# Dynamics of a Family of Nonlinear Delay Difference Equations

Authors: Qiuli He; Taixiang Sun; Hongjian Xi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the global asymptotic stability of the following difference equation:xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…, where 0≤k1<k2<⋯<ks and 0≤m1<m2<⋯<mt with {k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅, the initial values are positive, and f∈C(Es+t,(0,+∞)) with E∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibrium x- of that equation is globally asymptotically stable.


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## Body

## 1. Introduction

In this note, we consider a nonlinear difference equation and deal with the question of whether the unique positive equilibriumx- of that equation is globally asymptotically stable. Recently, there has been much interest in studying the global attractivity, the boundedness character, and the periodic nature of nonlinear difference equations; for example, see [1–22].Amleh et al. [1] studied the characteristics of the difference equation:
(E1)xn+1=p+xn-1xn.
They confirmed a conjecture in [13] and showed that the unique positive equilibrium x-=p+1 of (E1) is globally asymptotically stable provided p>1.Fan et al. [8] investigated the following difference equation:
(E2)xn+1=f(xn,xn-k).
They showed that the length of finite semicycle of (E2) is less than or equal to k and gave sufficient conditions under which every positive solution of (E2) converges to the unique positive equilibrium.Kulenović et al. [11] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the nonautonomous difference equation
(E3)xn+1=pn+xn-1xn,n=0,1,2,…,
where the initial values x-1,x0∈R+≡(0,+∞) and pn is the period-two sequence
(1)pn={α,ifniseven,β,ifnisodd,withα,β∈R+.Sun and Xi [20] studied the more general equation
(E4)xn+1=f(xn-s,xn-t),n=0,1,2,…,
where s,t∈{0,1,2,…} with s<t, the initial values x-t,x-t+1,…,x0∈R+ and gave sufficient conditions under which every positive solution of (E4) converges to the unique positive equilibrium.In this paper, we study the global asymptotic stability of the following difference equation:(2)xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…,
where 0≤k1<k2<⋯<ks and 0≤m1<m2<⋯<mt with {k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅, the initial values are positive and f∈C(Es+t,(0,+∞)) with E∈{(0,+∞),[0,+∞)} and a=inf(u1,u2,…,us,v1,v2,…,vt)∈Es+t×f(u1,u2,…,us;v1,v2,…,vt)∈E satisfies the following conditions:(H1) f(u1,u2,…,us;v1,v2,…,vt) is decreasing in ui for any i∈{1,2,…,s} and increasing in vj for any j∈{1,2,…,t}.(H2) Equation (2) has the unique positive equilibrium, denoted by x-.(H3) The function f(a,a,…,a;x,x,…,x) has only fixed point in the interval (a,+∞), denoted by A.(H4) For any y∈E, f(y,…,y;x,…,x)/x is nonincreasing in x∈(0,+∞).(H5) If (x,y)∈E×E is a solution of the system
(3)y=f(x,…,x;y,…,y),x=f(y,…,y;x,…,x),then x=y.


