# Part-Metric and Its Applications to Cyclic Discrete Dynamic Systems

Authors: Wanping Liu; Xiaofan Yang; Stevo Stević; Bratislav Iričanin
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We adapt the part metric and use it in studying positive solutions
of a certain family of discrete dynamic systems. Some examples are presented,
and we also compare some results in the literature.


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## Body

## 1. Introduction

There has been an increasing interest in studying discrete dynamic systems recently (see, e.g., [1–37]). For some recent papers on the systems of difference equations which are not derived from differential equations, see, for example, [4, 12, 14, 15], and the related references therein. In particular, in [4], were considered some cyclic systems of difference equations for the first time. Motivated by [4], in [12], the global attractivity of four k-dimensional systems of higher-order difference equations with two or three delays was investigated. The results in [12] can be easily extended to the corresponding systems with arbitrary number of delays by using the main results in [28].In [9], the authors used Thompson's part-metric [32] to investigate the behaviour of positive solutions to a difference equation from the William Lowell Putman Mathematical Competition [33] by applying a result on discrete dynamic systems in finite dimensional complete metric spaces. Further investigations devoted to applying various part-metric-related inequalities and some asymptotic methods in order to study (scalar) difference equations related to the equation in [33] can be found, for example, in [1, 3, 5, 18–22, 34–37] (see also the related references therein).In this paper, we adapt the part-metric and apply it in studying of the behaviour of positive solutions to the following family of discrete dynamic systems(1.1)Yn=Φ(Yn-k1,Yn-k2,…,Yn-kq),n∈N0,
where q∈ℕ∖{1}, 1≤k1<k2<⋯<kq, k1,…,kq∈ℕ, Yn=(yn(1),yn(2),…,yn(q))T, Y-kq,Y-kq+1,…,Y-1 are positive initial vectors and Φ:ℝ+q×q→ℝ+q is a continuous mapping which will be specified later.In Section2, we present some preliminary results which will be applied in the proofs of main results, given in Section 3. Some applications of the main result are given in Sections 4 and 5. In Section 6, we show that some recent results follow from a result in [9].


