Stability Criterion for Volterra Type Delay Difference Equations Including a Generalized Difference Operator

Murat Gevgesoglu∗; Yasar Bolat

doi:10.5666/KMJ.2020.60.1.163

Article

Kyungpook Mathematical Journal 2020; 60(1): 163-175

Published online March 31, 2020

Stability Criterion for Volterra Type Delay Difference Equations Including a Generalized Difference Operator

Murat Gevgesoglu∗ and Yasar Bolat

Department of Mathematics, Kastamonu University, Kastamonu, Turkey
e-mail : mgevgesoglu@kastamonu.edu.tr and ybolat@kastamonu.edu.tr

Received: January 17, 2018; Revised: May 10, 2019; Accepted: November 18, 2019

Abstract

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Abstract
1. Introduction
2. Basic Definitions, Theorems and Lemmas
3. Main Results
References

The stability of a class of Volterra-type difference equations that include a generalized difference operator Δ_a is investigated using Krasnoselskii’s fixed point theorem and some results are obtained. In addition, some examples are given to illustrate our theoretical results.

Keywords: stability, Volterra diﬀ,erence equations.

1. Introduction

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Abstract
1. Introduction
2. Basic Definitions, Theorems and Lemmas
3. Main Results
References

Difference equations are the discrete analogues of differential equations and they usually describe certain phenomena over the course of time. Difference equations have many applications in a wide variety of disciplines, such as economics, mathematical biology, social sciences and physics. We refer to [1, 2, 4, 6] for the basic theory and some applications of difference equations. Volterra difference equations are extensively used to model phenomena in engineering, economics, and in the natural and social sciences; their stability has been studied by many authors.

In [5], Khandaker and Raffoul considered a Volterra discrete system with nonlinear perturbation

x(n+1)=A(n)x(n)+∑s=0nB(n,s)x(s)+g(n,x(n))

and obtained necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with a variation of parameters formula.

In [7], Migda et al. investigated the boundedness and asymptotic stability of the zero solution of the discrete Volterra equation

x(n+1)=a(n)+b(n)x(n)+∑i=n0nK(n,i)x(i)

using fixed point theory.

In [3], Islam and Yankson studied the stability and boundedness of the nonlinear difference equation

x(t+1)=a(t)x(t)+c(t)Δx(t-g(t))+q(x(t),x(t-g(t)))

using fixed point theorems.

In [9], Yankson studied the asymptotic stability of the zero solution of the Volterra difference delay equation

x(n+1)=a(n)x(n)+c(n)Δx(n-g(n))+∑s=n-g(n)n-1k(n,s)h(x(s))

using Krasnoselskii’s fixed point theorem.

In this paper, motivated by [9], we investigate the asymptotic stability of the zero solution of neutral and Volterra type difference equations which include a generalized difference operator of the form

(1.1)Δa[x(n)-b(n)x(n-σ)]=c(n)x(n)+∑u=n-σn-1k(u,n)h(x(u),x(u-τ))

using Krasnoselskii’s fixed point theorem. Here b(n) : ℤ → ℝ and c(n) : ℤ → ℝ are discrete bounded functions, k(u, n) : ℤ × ℤ → ℝ⁺, h : ℝ × ℝ → ℝ, σ and τ are non-negative integers with lim(n − σ) = ∞ and lim(n − τ) = ∞.

The difference operator Δ and generalized difference operator Δ_a are defined as

Δx(n)=x(n+1)-x(n)

and

(1.2)Δa(x)(n)=x(n+1)-ax(n),a>0

respectively.

We assume that h(0, 0) = 0 and

(1.3)∣h(x1,y1)-h(x2,y2)∣≤K max{∣x1-x2∣,∣y1-y2∣}

for some positive constant K.

2. Basic Definitions, Theorems and Lemmas

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Abstract
1. Introduction
2. Basic Definitions, Theorems and Lemmas
3. Main Results
References

For any integer n₀ ≥ 0 we define Z₀ as the set of all integers in the interval [−σ − τ, n₀]. Let ω : Z₀ → ℝ be a discrete and bounded initial function.

Definition 2.1

x(n) = x(n, n₀, ω) is a solution of (1.1) if x(n) = ω(n) for n ∈ Z₀ and satisfies (1.1) for n ≥ n₀.

