# On the Dynamics of a Higher-Order Difference Equation

Authors: H. El-Metwally
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

This paper deals with the investigation of the following more general rational difference equation:yn+1=αyn/(β+γ∑i=0kyn-(2i+1)p∏i=0kyn-(2i+1)), n=0,1,2,…, where α,β,γ,p∈(0,∞) with the initial conditions x0,x-1,…,x-2k,x-2k-1∈(0,∞). We investigate the existence of the equilibrium points of the considered equation and then study their local and global stability. Also, some results related to the oscillation and the permanence of the considered equation have been presented.


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## Body

## 1. Introduction

In this paper we investigate the global stability character and the oscillatory of the solutions of the following difference equation:(1.1)yn+1=αynβ+γ∑i=0kyn-(2i+1)p∏i=0kyn-(2i+1),n=0,1,2,…,
where α, β,γ,p∈(0,∞) with the initial conditions x0,x-1,…,x-2k,x-2k-1∈(0,∞). Also we study the permanence of (1.1). The importance of permanence for biological systems was thoroughly reviewed by Huston and Schmidtt [1].In general, there are a lot of interest in studying the global attractivity, boundedness character, and periodicity of the solutions of nonlinear difference equations. In particular there are many papers that deal with the rational difference equations and that is because many researchers believe that the results about this type of difference equations are of paramount importance in their own right, and furthermore they believe that these results offer prototype towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one.Kulenović and Ladas [2] presented some known results and derived several new ones on the global behavior of the difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1) and of its special cases. Elabbasy et al. [3–5] established the solutions form and then investigated the global stability and periodicity character of the obtained solutions of the following difference equations:
(1.2)xn+1=axn-bxncxn-dxn-1,xn+1=αxn-kβ+γ∏i=0kxn-i,xn+1=dxn-lxn-kcxn-s-b+a.
El-Metwally [6] gave some results about the global behavior of the solutions of the following more general rational difference equations
(1.3)xn+1=axn-k0l0xn-k1l1…xn-kili+bxn-r0s0xn-r1s1…xn-rjsjcxn-k0l0xn-k1l1…xn-kili+dxn-r0s0xn-r1s1…xn-rjsj,yn+1=α0yn+α1yn-1+⋯+αtyn-tβ0yn+β1yn-1+⋯+βtyn-t.
Çinar [7–9] obtained the solutions form of the difference equations xn+1=xn-1/(1+xnxn-1), xn+1=xn-1/(-1+xnxn-1)andxn+1=axn-1/(1+bxnxn-1). Also, Cinar et al. [10] studied the existence and the convergence for the solutions of the difference equation xn+1=xn-3/(-1+xnxn-1xn-2xn-3). Simsek et al. [11] obtained the solution of the difference equation xn+1=xn-3/(1+xn-1). In [12] Yalcinkaya got the solution form of the difference equation xn+1=xn-(2k+1)/(1+xn-kxn-(2k+1)). In [13] Stević studied the difference equation xn+1=xn-1/(1+xn). Other related results on rational difference equations can be found in [14–19].LetI be some interval of real numbers and let
(1.4)f:Ik+1⟶I
be a continuously differentiable function. Then for every set of initial conditions x-k,x-k+1,…,x0∈I, the difference equation
(1.5)xn+1=f(xn,xn-1,…,xn-k),n=0,1,…
has a unique solution {xn}n=-k∞.Definition 1.1 (permanence).
The difference equation (1.5) is said to be permanent if there exist numbers m and M with 0<m≤M<∞ such that for any initial conditions x-k,x-k+1,…,x-1,x0∈(0,∞) there exists a positive integer N which depends on the initial conditions such that m≤xn≤Mfor alln≥N.Definition 1.2 (periodicity).
A sequence{xn}n=-k∞is said to be periodic with period p if xn+p=xnfor all n≥-k.Definition 1.3 (semicycles).
A positive semicycle of a sequence{xn}n=-k∞consists of a “string” of terms {xl,xl+1,…,xm} all greater than or equal to the equilibrium point x¯, with l≥-k and m≤∞ such that either l=-k or l>-k and xl-1<x¯; and, either m=∞ or m<∞ and xm+1<x¯. A negative semicycle of a sequence {xn}n=-k∞consists of a "string" of terms {xl,xl+1,…,xm} all less than the equilibrium point x¯, with l≥-k and m≤∞ such that: either l=-k or l>-k and xl-1≥x¯; and, either m=∞ or m<∞ and xm+1≥x¯.Definition 1.4 (oscillation).
A sequence{xn}n=-k∞is called nonoscillatory about the point x¯ if there is exists N≥-k such that either xn>x¯ for all n≥N or xn<x¯ for all n≥N. Otherwise {xn}n=-k∞ is called oscillatory about x¯.Recall that the linearized equation of (1.5) about the equilibrium x¯is the linear difference equation
(1.6)yn+1=∑i=0k∂f(x¯,x¯,…,x¯)∂xn-iyn-i.