## 2. Main Result

Theorem 1.
Assume that(H1)–(H5) hold. Then the unique positive equilibrium x- of (2) is globally asymptotically stable.Proof.
Letl=max{mt,ks}. Since
(4)a=inf(u1,u2,…,us,v1,v2,…,vt)∈Es+tf(u1,u2,…,us;v1,v2,…,vt)a∈E,
we have(5)x-=f(x-,x-,…,x-)>f(x-+1,x-,…,x-)≥a.
Claim  1.
f
(
A
,
…
,
A
;
a
,
…
,
a
)
<
x
-
<
A.
Proof of Claim  1. Assume on the contrary that x-≥A. Then it follows from (H1), (H3), and (H4) that
(6)A=f(a,…,a;A,…,A)>f(x-,…,x-;A,…,A)=f(x-,…,x-;A,…,A)AA≥f(x-,…,x-)x-A=A.
This is a contradiction. Therefore x-<A. Obviously
(7)f(A,…,A;a,…,a)<f(x-,…,x-;x-,…,x-)=x-.
Claim  1 is proven.
Claim  2. For any M≥A,J=[a,M] is an invariable interval of (2).
Proof of Claim  2. For any x0,x-1,…,x-l∈J, we have from (H4) that
(8)a≤x1=f(x-k1,x-k2,…,x-ks;x-m1,x-m2,…,x-mt)≤f(a,…,a;M,…,M)MM≤f(a,…,a;A,…,A)AM=M.
By induction, we may show that xn∈J for any n≥1. Claim  2 is proven.
Letm0=a,M0=M≥A and for any i≥0,
(9)mi+1=f(Mi,…,Mi;mi,…,mi),Mi+1=f(mi,…,mi;Mi,…,Mi).
Claim  3. For any n≥0, we have
(10)mn≤mn+1<x-<Mn+1≤Mn,limn→∞Mn=limn→∞mn=x-.
Proof of Claim  3. From Claim  2, we obtain
(11)m0≤m1=f(M0,…,M0;m0,…,m0)<f(x-,…,x-)=x-<f(m0,…,m0;M0,…,M0)=M1≤M0,m1=f(M0,…,M0;m0,…,m0)≤f(M1,…,M1;m1,…,m1)=m2<f(x-,…,x-)=x-<f(m1,…,m1;M1,…,M1)=M2≤f(m0,…,m0;M0,…,M0)=M1.
By induction, we have that for n≥0,
(12)mn≤mn+1<x-<Mn+1≤Mn.
Set
(13)β=limn→∞mnandα=limn→∞Mn.
Then
(14)β=f(α,…,α;β,…,β),α=f(β,…,β;α,…,α).
This with (H2) and (H5) implies α=β=x-. Claim  3 is proven.
Claim  4. The equilibrium x- of (2) is locally stable.
Proof of Claim  4. Let M=A and mn,Mn be the same as Claim  3. For any ε>0 with 0<ε<min{A-x-,x--a}, there exists n>0 such that
(15)x--ε<mn<x-<Mn<x-+ε.
Set 0<δ=min{x--mn,Mn-x-}. Then for any x0,x-1,…,x-l∈(x--δ,x-+δ), we have
(16)x1=f(x-k1,…,x-ks;x-m1,…,x-mt)≤f(mn,…,mn;Mn,…,Mn)=Mn+1≤Mn,x1=f(x-k1,…,x-ks;x-m1,…,x-mt)≥f(Mn,…,Mn;mn,…,mn)=mn+1≥mn.
In similar fashion,we can show that for any k≥1,
(17)xk∈[mn,Mn]⊂(x--ε,x-+ε).
Claim  4 is proven.
Claim  5.
x
- is the global attractor of (2).
Proof of Claim  5. Let {xn}n=-l∞ be a positive solution of (2), and let M=max{x1,…,xl+1,A} and mn,Mn be the same as Claim  3. From Claim  2, we have xn∈[m0,M0]=[a,M] for any n≥1. Moreover, we have
(18)xl+2=f(xl+1-k1,…,xl+1-ks;xl+1-m1,…,xl+1-mt)≤f(m0,…,m0;M0,…,M0)=M1,xl+2=f(xl+1-k1,…,xl+1-ks;xl+1-m1,…,xl+1-mt)≥f(M0,…,M0;m0,…,m0)=m1.
In similar fashion, we may show xn∈[m1,M1] for any n≥l+2. By induction, we obtain
(19)xn∈[mk,Mk]forn≥k(l+1)+1.
It follows from Claim  3 that limn→∞xn=x-. Claim  5 is proven.
From Claims  4 and 5, Theorem1 follows.


## 3. Applications

In this section, we will give two applications of Theorem1.Example 2.
Consider equation(20)xn+1=p+∑i=1taixn-mi∑k=1sbkxn-nk+∑i=1taixn-mi∑k=1sbkxn-nk,n=0,1,…,
where 0≤n1<n2<⋯<ns and 0≤m1<m2<⋯<mt with {n1,n2,…,ns}⋂‍{m1,m2,…,mt}=∅, p>0,ai>0 for any i∈{1,2,…,t} and bk>0 for any k∈{1,2,…,s}, and the initial conditions x-l,…,x0∈(0,∞) with l=max{mt,ns}. Write A=∑i=1t‍ai and B=∑k=1s‍bk. If pB>A, then the unique positive equilibrium x- of (20) is globally asymptotically stable.Proof.
LetE=(0,+∞). It is easy to verify that (H1), (H2), and (H4) hold for (20). Note that a=inf(u1,u2,…,us,v1,v2,…,vt)∈Es+tf(u1,u2,…,us;v1,v2,…,vt)=p. Then
(21)x=f(a,a,…,a;x,x,…,x)=p+AxBp+AxBp
has only solution
(22)x=[pAB+pAB+4p2B(Bp-A)]/2(Bp-a)
in the interval (p,+∞), which implies that (H3) holds for (20). In addition, let
(23)x=p+xAyB+xAyB,y=p+yAxB+yAxB,
then
(24)xy=p+xA/yB+xA/yBp+yA/xB+yA/xB.
Therefore x/y=1, which implies that (23) has unique solution
(25)x=y=x-=p+AB+A/B.
Thus (H5) holds for (20). It follows from Theorem 1 that the equilibrium x-=p+A/B+A/B of (20) is globally asymptotically stable.Example 3.
Consider equation(26)xn+1=q+∑i=1taixn-mip+∑k=1sbkxn-nk,n=0,1,…,
where 0≤n1<n2<⋯<ns and 0≤m1<m2<⋯<mt with {n1,n2,…,ns}⋂‍{m1,m2,…,mt}=∅, p>0,q>0, ai>0 for any 1≤i≤t and bj>0 for any 1≤j≤s, and the initial conditions x-l,…,x0∈(0,∞) with l=max{mt,ns}. Write A=∑i=1t‍ai and B=∑k=1s‍bk. If p>A, then the unique positive equilibrium x- of (26) is globally asymptotically stable.Proof.
LetE=[0,+∞). It is easy to verify that (H1)–(H4) hold for (26). In addition, the following equation
(27)x=q+xAp+yB,y=q+yAp+xB
has unique solution
(28)x=y=x-=A-p+(p-A)2+4Bq2B,
which implies that (H5) holds for (26). It follows from Theorem 1 that the equilibrium x-=(A-p+(p-A)2+4Bq)/2B of (26) is globally asymptotically stable.



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