1).This proves that solutions of (3.Then the zero solution of (3.1) is asymptotically stable.
Proof.We have already proved that the zero solution of (3.1) is Liapunov stable.Choose r(t 0 ) to be the δ of the Liapunov stability of the zero solution.
Let ψ(t) be any initial discrete function satisfying |ψ(t)| ≤ r(t 0 ).Define Thus for t > t 2 , we have Thus showing that the last term of (3.8) goes to zero as t goes to infinity.Therefore (P ϕ)(t) → 0 as t → ∞.
By the contraction mapping principle, P has a unique fixed point that solves (3.1) and goes to zero as t goes to infinity.Therefore, the zero solution of (3.1) is asymptotically stable.

Equi-boundedness
For this example we let t 0 = 0 thus Z 0 = [−2, 0].Let B 1 > 0 be given and ψ(t) : Z 0 → R be a given initial function with Let S be the set definded by (3.7).
In our example Therefore, φ(r) of (3.8) yields Now using (3.16) in (3.8) we define P ϕ (t) = ψ(t) on Z 0 and for t ≥ 0 To see that P defines a contraction mapping, we let ζ, η ∈ S. Then We now show that P maps from S into itself.Let ϕ ∈ S. Then Define the map P : S → S by (3.17).Then P is a contraction map and for any ϕ ∈ S, ||P ϕ|| ≤ .Therefore the zero solution of (3.14) is Liapunov stable at t 0 = 0.

Boundedness and Stability of Nonlinear Difference Equation Employing Fixed Point Theory
We continue our study of boundedness and stability using fixed point theory by considering the nonlinear difference equation with delay where a, c and g are defined as before.Here we assume that, q(0, 0) = 0 and q is locally Lipschitz in x and y.That is, there is a K > 0 so that if |x|, |y|, |z| and |w| ≤ K then |q(x, y) − q(z, w)| ≤ L|x − z| + E|y − w| for some positive constants L and E. Note that |q(x, y)| = |q(x, y) − q(0, 0) + q(0, 0)| ≤ |q(x, y) − q(0, 0)| + |q(0, 0)| ≤ L|x| + E|y|.
Lemma 4.1.Suppose that a(t) = 0 for all t ∈ Z. Then x is a solution of equation (4.1) if and only if )Φ(r) + q(x(r), x(r − g(r))) for M > 0. Also, suppose that there is an α ∈ (0, 1) such that Then solutions of (4.1) are equi-bounded.We first show that P maps from S into itself.
by (4.3).Thus showing that P maps from S into itself.
We next show that P is a contraction map.For ζ, η ∈ S, we get Thus, by the contraction mapping principle, P has a unique fixed point in S which solves (4.1).This proves that solutions of (4.1) are equi-bounded.
Theorem 4.2.Assume that the hypotheses of Theorem 4.1 hold.Then the zero solution of (4.1) is Liapunov stable.

. 1 . 4 . 1 .
), t ≥ t 0 where Φ(r) = c(r) − c(r − 1)a(r).EJQTDE, 2005 No. 26, p. 12Remark.The details of the inversion of equation (4.1) is similar to that of equation (3.1) discussed in the previous section and so we omit the proof of lemma 4Theorem Suppose a(t) = 0 for t ≥ t 0 and a satisfies
.5) Let ψ : Z 0 → R be an initial discrete function satisfying |ψ(t)| ≤ δ, t ∈ Z 0 .Let S be the set defined by(3.10).Define the map P : S → S by(4.4).It follows from the proof of Theorem 4.1 that P is a contraction map and for any ψ ∈ S, ||P ϕ|| ≤ .This proves that the zero solution of (4.1) is Liapunov stable.Proof We have already proved that the zero solution of (4.1) is Liapunov stable.Choose r(t 0 ) to be the δ of the Liapunov stability of the zero solution.Let ψ be any initial discrete function satisfying |ψ(t)| ≤ r, t ∈ Z 0 .Let S * be the set defined by(3.13).Define the map P : S * → S * by (4.4).From the proof of Theorem 4.1, the map P is a contraction and it maps S * into itself.Next we show that (P ϕ)(t) goes to zero as t goes to infinity.The first term on the right of (4.4) goes to zero because of condition (4.6).The second term on the right goes to zero because of condition (4.7) and the fact that ϕ ∈ S * .Finally we show that the last term on the right of (4.4) goes to zero as t → ∞.Let ϕ ∈ S * then |ϕ(t−g(t))| ≤ .Also, since ϕ(t−g(t)) → 0 as t−g(t) → ∞, there exists a t 1 > 0 such that for t > t 1 , |ϕ(t − g(t))| < 1 for 1 > 0. Due to condition (4.6) there exists a t 2 > t 1 such that for t > t 2 implies that − Φ(r)ϕ(r − g(r)) + q(ϕ(r), ϕ(r − g(r)))