Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 8, 1-7; http://www.math.u-szeged.hu/ejqtde/

Stability in discrete equations with variable delays

ERNEST YANKSON Department of Mathematics and Statistics University of Cape Coast Ghana. email: ernestoyank@yahoo.com

Abstract

In this paper we study the stability of the zero solution of difference equations with variable delays. In particular we consider the scalar delay equation

Δx(n) = −a(n)x(n − τ(n))

and its generalization

Δx(n) = − ∑_{j=1}^{N} a_{j}(n)x(n − τ_{j}(n)).

Fixed point theorems are used in the analysis.

Key words: Fixed point; Contraction mapping; Asymptotic stability Mathematics subject classifications: 34K13,34C25, 34G20.

1 Introduction

Let ℝ denote the real numbers, ℝ^{+} = [0, ∞), ℤ the integers, ℤ^{−} the negative integers, and ℤ^{+} = {x ∈ ℤ | x ≥ 0}. In this paper we study the asymptotic stability of the zero solution of the scalar delay equation

Δx(n) = −a(n)x(n − τ(n))  (1.1)

and its generalization

Δx(n) = − ∑_{j=1}^{N} a_{j}(n)x(n − τ_{j}(n)).  (1.2)

where a, a_{j} : ℤ^{+} → ℝ and τ, τ_{j} : ℤ^{+} → ℤ^{+} with n − τ(n) → ∞ as n → ∞. For each n_{0}, define m_{j}(n_{0}) = inf{s − τ_{j}(s) : s ≥ n_{0}}, m(n_{0}) = min{m_{j}(n_{0}) : 1 ≤ j ≤ N}. Note that (1.2) becomes (1.1) for N = 1.

Recently, in [11], Raffoul studied the stability of the zero solution of (1.1) when τ(n) = r. Our objective in this research is to generalize the stability results in [11] to (1.2) for variable τ_{j}(n)’s. For more on stability using fixed point theory we refer to [1],[7],[9],[11],[12] and for basic results on difference calculus we refer to [2] and [8]. We also refer to [3],[4],[5],[6] and [10] for other results on stability for difference equations.

Remark 1.1 In [7], the author and Islam showed that the zero solution of the equation

x(n + 1) = b(n)x(n) + a(n)x(n − τ(n))

is asymptotically stable with one of the assumptions being that

∏_{s=0}^{n−1} b(s) → 0 as n → ∞.  (1.3)

However, as pointed out in [11], condition (1.3) cannot hold for (1.2) since b(n) = 1, for all n ∈ ℤ. The results we obtain in this paper overcome the requirement of (1.3).

Let D(n_{0}) denote the set of bounded sequences ψ : [m(n_{0}), n_{0}] → ℝ with the maximum norm ||.||. Also, let (B, ||.||) be the Banach space of bounded sequences φ : [m(n_{0}), ∞) → ℝ with the maximum norm. Define the inverse of n − τ_{i}(n) by g_{i}(n) if it exists and set

Q(n) = ∑_{j=1}^{N} b(g_{j}(n)),

where

∑_{j=1}^{N} b(g_{j}(n)) = 1 − ∑_{j=1}^{N} a(g_{j}(n)).

For each (n_{0}, ψ) ∈ ℤ^{+} × D(n_{0}), a solution of (1.2) through (n_{0}, ψ) is a function x : [m(n_{0}), n_{0} + α) → ℝ^{n} for some positive constant α > 0 such that x(t) satisfies (1.2) on [n_{0}, n_{0}+α) and x(n) = ψ(n) for n ∈ [m(n_{0}), n_{0}]. We denote such a solution by x(n) = x(n, n_{0}, ψ). For a fixed n_{0}, we define

||ψ|| = max{|ψ(n)| : m(n_{0}) ≤ n ≤ n_{0}}.

2 Stability

In this section we obtain conditions for the zero solution of (1.2) to be asymptotically stable.