## 2. Auxiliary Results

Letℝ be the whole set of reals and let ℝ+=(0,+∞). Denote by ℝ+n the set of all positive n-dimensional vectors and by ℝ+m×n the set of all m×n matrices with positive entries, that is, ℝ+m×n={(aij)m×n∣aij∈ℝ+}, m,n∈ℕ.The following theorem was proved in [9, Theorem 1].Theorem 2.1.
Let(M,d) be a complete metric space, where d denotes a metric and M is an open subset of ℝn, and let 𝒯:M→M be a continuous mapping with the unique equilibrium x*∈M. Suppose that for the discrete dynamic system
(2.1)xn+1=Txn,n∈N0,
there is a k∈ℕ such that for the kth iterate of 𝒯, the next inequality holds
(2.2)d(Tkx,x*)<d(x,x*)
for all x≠x*. Then, x* is globally asymptotically stable with respect to metric d.The part-metric (see [9, 32]) is a metric defined on ℝ+n by (2.3)p(X,Y)=log⁡2max⁡1≤i≤n{xiyi,yixi},
for arbitrary vectors X=(x1,x2,…,xn)T∈ℝ+n and Y=(y1,y2,…,yn)T∈ℝ+n.Recall that the part-metricp has the following properties [9, 10]:(1)
p is a continuous metric on ℝ+n,(2)
(ℝ+n,p) is a complete metric space,(3)
the distances induced by the part-metric and by the Euclidean norm∥·∥ are equivalent on ℝ+n.Based on these properties and Theorem2.1, Kruse and Nesemann in [9] obtained the following result.Lemma 2.2 (see [9, Corollary 2]).
Let𝒯:ℝ+n→ℝ+n be a continuous mapping with a unique equilibrium x*∈ℝ+n. Suppose that for the discrete dynamic system (2.1) there is some k∈ℕ such that for the part-metric p inequality p(𝒯kx,x*)<p(x,x*) holds for all x≠x*. Then, x* is globally asymptotically stable.Our idea is to adapt the part-metric to matrices. For any two matrices with positive entriesA=(aij)m×n∈ℝ+m×n and B=(bij)m×n∈ℝ+m×n, we define the part-metric in the following natural way:(2.4)P(A,B)=log⁡2max⁡1≤i≤m,1≤j≤n{aijbij,bijaij}.
Note that an m×n matrix (aij)m×n is equivalent to a vector with mn elements, such as (2.5)(aij)m×n⟷(a1,1,a2,1,…,am,1,a1,2,a2,2,…,am,2,…,a1,n,a2,n,…,am,n)T.
Thus, for the above matrices A and B, we have that(2.6)P(A,B)=p((A1T,A2T,…,AnT)T,(B1T,B2T,…,BnT)T)=max⁡1≤j≤n{p(Aj,Bj)},
where Aj=(a1j,a2j,…,amj)T, Bj=(b1j,b2j,…,bmj)T, j=1,2,…,n.From this and the above-mentioned properties for the part-metric we have the following:(1)
the part-metric𝒫 is a continuous metric on ℝ+m×n,(2)
(ℝ+m×n,𝒫) is a complete metric space,(3)
the distances induced by the part-metric𝒫 and the Euclidean norm are equivalent on ℝ+m×n.From this and by Lemma2.2, we have that the next result holds.Theorem 2.3.
Let𝒯:ℝ+m×n→ℝ+m×n be a continuous mapping with the unique equilibrium C∈ℝ+m×n. Suppose that for the discrete dynamic system
(2.7)Xn+1=TXn,n∈N0,
there is a k∈ℕ such that for metric 𝒫, the inequality 𝒫(𝒯kX,C)<𝒫(X,C) holds for each X≠C. Then, C is globally asymptotically stable.Remark 2.4.
Note that if we do not assume in Theorem2.3 that C is the unique equilibrium of (2.7), then if C̃ is another equilibrium it must be
(2.8)P(C̃,C)=P(TkC̃,C)<P(C̃,C),
which is impossible. Hence, there is only one equilibrium of (2.7).