Definition 2.2

The zero solution of (1.1) is stable if for any ɛ > 0 and any integer n₀ ≥ 0 there exists a δ = δ (ɛ) such that |ω(n)| < δ for n ∈ Z₀ implies |x(n, n₀, ω)| < ɛ for n ≥ n₀.

Definition 2.3

The zero solution of (1.1) is asymptotically stable if it is stable and for any integer n₀ ≥ 0 there exists a δ = δ (n₀) such that |ω(n)| < δ for n ∈ Z₀ implies limn→∞x(n)=0.

Lemma 2.1

Where the generalized difference operator Δ_a is as defined in(1.2), we have

Δax(n)=an+1Δ(x(n)an).

Proof

It is obvious.

Now below we state Krasnoselskii’s theorem. For the proof we refer to [8].

Theorem 2.1

Let M be a closed convex nonempty subset of a Banach space (B, ||.||). Suppose that A and Q map M into B such that

x, y ∈ M implies Ax + Qy ∈ M,
A is continuous and AM is contained in a compact set,
Q is a contraction mapping.

Then, there exits z ∈ M with z = Az + Qz.

Theorem 2.2.(Ascoli-Arzela Theorem)

Let (X, d) be a compact metric space and C(X) be a vector space consisting of all continuous function f : X → ℝ. A subset F of C(X) is relatively compact if and only if F is equibounded and equicontinuous.

3. Main Results

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Abstract
1. Introduction
2. Basic Definitions, Theorems and Lemmas
3. Main Results
References

Lemma 3.1

Assume that (a + c(n)) ≠ 0 for all n ∈ ℤ. Necessary and sufficient condition for x (n) to be the solution of(1.1)are

x(n)=(x(n0)-b(n0)x(n0-σ))∏u=n0n-1(a+c(u))+b(n)x(n-σ)+∑r=n0n-1[c (r) b (r) x (r-σ)+∑u=r-σr-1k (u,r) h (x (u),x (u-τ))]∏s=r+1n-1(a+c(s)),n≥n0.

Proof

From (1.1) we can write

(3.1)Δax (n)-c (n) x (n)=Δa(b (n) x (n-σ))+∑u=n-σn-1k(u,n) h (x (u),x (u-τ)).

Using the definition of the operator Δ_a in the left-hand side of (3.1) and multiplying both sides of (3.1) with ∏s=n0n(a+c(s))-1 we have

(3.2)Δ(x (n)∏s=n0n-1(a+c(s))-1)=[Δa (b (n) x (n-σ))+∑u=n-σn-1k(u,n) h (x (u),x (u-τ))]∏s=n0n(a+c(s))-1.

By summing both sides of (3.2) from n₀ to n − 1, we obtain

x (n)∏s=n0n-1(a+c(s))-1=x(n0)+∑r=n0n-1[Δa(b (n) x (n-σ))+∑u=n-σn-1k(u,n)h (x (u),x (u-τ))]∏s=n0r(a+c(s))-1

from this last equality, we write

x(n)=x(n0)∏s=n0n-1(a+c(s))+{∑r=n0n-1[Δa(b (r) x (r-σ))+∑u=r-σr-1k(u,r)h (x (u),x (u-τ))]∏s=n0r(a+c(s))-1}∏s=n0n-1(a+c(s)).

Because

∏s=n0r(a+c(s))-1∏s=n0n-1(a+c(s))=∏s=r+1n-1(a+c(s)),

we can write

x(n)=x (n0)∏s=n0n-1(a+c(s))+∑r=n0n-1[Δa(b (r) x (r-σ))+∑u=r-σr-1k(u,r)h (x (u),x (u-τ))]∏s=r+1n-1(a+c(s))

or

(3.3)x(n)=x (n0)∏s=n0n-1(a+c(s))+∑r=n0n-1Δa(b (r) x (r-σ))∏s=r+1n-1(a+c(s))+∑r=n0n-1[∑u=r-σr-1k(u,r)h (x (u),x (u-τ))]∏s=r+1n-1(a+c(s)).