We begin by rewriting (1.2) as

Δx(n) = − ∑_{j=1}^{N} a_{j}(g_{j}(n))x(n) + Δ_{n} ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))x(s)  (2.1)

where Δ_{n} represents that the difference is with respect to n. If we let

∑_{j=1}^{N} b_{j}(g_{j}(n)) = 1 − ∑_{j=1}^{N} a_{j}(g_{j}(n)),

then (2.1) is equivalent to

x(n + 1) = ∑_{j=1}^{N} b_{j}(g_{j}(n))x(n) + Δ_{n} ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))x(s)  (2.2)

Lemma 2.1 Suppose that Q(n) ≠ 0 for all n ∈ ℤ^{+} and the inverse function g_{j}(n) of n − τ_{j}(n) exists. Then x(n) is a solution of (2.2) if and only if

x(n) = (x(n_{0}) − ∑_{j=1}^{N} ∑_{s=n_{0}−τ_{j}(n_{0})}^{n_{0}−1} a_{j}(g_{j}(s))x(s)) ∏_{s=n_{0}}^{n−1} Q(s) + ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))x(s)

− ∑_{s=n_{0}}^{n−1} ([1 − Q(s)] ∏_{k=s+1}^{n−1} Q(k) ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} a_{j}(g_{j}(u))x(u)), n ≥ n_{0}.

Proof. By the variation of parameters formula we obtain

x(n) = x(n_{0}) ∏_{s=n_{0}}^{n−1} Q(s) + ∑_{k=0}^{n−1} ( ∏_{s=k}^{n−1} Q(s)Δ_{k} ∑_{j=1}^{N} ∑_{s=k−τ_{j}(k)}^{k−1} a_{j}(g_{j}(s))x(s)). (2.3)

Using the summation by parts formula we obtain

∑_{k=0}^{n−1} ( ∏_{s=k}^{n−1} Q(s)Δ_{k} ∑_{j=1}^{N} ∑_{s=k−τ_{j}(k)}^{k−1} a_{j}(g_{j}(s))x(s))

= ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))x(s)

− ∏_{s=n_{0}}^{n−1} Q(s) ∑_{j=1}^{N} ∑_{s=n_{0}−τ_{j}(n_{0})}^{n_{0}−1} a_{j}(g_{j}(s))x(s)

− ∑_{s=n_{0}}^{n−1} ([1 − Q(s)] ∏_{k=s+1}^{n−1} Q(k) ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} a_{j}(g_{j}(u))x(u)).  (2.4)

Substituting (2.5) into (2.3) gives the desired result. This completes the proof of Lemma 2.1.

We next state and prove our main results.

Theorem 2.1 Suppose that the inverse function g_{j}(n) of n − τ_{j}(n) exists, and assume there exists a constant α ∈ (0, 1) such that

∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} |a_{j}(g_{j}(s))| +

∑_{s=n_{0}}^{n−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))|) ≤ α. (2.5)

Moreover, assume that there exists a positive constant M such that

| ∏_{s=n_{0}}^{n−1} Q(s)| ≤ M.

Then the zero solution of (1.2) is stable.

Proof. Let ϵ > 0 be given. Choose δ > 0 such that

(M + Mα)δ + αϵ ≤ ϵ.

Let ψ ∈ D(n_{0}) such that |ψ(n)| ≤ δ. Define

𝕊 = {φ ∈ B : φ(n) = ψ(n) if n ∈ [m(n_{0}), n_{0}], ||φ|| ≤ ϵ}.

Then (𝕊, ||.||) is a complete metric space where, ||.|| is the maximum norm.

Define the mapping P : S → S by

(P φ)(n) = ψ(n) for n ∈ [m(n_{0}), n_{0}]

and

(P φ)(n) = (ψ(n_{0}) − ∑_{j=1}^{N} ∑_{s=n_{0}−τ_{j}(n_{0})}^{n_{0}−1} a_{j}(g_{j}(s))ψ(s)) ∏_{s=n_{0}}^{n−1} Q(s)

+ ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))φ(s)

− ∑_{s=n_{0}}^{n−1} ([1 − Q(s)] ∏_{k=s+1}^{n−1} Q(k) ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} a_{j}(g_{j}(u))φ(u)). (2.6)

Clearly, ℙφ is continuous. We first show that P maps from S to S. By (2.6)

|(P φ)(n)| ≤ Mδ + Mαδ + { ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} a_{j}(g_{j}(s))

+ ∑_{s=n_{0}}^{n−1} ([1 − Q(s)] ∏_{k=s+1}^{n−1} Q(k) ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} a_{j}(g_{j}(u))φ(u)}||φ||

≤ (M + Mα)δ + αϵ
≤ ϵ.