## 3. Main Result

LetY=(Y1,Y2,…,Yq) be a square q×q matrix, where Yi=(yi(1),yi(2),…,yi(q))T, i=1,2,…,q, and Φ is defined by (3.1)Φ(Y)=Φ(Y1,Y2,…,Yq)=(ϕ(y1(1),y2(2),…,yq(q))ϕ(y1(2),y2(3),…,yq(1))⋮ϕ(y1(q),y2(1),…,yq(q-1))),
where ϕ:ℝ+q→ℝ+ is a continuous mapping. Clearly, Φ is a continuous mapping and our system becomes(3.2)Yn=(yn(1)yn(2)⋮yn(q))=(ϕ(yn-k1(1),yn-k2(2),…,yn-kq(q))ϕ(yn-k1(2),yn-k2(3),…,yn-kq(1))⋮ϕ(yn-k1(q),yn-k2(1),…,yn-kq(q-1))),n∈N0,
where q∈ℕ∖{1}, 1≤k1<k2<⋯<kq, k1,k2,…,kq∈ℕ.As an application of Theorem2.3, we will establish a theorem regarding the global asymptotic stability of cyclic system of difference equations in (3.2), as follows.Theorem 3.1.
Consider system (3.2), where ϕ:ℝ+q→ℝ+,q≥2 is a continuous mapping. Let C¯=(c,c,…,c)T, c>0, be an equilibrium of (3.2). If for (x1,x2,…,xq)T≠C¯,
(3.3)min⁡1≤i≤q{xi,c2xi}<ϕ(x1,x2,…,xq)<max⁡1≤i≤q{xi,c2xi},
then C¯ is globally asymptotically stable.Proof.
Define a matrix mapping𝒯:ℝ+q×kq→ℝ+q×kq such that
(3.4)T(X1,X2,…,Xkq)=(Φ(Xk1,Xk2,…,Xkq),X1,X2,…,Xkq-1),
where Xi=(x1i,x2i,…,xqi)T, i=1,2,…,kq. Then, (3.2) can be converted into the first-order recursive q×kq matrix equation
(3.5)Mn=T(Mn-1),n∈N,
with M0 initial matrix, with positive entries.
ClearlyC=(C¯,C¯,…,C¯) is an equilibrium of (3.5).
Let(Yn)n=-kq+∞ be an arbitrary positive solution to (3.2), and denote(3.6)Mn=(Yn-1,Yn-2,…,Yn-kq),n∈N0,
then we get a matrix sequence (Mn)n=0∞. Apparently, the matrix sequence (Mn)n=0∞ is a solution to (3.5).
WhenM0=C, it is clear that Mn=C holds for n∈ℕ0. Hence, in what follows, we assume that M0≠C.
Let the relation “≤j” be either “=” or “<” for each j∈ℕ0. Since(3.7)T(Mn)=Mn+1=(Yn,Yn-1,…,Yn-kq+1),n∈N0,
then
(3.8)Yn=Φ(Yn-k1,Yn-k2,…,Yn-kq).
By (3.2) and condition (3.3), we get that for each j∈{1,2,…,q}(3.9)yn(j)c=ϕ(yn-k1(j),…,yn-kq(θ(j+q-1)))c≤jmax⁡1≤i≤q{yn-ki(θ(j+i-1))c,cyn-ki(θ(j+i-1))},cyn(j)=cϕ(yn-k1(j),…,yn-kq(θ(j+q-1)))≤jmax⁡1≤i≤q{yn-ki(θ(j+i-1))c,cyn-ki(θ(j+i-1))},
where θ(n)≡n(modq) with θ(q)=q.
LetBn=(Yn-k1,Yn-k2,…,Yn-kq).
Case 1.
IfBn≠(C¯,C¯,…,C¯), then there exists at least one j∈{1,2,…,q} such that the relation “≤j” in (3.9) is “<”. Thus,
(3.10)p(Yn,C¯)<max⁡{p(Yn-k1,C¯),p(Yn-k2,C¯),…,p(Yn-kq,C¯)},n∈N0.Case 2.
IfBn=(C¯,C¯,…,C¯), then Yn=C¯ and “≤j” is always “=” for each j∈{1,2,…,q}, which implies that
(3.11)p(Yn,C¯)=max⁡{p(Yn-k1,C¯),p(Yn-k2,C¯),…,p(Yn-kq,C¯)}=0,n∈N0.From relations (3.10) and (3.11), we obtain that
(3.12)p(Yn,C¯)≤nmax⁡{p(Yn-k1,C¯),p(Yn-k2,C¯),…,p(Yn-kq,C¯)},n∈N0.
From the following set of inequalities
(3.13)p(Y0,C¯)≤0max⁡{p(Y-k1,C¯),p(Y-k2,C¯),…,p(Y-kq,C¯)},p(Y1,C¯)≤1max⁡{p(Y1-k1,C¯),p(Y1-k2,C¯),…,p(Y1-kq,C¯)},⋮p(Ykq-1,C¯)≤kq-1max⁡{p(Ykq-1-k1,C¯),p(Ykq-1-k2,C¯),…,p(Ykq-1-kq,C¯)},
and since M0≠C, it follows that there exists at least one index j∈{0,1,…,kq-1} such that the relation “≤j” is “<”, which implies
(3.14)max⁡0≤i≤kq-1{p(Yi,C¯)}<max⁡-kq≤i≤-1{p(Yi,C¯)}.
From the definition of the part-metric, we have that
(3.15)P(Mn,C)=log⁡2max⁡1≤i≤kq,1≤j≤q{cyn-i(j),yn-i(j)c}=P((Yn-1,…,Yn-kq),C).
Then, we derive
(3.16)P(Tkq(M0),C)=P(Mkq,C)=P((Ykq-1,Ykq-2,…,Ykq-kq),C)=max⁡0≤i≤kq-1{p(Yi,C¯)}<max⁡-kq≤i≤-1{p(Yi,C¯)}=P(M0,C).
Because M0 is arbitrary and M0≠C, then by Theorem 2.3 (see also Remark 2.4) we have that C is a globally asymptotically stable equilibrium of (3.5), which implies that the equilibrium C¯ of system (3.2) is globally asymptotically stable, as desired.