## 2. Dynamics of (1.1)

The change of variablesyn=(β/γ)1/(p+k+1)xn reduces (1.1) to the following difference equation
(2.1)xn+1=rxn1+∑i=0kxn-(2i+1)p∏i=0kxn-(2i+1),n=0,1,2,…,
where r=α/β.In this section we study the local stability character and the global stability of the equilibrium points of the solutions of (2.1). Also we give some results about the oscillation and the permanence of (2.1).Recall that the equilibrium point of (2.1) are given by
(2.2)x¯=rx¯1+(k+1)x¯p+k+1.
Then (2.1) has the equilibrium points x¯=0 and whenever r>1, (2.1) possesses the unique equilibrium point x¯=((r-1)/(k+1))1/(p+k+1).The following theorem deals with the local stability of the equilibrium pointx¯=0 of (2.1).Theorem 2.1.
The following statements are true:
(i) ifr<1, then the equilibrium point x¯=0 of (2.1) is locally asymptotically stable,
(ii) ifr>1, then the equilibrium point x¯=0 of (2.1) is a saddle point.Proof.
The linearized equation of (2.1) about x¯=0 is un+1-run=0. Then the associated eigenvalues are λ=0 and λ=r. Then the proof is complete.Theorem 2.2.
Assume thatr<1, then the equilibrium point x¯=0 of (2.1) is globally asymptotically stable.Proof.
Let{xn}n=-2k+1∞ be a solution of (2.1). It was shown by Theorem 2.1 that the equilibrium point x¯=0 of (2.1) is locally asymptotically stable. So, it is suffices to show that
(2.3)lim⁡xnn→∞=0.
Now it follows from (2.1) that
(2.4)xn+1=rxn1+xn-1p+1xn-3⋯xn-2k+1+xn-1xn-3p+1⋯xn-2k+1+⋯+xn-1xn-3⋯xn-2k+1p+1≤rxn<xn.
Then the sequence {xn}n=0∞ is decreasing and this completes the proof.Theorem 2.3.
Assume thatr>1. Then every solution of (2.1) is either oscillatory or tends to the equilibrium point x¯=((r-1)/(k+1))1/(p+k+1).Proof.
Let{xn}n=-2k+1∞ be a solution of (2.1). Without loss of generality assume that {xn}n=-2k+1∞ is a nonoscillatory solution of (2.1), then it suffices to show that limn→∞xn=x¯. Assume that xn≥x¯ for n≥n0 (the case where xn≤x¯ for n≥n0 is similar and will be omitted). It follows from (2.1) that
(2.5)xn+1=rxn1+xn-1p+1xn-3⋯xn-2k-1+xn-1xn-3p+1⋯xn-2k-1+⋯+xn-1xn-3⋯xn-2k-1p+1≤xn(r1+(k+1)x¯p+k+1)=xn.
Hence {xn} is monotonic for n≥n0+2k+1, therefore it has a limit. Let limn→∞xn=μ, and for the sake of contradiction, assume that μ>x¯. Then by taking the limit of both side of (2.1), we obtain μ=rμ/(1+(k+1)μp+k+1), which contradicts the hypothesis that x¯=((r-1)/(k+1))1/(p+k+1) is the only positive solution of (2.2).Theorem 2.4.
Assume that{xn}n=-2k+1∞ is a solution of (2.1) which is strictly oscillatory about the positive equilibrium point x¯=((r-1)/(k+1))1/(p+k+1) of (2.1). Then the extreme point in any semicycle occurs in one of the first 2(k+1) terms of the semicycle.Proof.