Thus P maps from S into itself. We next show that P is a contraction. Let ζ, η ∈ S. Then

|(P ζ)(t) − (P η)(t)| ≤ { ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} |a_{j}(g_{j}(s))|

+ ∑_{s=n_{0}}^{n−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))|)}||ζ − η||

≤ α||ζ − η||

This shows that P is a contraction. Thus, by the contraction mapping principle, P has a unique fixed point in S which solves (1.2) and for any φ ∈ S, ||P φ|| ≤ ϵ. This proves that the zero solution of (1.2) is stable.

Theorem 2.2 Assume that the hypotheses of Theorem 2.1 hold. Also assume that

∏_{k=n_{0}}^{n−1} Q(k) → 0 as n → ∞.  (2.7)

Then the zero solution of (1.2) is asymptotically stable.

Proof. We have already proved that the zero solution of (1.2) is stable. Let ψ ∈ D(n_{0}) such that |ψ(n)| ≤ δ and define

𝕊^{∗} = {φ ∈ B : φ(n) = ψ(n) if n ∈ [m(n_{0}), n_{0}], ||φ|| ≤ ϵ and φ(n) → 0, as n → ∞}.

Define P : S^{∗} → S^{∗} by (2.6). From the proof of Theorem 2.2, the map P is a contraction and for every φ ∈ S^{∗}, ||(P φ)|| ≤ ϵ.

We next show that (P φ)(n) → 0 as n → ∞. The first term on the right side of (2.6) goes to zero because of condition (2.7). It is clear from (2.5) and the fact that φ(n) → 0 as n → ∞ that ∑_{j=1}^{N} ∑_{s=n−τ_{j}(n)}^{n−1} |a_{j}(g_{j}(s))||φ(s)| → 0 as n → ∞.

Now we show that the last term on the right side of (2.6) goes to zero as n → ∞. Since φ(n) → 0 and n − τ_{j}(n) → ∞ as n → ∞, for each ϵ_{1} > 0, there exists a N_{1} > n_{0} such that s ≥ N_{1} implies |φ(s − τ_{j}(s))| < ϵ_{1} for j = 1, 2, 3, ..., N. Thus for n ≥ N_{1}, the last term, I_{3} in (2.6) satisfies

|I_{3}| = | ∑_{s=n_{0}}^{n−1} ([1 − Q(s)] ∏_{k=s+1}^{n−1} Q(k) ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} a_{j}(g_{j}(u))φ(u))|

≤ ∑_{s=n_{0}}^{N_{1}−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))||φ(u)|)

+ ∑_{s=N_{1}}^{n−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))||φ(u)|)

≤ max_{σ≥m(n_{0})} |φ(σ)| ∑_{s=n_{0}}^{N_{1}−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))|)

+ ϵ_{1} ∑_{s=N_{1}}^{n−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))|)

By (2.7), there exists N_{2} > N_{1} such that n ≥ N_{2} implies

max_{σ≥m(n_{0})} |φ(σ)| ∑_{s=n_{0}}^{N_{1}−1} (|[1 − Q(s)]|| ∏_{k=s+1}^{n−1} Q(k)| ∑_{j=1}^{N} ∑_{u=s−τ_{j}(s)}^{s−1} |a_{j}(g_{j}(u))|) < ϵ_{1}.

Apply (2.5) to obtain |I_{3}| ≤ ϵ_{1} + ϵ_{1}α < 2ϵ_{1}. Thus, I_{3} → 0 as n → ∞. Hence (P φ)(n) → 0 as n → ∞, and so P φ ∈ S^{∗}.

By the contraction mapping principle, P has a unique fixed point that solves (1.2) and goes to zero as n goes to infinity. Therefore, the zero solution of (1.2) is asymptotically stable.

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(Received November 14, 2008)