## 4. On Some Symmetric Discrete Dynamic Systems

For the sake of convenience, first we define two continuous mappingsf,g:ℝ+q→ℝ+, q≥2, as follows:(4.1)f(x1,x2,…,xq)=∏j=1q(xjr+1)-∏j=1q(xjr-1)∏j=1q(xjr+1)+∏j=1q(xjr-1),g(x1,x2,…,xq)=∏j=1q(xjr+1)+∏j=1q(xjr-1)∏j=1q(xjr+1)-∏j=1q(xjr-1),
where r is a real parameter belonging to the interval (0,1].Many researchers have studied the symmetric difference equation(4.2)xn=ϕ(xn-k1,xn-k2,…,xn-kq),n∈N0,
where q∈ℕ∖{1}, 1≤k1<k2<⋯<kq, k1,k2,…,kq∈ℕ, and ϕ∈{f,g}.In the following, we mainly investigate the behaviour of positive solutions to the following class of cyclic difference equation systems(4.3)(yn(1)yn(2)⋮yn(q))=(ϕ(yn-k1(1),yn-k2(2),…,yn-kq(q))ϕ(yn-k1(2),yn-k2(3),…,yn-kq(1))⋮ϕ(yn-k1(q),yn-k2(1),…,yn-kq(q-1))),n∈N0,
where q∈ℕ∖{1},1≤k1<k2<⋯<kq, k1,k2,…,kq∈ℕ, and ϕ∈{f,g}.In order to establish the main result concerning (4.3), we need some preliminary lemmas.Lemma 4.1.
System (4.3) has unique positive equilibrium (1,1,…,1︸q)T.Proof.
Let(c1,c2,…,cq)T be an arbitrary positive equilibrium of system (4.3). Since the mappings f and g are both symmetric, then we derive that
(4.4)ci=ϕ(c1,c2,…,cq),i=1,2,…,q,
from which it follows that ci=c>0, i=1,2,…,q, and then
(4.5)c=ϕ(c,c,…,c).
By Lemma 2.1 in [13], we obtain c=1, as desired.Lemma 4.2.
Leta1 and a2 be positive real numbers with (a1,a2)≠(1,1), and ϕ∈{f,g}. Then,
(4.6)min⁡{a1,a2,1a1,1a2}<ϕ(a1,a2)<max⁡{a1,a2,1a1,1a2}.Proof.
From the next identities(4.7)x-x+y1+xy=y(x2-1)1+xy,y-x+y1+xy=x(y2-1)1+xy,1x-x+y1+xy=1-x2x(1+xy),1y-x+y1+xy=1-y2y(1+xy),
it is easy to see that when (a1,a2)≠(1,1) the following inequalities hold:
(4.8)min⁡{a1,a2,1a1,1a2}<a1+a21+a1a2<max⁡{a1,a2,1a1,1a2}.
Because r∈(0,1], then for the case ϕ=f, we easily obtain that
(4.9)min⁡{a1,a2,1a1,1a2}≤min⁡{a1r,a2r,1a1r,1a2r}<f(a1,a2)=a1r+a2r1+a1ra2r<max⁡{a1r,a2r,1a1r,1a2r}≤max⁡{a1,a2,1a1,1a2}.
The case ϕ=g follows immediately from the case ϕ=f due to the fact that fg≡1.Lemma 4.3.
Letq≥2 be an integer and ϕ∈{f,g}. Let a1,a2,…,aq be positive real numbers with (a1,a2,…,aq)≠(1,1,…,1). Then,
(4.10)min⁡1≤i≤q{ai,1ai}<ϕ(a1,a2,…,aq)<max⁡1≤i≤q{ai,1ai}.Proof.