Assume that{xn}n=-2k-1∞ is a strictly oscillatory solution of (2.1). Let N≥2k+2 and let {xN,xN+1,…,xM} be a positive semicycle followed by the negative semicycle {xM,xM+1,…,xM}. Now it follows from (2.1) that
(2.6)xN+2k+2-xN=rxN+2k+11+xN+2kp+1xN+2k-2⋯xN+xN+2kxN+2k-2p+1⋯xN+⋯+xN+2k⋯xNp+1-xN≤xN+2k+1(r1+(k+1)x¯p+k+1)-xN=xN+2k+1-xN=rxN+2k1+xN+2k-1p+1xN+2k-3⋯xN-1+⋯+xN+2k-1⋯xN-1p+1-xN≤xN+2k(r1+(k+1)x¯p+k+1)-xN=xN+2k-xN≤xN+2k-1-xN≤⋯≤xN+1-xN≤xN-xN=0.
ThenxN≥xN+2(k+1) for all N≥2(k+1).
Similarly, we see from (2.1) that
(2.7)xM+2k+2-xM=rxM+2k+11+xM+2kp+1xM+2k-2⋯xM+xM+2kxM+2k-2p+1⋯xM+⋯+xM+2k⋯xMp+1-xM≥xM+2k+1(r1+(k+1)x¯p+k+1)-xM=xM+2k+1-xM=rxM+2k1+xM+2k-1p+1xM+2k-3⋯xM-1+⋯+xM+2k-1⋯xM-1p+1-xM≥xM+2k(r1+(k+1)x¯p+k+1)-xM=xM+2k-xM≤xM+2k-1-xM≥⋯≥xM+1-xM≥xM-xM=0.
ThereforexM+2(k+1)≥xM for all M≥2(k+1). The proof is so complete.Theorem 2.5.
Equation (2.1) is permanent.Proof.
Let{xn}n=-2k+1∞ be a solution of (2.1). There are two cases to consider:
(i){xn}n=-2k+1∞ is a nonoscillatory solution of (2.1). Then it follows from Theorem 2.3 that
(2.8)lim⁡xnn→∞=x¯,
that is there is a sufficiently large positive integer N such that |xn-x¯|<ɛ for all n≥N and for some ɛ>0. So, x¯-ɛ<xn<x¯+ɛ, this means that there are two positive real numbers, say C and D, such that
(2.9)C≤xn≤D,(ii){xn}n=-2k+1∞ is strictly oscillatory about x¯=((r-1)/(k+1))1/(p+k+1).
Now let{xs+1,xs+2,…,xt} be a positive semicycle followed by the negative semicycle {xt+1,xt+2,…,xu}. If xV and xW are the extreme values in these positive and negative semicycle, respectively, with the smallest possible indices V and W, then by Theorem 2.4 we see that V-s≤2(k+1) and W-u≤2(k+1). Now for any positive indices μ and L with μ<L, it follows from (2.1) for n=μ,μ+1,…,L-1 that
(2.10)xL=xL-1(r1+xL-2p+1xL-4⋯xL-2k-2+xL-2xL-4p+1⋯xL-2k-2+⋯+xL-2xL-4⋯xL-2k-2p+1)=r2xL-2(1+xL-3p+1⋯xL-2k-3+⋯+xL-3⋯xL-2k-3p+1)(1+xL-2p+1⋯xL-2k-2+⋯+xL-2⋯xL-2k-2p+1)⋮=xL-ζrζ∏η=1ζ(11+∑i=0kxL-(2i+1)-ηp∏i=0kxL-(2i+1)-η)=xμrL-μ∏η=μL-1(11+∑i=0kxη-(2i+1)p∏i=0kxη-(2i+1)).
Therefor for V=L and s=μ we obtain
(2.11)xV=xsrV-s∏η=sV-1(11+∑i=0kxη-(2i+1)p∏i=0kxη-(2i+1))≤x¯r2k+1=H.
Again wheneverW=L and μ=t, we see that
(2.12)xW=xtrW-t∏η=tW-1(11+∑i=0kxη-(2i+1)p∏i=0kxη-(2i+1))≥x¯rW-t∏η=tW-1(11+(k+1)Hp+k+1)=x¯rW-t(11+(k+1)Hp+k+1)W-t-1≥x¯(11+(k+1)Hp+k+1)2k+1=G.
That is, G≤xn≤H. It follows from (i) and (ii) that
(2.13)min⁡{C,G}≤xn≤max⁡{D,H}.
Then the proof is complete.



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*Source: 263053-2012-03-15.xml*