Now, using Lemma 2.1 in the second term on the right-hand side of (3.3), we have

∑r=n0n-1Δa(b (r) x (r-σ))∏s=r+1n-1(a+c(s))=∑r=n0n-1ar+1Δ(b (r) x (r-σ)ar)∏s=r+1n-1(a+c(s))=∑r=n0n-1[Δ(b (r) x (r-σ)∏s=rn-1(a+c(s)))-Δ(∏s=rn-1(a+c(s)) ar)b (r) x (r-σ)ar]=∣b (r) x (r-σ)∏s=rn-1(a+c(s))∣r=n0r=n-∑r=n0n-1[Δ(∏s=rn-1(a+c(s))ar)b(r) x (r-σ)ar]=b (n) x (n-σ)-b (n0) x (n0-σ)∏s=n0n-1(a+c(s))-∑r=n0n-1[Δ(∏s=rn-1(a+c(s)) ar)b (r) x (r-σ)ar].

Hence, by putting this last equality in (3.3), we reach

(3.4)x(n)=x (n0)∏s=n0n-1(a+c(s))+∑r=n0n-1[∑u=r-σr-1k(u,r)h (x (u),x (u-τ))]∏s=r+1n-1(a+c(s))+b (n) x (n-σ)-b (n0) x (n0-σ)∏s=n0n-1(a+c(s))-∑r=n0n-1[Δ(∏s=rn-1(a+c(s))ar)b(r) x (r-σ)ar].

Because in the last term on the right-hand side of (3.4)

Δ(∏s=rn-1(a+c(s)) ar)=∏s=r+1n-1(a+c(s)) ar+1-∏s=rn-1(a+c(s)) ar=-c(r)∏s=r+1n-1(a+c(s)) ar,

from (3.4) we obtain

x (n)=[x (n0)-b (n0) x (n0-σ)]∏s=n0n-1(a+c(s))+b (n) x (n-σ)+∑r=n0n-1[c(r)b(r)x(r-σ)+∑u=r-σr-1k(u,r)h (x (u),x (u-τ))]∏s=r+1n-1(a+c(s)), n≥n0.

This completes the proof.

Now let φ(n) be a real sequence defined on ℤ and define the set S as

S={φ:ℤ→ℝ∣ ‖φ‖→0, n→∞}

where

‖φ‖=max∣φ(n)∣,n∈ℤ.

Then, we can see that (S, ||.||) is a Banach space. We then define the mapping H : S → S on Z₀ by

(Hφ) (n)=ω (n)

and for n ≥ n₀ by

(3.5)(Hφ) (n)=[ω (n0)-b (n0) ω (n0-σ)]∏s=n0n-1(a+c(s))+b(n) φ (n-σ)+∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s)).

Lemma 3.2

Let(1.3)hold. Suppose that

(3.6)∏s=n0n-1(a+c(s))→0 as n→∞

and there exists α ∈ (0, 1) such that for n ≥ n₀

(3.7)∑r=n0n-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣≤α.

The mapping H defined by(3.5)approaches 0 as n→∞.

Proof

Due to the condition (3.6) the first term of right-hand side of equation (3.5) approaches to zero as n→∞. Because b(n) is bounded and φ ∈ S is also the second term of right-hand side of equation (3.5) approaches to zero as n → ∞. Now, we show that the last term on the right-hand side of equation (3.5) approaches to zero as n→∞.

Given ɛ₁ > 0 and let n₁ be a positive integer such that for n > n₁ and φ ∈ S, |φ(n − σ)| < ɛ₁. Because φ(n−σ) → 0, for given ɛ₂ > 0 we can find a n₂ > n₁ such that for n > n₂ |φ(n − σ)| < ɛ₂. Furthermore, because of condition (3.6) we can find a n₃ > n₂ such that for n>n3∣∏s=n2n-1(a+c(s))∣<ɛ2αɛ1.

Hence, for n > n₃ from the last term of right-hand side of (3.5) we have

∣∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ(u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣∣≤∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ(u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣∣≤∑r=n0n2-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ(u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣∣+∑r=n2n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ(u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣≤ɛ1∑r=n0n2-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣+ɛ2∑r=n0n-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣=ɛ1∑r=n0n2-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣+ɛ2α=ɛ1∑r=n0n2-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n2-1(a+c(s))∣ ∣∏s=n2n-1(a+c(s))∣+ɛ2α≤ɛ1α∣∏s=n2n-1(a+c(s))∣+ɛ2α≤ɛ2(1+α).

This completes the proof.

To use Krasnoselskii’s theorem, we construct two mappings Q and A expressing (3.5) as

(Hφ) (n)=(Qφ) (n)+(Aφ) (n)

where Q, A : S → S are mappings with

(3.8)(Qφ) (n)=[ω (n0)-b (n0) ω (n0-σ)]∏s=n0n-1(a+c(s))+b(n) φ (n-σ)

and

(3.9)(Aφ) (n)=∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s))

respectively.