For the caseq=2, the assertion follows from Lemma 4.2. Next, we argue by the induction and assume that the assertion is true for q=k(k≥2). Then, it suffices to prove that the assertion holds when q=k+1. Now, let a1,a2,…,ak+1 be positive real numbers with (a1,a2,…,ak+1)≠(1,1,…,1). Consider the following function h in a variable x(4.11)h(x;a1,a2,…,ak)=(xr+1)∏j=1k(ajr+1)-(xr-1)∏j=1k(ajr-1)(xr+1)∏j=1k(ajr+1)+(xr-1)∏j=1k(ajr-1),
where a1,a2,…,ak are arbitrary (but fixed) positive numbers. Clearly,
(4.12)h(ak+1;a1,a2,…,ak)=f(a1,a2,…,ak+1).
The first derivative of the function h regarding the variable x is equal to
(4.13)h′(x;a1,a2,…,ak)=-4rxr-1∏j=1k(aj2r-1)((xr+1)∏j=1k(ajr+1)+(xr-1)∏j=1k(ajr-1))2.
In the following, we distinguish three possibilities.Case 1 (∏j=1k(aj2r-1)<0).
Then,h′(x;a1,a2,…,ak)>0 holds for all x>0, which implies that the function h(x;a1,a2,…,ak) is strictly increasing in variable x. From this, we obtain
(4.14)h(ak+1;a1,a2,…,ak)<lim⁡x→+∞h(x;a1,a2,…,ak)=∏j=1k(ajr+1)-∏j=1k(ajr-1)∏j=1k(ajr+1)+∏j=1k(ajr-1)=f(a1,a2,…,ak)≤max⁡1≤i≤k{ai,1ai}≤max⁡1≤i≤k+1{ai,1ai},h(ak+1;a1,a2,…,ak)>lim⁡x→0+h(x;a1,a2,…,ak)=∏j=1k(ajr+1)+∏j=1k(ajr-1)∏j=1k(ajr+1)-∏j=1k(ajr-1)=g(a1,a2,…,ak)≥min⁡1≤i≤k{ai,1ai}≥min⁡1≤i≤k+1{ai,1ai}.Case 2 (∏j=1k(aj2r-1)>0).
Then,h′(x;a1,a2,…,ak)<0 holds for all x>0, implying that the function h(x;a1,a2,…,ak) is strictly decreasing in variable x. From this, we obtain that
(4.15)h(ak+1;a1,a2,…,ak)<lim⁡x→0+h(x;a1,a2,…,ak)=∏j=1k(ajr+1)+∏j=1k(ajr-1)∏j=1k(ajr+1)-∏j=1k(ajr-1)=g(a1,a2,…,ak)≤max⁡1≤i≤k{ai,1ai}≤max⁡1≤i≤k+1{ai,1ai},h(ak+1;a1,a2,…,ak)>lim⁡x→+∞h(x;a1,a2,…,ak)=∏j=1k(ajr+1)-∏j=1k(ajr-1)∏j=1k(ajr+1)+∏j=1k(ajr-1)=f(a1,a2,…,ak)≥min⁡1≤i≤k{ai,1ai}≥min⁡1≤i≤k+1{ai,1ai}.Case 3 (∏j=1k(aj2r-1)=0).
This impliesh(x;a1,a2,…,ak)=1 for all x>0. From this relation and the inspection that the condition (a1,a2,…,ak+1)≠(1,1,…,1) implies max1≤i≤k+1{ai,1/ai}>1 and min1≤i≤k+1{ai,1/ai}<1, we obtain that
(4.16)min⁡1≤i≤k+1{ai,1ai}<h(ak+1;a1,a2,…,ak)<max⁡1≤i≤k+1{ai,1ai}.
Hence, by induction, the assertion immediately holds for ϕ=f, and then the case ϕ=g follows directly from the case ϕ=f because f·g≡1By Lemmas4.1 and 4.3 and Theorem 3.1, we obtain the following theorem.Theorem 4.4.
The unique equilibrium of system (4.3) is globally asymptotically stable.Remark 4.5.
Note that the following two systems (particular casesq=2 and q=3 of the system (4.3), resp.)
(4.17)un=ϕ(un-k,vn-l),vn=ϕ(vn-k,un-l),n∈N0,un=ϕ(un-k,vn-l,wn-m),vn=ϕ(vn-k,wn-l,un-m),wn=ϕ(wn-k,un-l,vn-m),n∈N0,
where 1≤k<l<m, ϕ∈{f,g}, were studied in [12].


## 5. On a System of Difference Equations

Letμ:ℝ+q→ℝ+, q≥4 be a continuous mapping defined by(5.1)μ(t1,t2,…,tq)=∑i=1q-2ti+tq-1tqt1t2+∑i=3qti.
Then, the following difference equation(5.2)xn+1=μ(xn,xn-1,xn-2,xn-3),n∈N0,
is an extension of the difference equation in [33], which was studied in [9].First, we consider the next four-dimensional system of difference equations:(5.3)(wnxnynzn)=(μ(wn-k1,xn-k2,yn-k3,zn-k4)μ(xn-k1,yn-k2,zn-k3,wn-k4)μ(yn-k1,zn-k2,wn-k3,xn-k4)μ(zn-k1,wn-k2,xn-k3,yn-k4)),n∈N0,
where 1≤k1<k2<k3<k4, ki∈ℕ for i∈{1,2,3,4}.Lemma 5.1.
System (5.3) has unique positive equilibrium (1,1,1,1)T.Proof.
Let(a,b,c,d)T be an arbitrary positive equilibrium of the system (5.3). Then, we get
(5.4)a=a+b+cdab+c+d,b=b+c+dabc+d+a,c=c+d+abcd+a+b,d=d+a+bcda+b+c,
which imply
(5.5)ac=1,bd=1.
Applying (5.5), the system in (5.4) is reduced to
(5.6)a(a2-1)b2+a(1-a)b+a2-1=0,b3+a(a-1)b2+(a-1)b-a2=0.
If a=1, then it follows from the second identity that b=1. Now, assume a∈ℝ+∖{1}. By solving the first equation in (5.6) with respect to variable b, we get that the discriminant
(5.7)Δ=a2(1-a)2-4a(a2-1)2=a(a-1)2(-4a2-7a-4)<0,
which implies the first equation in (5.6) has no real roots. This contradicts b>0. Hence, a=b=1, which along with (5.5) implies c=d=1, finishing the proof.The following lemma follows directly from Lemma 3.3 in [34] or the proof of Lemma 4 in [35].Lemma 5.2.
Leta1,a2,a3, and a4 be positive real numbers with (a1,a2,a3,a4)≠(1,1,1,1). Then,
(5.8)min⁡1≤i≤4{ai,1ai}<μ(a1,a2,a3,a4)<max⁡1≤i≤4{ai,1ai}.By Lemmas5.1 and 5.2, and Theorem 3.1, we easily derive the next theorem.Theorem 5.3.
Unique positive equilibrium(1,1,1,1)T of system (5.3) is globally asymptotically stable.In the following, we consider the nextq-dimensional (q≥5) generalization of system (5.3)(5.9)(yn(1)yn(2)⋮yn(q))=(μ(yn-k1(1),yn-k2(2),…,yn-kq(q))μ(yn-k1(2),yn-k2(3),…,yn-kq(1))⋮μ(yn-k1(q),yn-k2(1),…,yn-kq(q-1))),n∈N0,
where 5≤q∈ℕ, 1≤k1<k2<⋯<kq, k1,k2,…,kq∈ℕ.It is easy to see that(1,1,…,1)T is a positive equilibrium of system (5.9), but it is not so easy to confirm its uniqueness as in the proof of Lemma 5.1. However, we have the following lemma which follows directly from Lemmas 3.4 and 3.5 in [34].Lemma 5.4.
Letr be an integer with r≥4. Let a1,a2,…,ar be positive numbers with (a1,a2,…,ar)≠(1,1,…,1). Then,
(5.10)min⁡1≤i≤r{ai,1ai}<∑i=1r-2ai+ar-1ara1a2+∑i=3rai<max⁡1≤i≤r{ai,1ai}.Hence, we can apply Theorem3.1 to establish the following theorem.Theorem 5.5.
The positive equilibrium(1,1,…,1)T of system (5.9) is globally asymptotically stable.