Lemma 3.3

Assume that(1.3), (3.6) and(3.7)hold and suppose that there exists a positive constant ξ such that

(3.10)a+c(n)≤1 and maxn∈ℤ∣a+c(n)∣=ξ

Then, the mapping A defined by(3.9)is continuous and compact.

Proof

First, we show that the mapping A defined by (3.9) is continuous. Let φ, φ̄ ∈ S. For a given ɛ > 0 choose δ=ɛα such that ||φ − φ̄|| < δ holds. Then, we have

‖(Aφ)-(Aφ¯)‖=maxn∈ℤ∣{∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ(u),φ (u-τ))]∏s=r+1n-1(a+c(s))}-{∑r=n0n-1[c(r)b(r)φ¯(r-σ)+∑u=r-σr-1k(u,r)h (φ¯ (u),φ¯ (u-τ))]∏s=r+1n-1(a+c(s))}∣≤∑r=n0n-1[∣c(r)b(r)∣ ∣φ(r-σ)-φ¯(r-σ)∣] ∣∏s=r+1n-1(a+c(s))∣+∑r=n0n-1∣∑u=r-σr-1k(u,r)h (φ (u),φ (u-τ))-∑u=r-σr-1k(u,r)h (φ¯ (u),φ¯ (u-τ))∣ ∣∏s=r+1n-1(a+c(s))∣≤∑r=n0n-1∣c(r)b(r)∣ ∣∏s=r+1n-1(a+c(s))∣ ‖φ-φ¯‖+∑r=n0n-1∣∑u=r-σr-1k(u,r) [h (φ (u),φ (u-τ))-h (φ¯ (u),φ¯ (u-τ))]∣ ∣∏s=r+1n-1(a+c(s))∣≤∑r=n0n-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣ ‖φ-φ¯‖≤α‖φ-φ¯‖≤ɛ

which shows that the mapping A is continuous. Now we show that A is compact. For this we use Arzela-Ascoli theorem. Let {φ_n} ⊂ S be a sequence of uniformly bounded functions where ||φ_n|| ≤ m for m > 0 and n is a positive integer. Then using (1.3) we have

‖Aφn‖=maxn∈ℤ∣∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣≤∑r=n0n-1∣[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r) h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣≤∑r=n0n-1∣[∣c(r)b(r)∣+L∑u=r-σr-1k(u,r)]∣ ∣∏s=r+1n-1(a+c(s))∣ ‖φ‖≤α‖φ‖≤αm

which shows that (Aφ_n) is uniformly bounded. Furthermore,

‖Δ (Aφ)‖=maxn∈ℤ∣(Aφ) (n+1)-(Aφ) (n)∣≤∣a+c(n)∣ ∣c(n)b(n)φ(n-σ)+∑u=n-σn-1k(u,r)h (φ(u),φ (u-τ))∣ ∣∏s=r+1n-1(a+c(s))∣≤ξ(∣c(n)b(n)∣+K∑u=n-σn-1k(u,r)) ‖φ‖≤ξαm≤γ

for some positive constant γ. This shows that (Aφ_n) is equi-continuous. Hence, by Arzela-Ascoli’s theorem, the mapping A is compact.

Lemma 3.4

Consider the mapping Q defined by(3.8)and assume that

(3.11)∣b(n)∣≤μ<1

holds for some positive constant μ. Then, Q is a contraction.

Proof

Take any two functions φ, φ̄ ∈ S. We then have

‖(Qφ)-(Qφ¯)‖=maxn∈ℤ∣[ω (n0)-b (n0) ω (n0-σ)]∏s=n0n-1(a+c(s))+b (n) φ (n-σ)-[ω (n0)-b (n0) ω (n0-σ)]∏s=n0n-1(a+c(s))-b (n) φ (n-σ)∣≤∣b(n)∣ ‖φ-φ¯‖≤μ‖φ-φ¯‖

which shows that Q is a contraction mapping.

Theorem 3.1

Suppose that(1.3), (3.6), (3.7), (3.10) and(3.11)hold. Also suppose that there exists positive constants c and β ∈ (0, 1) such that

(3.12)∣∏s=n0n-1(a+c(s))∣≤c

and

(3.13)∣b(n)∣+∑r=n0n-1[∣c(r)b(r)∣+K∑u=r-σr-1k(u,r)] ∣∏s=r+1n-1(a+c(s))∣≤β, n≥n0

hold. Then, the zero solution of(1.1)is asymptotically stable.