## 6. An Application of a Kruse-Nesseman Result

Numerous papers studied particular cases of (4.2) by using semi-cycle analysis of their solutions. It was shown by Berg and Stević in [1] that this analysis is unnecessarily complicated and useful only for lower-order difference equations. They also described some methods for determining rules of semi-cycles which can be used in many classes of difference equations. On the other hand, it has been noticed in several papers (see, e.g., [18]) that the stability results in many of these papers follow from the following result by Kruse and Nesemann in [9].Theorem 6.1.
Consider the difference equation(6.1)yn=F(yn-1,…,yn-m),n∈N0,
where F:ℝ+m→ℝ+ is a continuous function with a unique equilibrium x*∈ℝ+. Suppose that there is a k0∈ℕ such that for each solution (yn) of (6.1),
(6.2)(yn-yn-k0)(yn-(x*)2yn-k0)≤0
with equality if and only if yn=x*. Then, x* is globally asymptotically stable.Motivated by [18], in recent paper [2], Berg and Stević also applied Theorem 6.1 by proving the next result, which covers numerous particular cases appearing in the literature. We formulate the proposition here as a useful information to the reader. Before we formulate it we need some notation. Let Sj={1,2,…,j}, j=1,…,k, let (6.3)Trk:=∑{t1,t2,…,tr}⊆Skt1<t2<⋯<tr⁡xt1xt2⋯xtr,
for r=0,1,…,k, where we define T0k=1 and T-1k=Tkk-1=0, and let (6.4)∑′Trk=∑r=1krodd⁡Trk,∑′′Trk=∑r=0kreven⁡Trk,
or reversed, ∑′Trk is the sum over the even, and ∑′′Trk is the sum over the odd r.Theorem 6.2.
Supposeχ is a nonnegative continuous function on ℝ+k, k∈ℕ, and 1≤i1<i2<⋯<ik. If a sequence (yi) satisfies the difference equation
(6.5)yn=f(yn-i1,yn-i2,…,yn-ik)g(yn-i1,yn-i2,…,yn-ik),n∈N0,
where
(6.6)f(x1,x2,…,xk)=χ+∑′Trk,g(x1,x2,…,xk)=χ+∑′′Trk,
with y-ik,…,y-1∈ℝ+, then it converges to the unique positive equilibrium 1.Another proof of the previous result, in the caseχ=0, can be also find in recent paper [28] by Stević.Recently Sun and Xi in [31] gave an interesting proof of the following result. At first sight their result looked new and not so closely related to Theorem 6.1. However, we prove here that it is also a consequence of Theorem 6.1.Theorem 6.3.
Letf∈C(R+k,R+) and g∈C(R+l,R+) with k,l∈ℕ, 0≤r1<⋯<rk and 0≤m1<⋯<ml and satisfy the following two conditions:(H1)
[f(u1,u2,…,uk)]*=f(u1*,u2*,…,uk*) and [g(u1,u2,…,ul)]*=g(u1*,u2*,…,ul*).(H2)
f(u1*,u2*,…,uk*)≤u1*.Then, x¯=1 is the unique positive equilibrium of the difference equation
(6.7)xn=f(xn-r1-1,…,xn-rk-1)g(xn-m1-1,…,xn-ml-1)+1f(xn-r1-1,…,xn-rk-1)+g(xn-m1-1,…,xn-ml-1),n∈N,
which is globally asymptotically stable (here u*=max{u,1/u}).Proof.
Let(6.8)fn=f(xn-r1-1,…,xn-rk-1),gn=g(xn-m1-1,…,xn-ml-1).
We should determine the sign of the product of the next expressions
(6.9)Pn:=fngn+1fn+gn-xn-r1-1=1fn+gn(fngn(1-xn-r1-1fn)+1-xn-r1-1fn),(6.10)Qn:=fngn+1fn+gn-1xn-r1-1=1xn-r1-1(fn+gn)(gn(xn-r1-1fn-1)+fn(xn-r1-1fn-1)).
From (6.9) and (6.10), we see if we show that xn-r1-1fn-1 and (xn-r1-1/fn)-1 have the same sign for n∈ℕ, then PnQn will be nonpositive.
There are four cases to be considered.
Case 1 (xn-r1-1≥1, fn≥1).
Clearly, in this case,xn-r1-1fn-1≥0. By (H1) and (H2), we have that
(6.11)1≤fn=(fn)*=f(xn-r1-1*,…,xn-rk-1*)≤xn-r1-1*=xn-r1-1.
Hence, (xn-r1-1/fn)-1≥0 and consequently
(6.12)(xn-r1-1fn-1)(xn-r1-1fn-1)≥0.Case 2 (xn-r1-1≥1, fn≤1).
Since1/fn≥1 we obtain (xn-r1-1/fn)-1≥0. On the other hand, by (H1) and (H2) we have
(6.13)1fn=(fn)*=f(xn-r1-1*,…,xn-rk-1*)≤xn-r1-1*=xn-r1-1,
so that xn-r1-1fn-1≥0. Hence, (6.12), follows in this case.Case 3 (Casexn-r1-1≤1, fn≥1).
Then we have that1/fn≤1 and consequently (xn-r1-1/fn)-1≤0. On the other hand, we have
(6.14)fn=(fn)*=f(xn-r1-1*,…,xn-rk-1*)≤xn-r1-1*=1xn-r1-1,
so that xn-r1-1fn-1≤0. Hence, (6.12) follows in this case too.Case 4 (Casexn-r1-1≤1, fn≤1).
Thenxn-r1-1fn-1≤0. On the other hand, we have
(6.15)1fn=(fn)*=f(xn-r1-1*,…,xn-rk-1*)≤xn-r1-1*=1xn-r1-1,
so that (xn-r1-1/fn)-1≤0. Hence, (6.12) also holds in this case. Thus PnQn≤0, for every n∈ℕ.
Assume thatPnQn=0, then, Pn=0 or Qn=0. Using (6.9) or (6.10) along with (6.12) in any of these two cases, we have that(6.16)fn=1xn-r1-1=xn-r1-1,n∈N.
Hence, xn-r1-1=1, n∈ℕ.
Finally, lety* be a solution of the equation (6.17)0=f(y⃗k*)g(y⃗l*)+1f(y⃗k*)+g(y⃗l*)-y*=1f(y⃗k*)+g(y⃗l*)(f(y⃗k*)g(y⃗l*)(1-y*f(y⃗k*))+1-y*f(y⃗k*)),
where y⃗j*=(y*,…,y*) denotes the vector consisting of j copies of y*. Then according to the considerations in Cases 1–4 it follows that f(y⃗k*)=y*=1/y*, so that y*=1. Hence y*=1 is a unique positive equilibrium of (6.7).
From all above mentioned and by Theorem6.1, we get the result.



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*Source: 534974-2011-06-09.xml*