Proof

Given ɛ > 0. Choose δ such that

∣1-b(n0)∣δc<ɛ(1-β)

Let ω be a given initial function such that |ω(n)| < δ. Let us define the set M as

M={φ∈S:‖φ‖<ɛ}

and take any φ, ϕ ∈ M. Then, we have

‖(Qϕ)+(Aφ)‖=maxn∈ℤ∣[ω (n0)-b (n0) ω (n0-σ)]∏u=n0n-1(a+c(u))+b (n) ϕ (n-σ)+∑r=n0n-1[c(r)b(r)φ(r-σ)+∑u=r-σr-1k(u,r)h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣≤∣[ω (n0)-b (n0) ω (n0-σ)]∏u=n0n-1(a+c (u))∣+∣b (n) ϕ (n-σ)∣+∑r=n0n-1∣[c(r) b (r) φ (r-σ)+∑u=r-σr-1k(u,r) h (φ (u),φ (u-τ))]∏s=r+1n-1(a+c(s))∣≤∣1-b(n0)∣ δc+∣b(n)∣ɛ+ɛ∑r=n0n-1[∣c (r) b (r)∣+K∑u=r-σr-1k (u,r)] ∣∏s=r+1n-1(a+c(s))∣≤∣1-b(n0)∣ δc+{∣b(n)∣+∑r=n0n-1[∣c (r) b (r)∣+K∑u=r-σr-1k (u,r)] ∣∏s=r+1n-1(a+c(s))∣}ɛ≤∣1-b(n0)∣ δc+βɛ<ɛ

which shows that (Qϕ) + (Aφ) ∈ M.

By the last result, Lemma 4 and Lemma 5 all conditions of Theorem 1 are satisfied onM. Consequently, there exits a fixed point x ∈ M such that x = Qx+Ax holds. Lemma 2 implies that this fixed point x(n) is a solution of (1.1). Furthermore the solution x(n) is stable because ||x|| < ɛ for a given ɛ > 0. By Lemma 3 the solution x(n) is asymptotically stable.

Example 3.1

Consider the difference equation

(3.14)Δ2[x (n)-132 (n+1)!x(n-2)]=-2nn+1x(n)+∑u=n-2n-12n16 (n+1)! (u2+4)h (x (u),x (u-3)),n≥1

Here,

a=σ=2, τ=3, n0=1,c(n)=-2nn+1, b(n)=132 (n+1)!,K(u,n)=2n16 (n+1)! (u2+4).

We see that

∏s=1n-1(2-2nn+1)=2n-1n!→0 as n→∞,

so (3.6) holds. Because

∑r=1n-1[2rr+1132 (n+1)!+∑u=r-2r-12r16 (r+1)! (u2+4)] ∣∏s=r+1n-1(2s+1)∣≤316<1,

(3.7) holds. Because

a+c(n)=2-2nn+1≤1 and maxn∈ℤ∣a+c(n)∣=maxn∈ℤ∣2-2nn+1∣=1

(3.10) holds. Because

∣132 (n+1)!∣≤132<1,

(3.11) holds. Because

∣∏s=1n-1(2-2ss+1))∣=∣∏s=1n-1(2s+1))∣≤1,

(3.12) holds. Also, because

∣132 (n+1)!∣+∑r=1n-1[2rr+1132 (n+1)!+∑u=r-2r-12r16 (r+1)! (u2+4)] ∣∏s=r+1n-1(2s+1)∣≤1364<1,

(3.13) holds. So, by Theorem 3 the zero solution of (3.14) is asymptotically stable.

The solution is of the form

x(n)=(x(1)-164x (-1))∏u=1n-1(2-2uu+1)+132 (n+1)!x (n-2)+∑r=1n-1[-2rr+1132 (r+1)!x (r-2)+∑u=r-2r-12r16 (r+1)! (u2+4)h (x (u),x (u-3))]∏s=r+1n-1(2-2ss+1),n≥1.

References

Go to

Abstract
1. Introduction
2. Basic Definitions, Theorems and Lemmas
3. Main Results
References

RP. Agarwal. Difference equations and inequalities: theory, methods and applications. 2nd ed, , Marcel Dekker, Inc, New York, NY, 2000.
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