# On a Higher-Order Difference Equation

Authors: Bratislav D. Iričanin; Wanping Liu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We describe in an elegant and short way the behaviour of positive solutions of the higher-order difference equationxn=cxn−pxn−p−q/xn−q, n∈ℕ0, where p,q∈ℕ and c>0, extending some recent results in the literature.


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## Body

## 1. Introduction

Studying difference equations has attracted a considerable interest recently, see, for example, [1–39] and the references listed therein. The study of positive solutions of the following higher-order difference equations: (1.1)xn=max{A,Bxn-p1r1xn-p2r2⋯xn-pkrkxn-q1s1xn-q2s2⋯xn-qlsl},n∈ℕ0,
and(1.2)xn=A+Bxn-p1r1xn-p2r2⋯xn-pkrkxn-q1s1xn-q2s2⋯xn-qlsl,n∈ℕ0,
where A,B>0,pi,qi are natural numbers such that p1<p2<⋯<pk,q1<q2<⋯<ql, ri,si∈ℝ+, and k∈ℕ was proposed by Stević in several talks, see, for example, [21, 26]. For some results concerning equations related to (1.1) see, for example, [6, 7, 10, 29, 31, 32, 34, 38], while some results on equations related to (1.2) can be found, for example, in [3, 8, 9, 11–14, 18–20, 22, 25, 29, 32, 33, 35] (see also related references cited therein).CaseA=0 is of some less interest, since in this case positive solutions of (1.1) and (1.2), by using the change yn=lnxn, become solutions of a linear difference equation with constant coefficients. However, some particular results for the case recently appeared in the literature, see [16, 17, 39].Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order difference equation(1.3)xn=cxn-pxn-p-qxn-q,n∈ℕ0,
where p,q∈ℕ and c>0, in, let us say, an elegant and short way.Let us introduce the following.Definition 1.1.
A solution(xn)n=-(p+q)∞ of (1.3) is said to be eventually periodic with period τ if there is n0∈{-(p+q),…,-1,0,1,…} such that xn+τ=xn for all n≥n0. If n0=-(p+q), then we say that the sequence (xn)n=-(p+q)∞ is periodic with period τ.For some results on equations all solutions of which are eventually periodic see, for example, [2, 4, 8, 15, 28, 37] and the references therein.Definition 1.2.
One says that a solution(xn)n=n0∞ of a difference equation converges geometrically to x* if there exist L∈ℝ+ and θ∈[0,1) such that
(1.4)|xn-x*|≤Lθn,∀n≥n0.Now we return to (1.3).First, note that ifp=q, then (1.3) becomes (1.5)xn=cxn-2p,n∈ℕ0,
from which easily follow the following results:(a)
ifc=1, then all positive solutions of (1.5) are periodic with period 2p;(b)
ifc∈(0,1), then each positive solution of (1.5) converges to zero. Moreover, its subsequences (x2pm-i)m∈ℕ0,i=1,2,…,2p, converges decreasingly to zero as m→∞;(c)
ifc∈(1,∞), then each positive solution of (1.5) tends to infinity as n→∞. Moreover, its subsequences (x2pm-i)m∈ℕ0,i=1,2,…,2p, tend increasingly to infinity as m→∞.We may assume thatp and q are relatively prime integers, that is, gcd(p,q)=1 (the greatest common divisor of numbers p and q). Namely, if gcd(p,q)=r>1, then by using the changes zm(i)=xmr+i,i=0,1,…,r-1, (1.3) is reduced to r copies of the following equation: (1.6)zn=czn-p1zn-p1-q1zn-q1,n∈ℕ0,
where p1=p/r,q1=q/r,c>0, and gcd(p1,q1)=1.Further, note that from (1.3), we have that (1.7)xnxn-q=cxn-pxn-p-q,n∈ℕ0,
which implies that the sequence un=xnxn-q,n≥-p, satisfies the following simple difference equation: (1.8)un=cun-p,n∈ℕ0.


## 2. Main Results

Here we formulate and prove our main results.Theorem 2.1.
Assume thatc=1, gcd(p,q)=1, and p is odd, then all positive solutions of (1.3) are eventually periodic with period τ=2pq.Proof.
By using repeatedly relation (1.7) p-times, we obtain
(2.1)xn=unxn-q=unun-qxn-2q=⋯=unun-qun-2qun-3q⋯un-2q(p-1)un-q(2p-1)xn-2pq.
Now, note that from (1.8), it follows that in this case un is periodic with period p. On the other hand, since gcd(p,q)=1 for each i,j∈{0,1,…,p-1},i≠j, we have that
(2.2)(n-(2i+1)q)-(n-(2j+1)q)=(j-i)2q≢0(modp),(n-(2i+2)q)-(n-(2j+2)q)=(j-i)2q≢0(modp).
Hence, the indices (n-(2i+1)q),i∈{0,1,…,p-1}, and (n-(2i+2)q),i∈{0,1,…,p-1}, belong to p different subsequences. From this and the periodicity of un, it follows that
(2.3)unun-2q⋯un-2q(p-1)=un-qun-3q⋯un-q(2p-1),
from which the theorem follows.By taking the logarithm of (1.3) and using the change vn=lnxn, we get (2.4)vn+vn-q-vn-p-vn-p-q=lnc,n∈ℕ0.
The characteristic polynomial of the homogeneous part of (2.4) is (2.5)λp+q+λp-λq-1=(λq+1)(λp-1)=0,
from which it follows that all its roots are expressed by (2.6)exp((2k+1)πiq),k=0,1,…,q-1,exp(2lπip),l=0,1,…,p-1.
These roots are simple if and only if (2.7)2k+1q≠2lp,foreachk,l∈ℕ0.
Clearly, if p is odd, inequality (2.7) holds. If p is even, that is, p=2sr, for some s,r∈ℕ, then, since gcd(p,q)=1,q must be odd. Then, for k=(q-1)/2 and l=r, we will get that inequality (2.7) does not hold.From the above consideration and Theorem2.1, we get the next corollary.Corollary 2.2.
Assume thatc=1 and gcd(p,q)=1. Then all positive solutions of (1.3) are eventually periodic if and only if p is odd. Moreover, if p is odd, then the period is τ=2pq.Since the rootλ=1 of characteristic polynomial (2.5) is a simple one, a particular solution of nonhomogeneous (2.4) has the form (2.8)vnP=An,
from which, by a direct calculation, we easily get that A=lnc/2p.Hence, ifp is odd, the general solution of (1.3) is (2.9)xn=evn=cn/2pexp(∑k=0q-1(ck,1cos(2k+1)πnq+ck,2sin(2k+1)πnq)+∑l=1p-1(dk,1cos2lπnp+dk,2sin2lπnp)).
Note that from (2.9), it follows that (2.10)xn=cn/2px̂n,
and that x̂n is a positive solution of (1.3) with c=1.From (2.9), (2.10), and Theorem 2.1 the following results directly follow.Theorem 2.3.
Assume thatc∈(0,1), gcd(p,q)=1, and p is odd, then every positive solution of (1.3) converges geometrically to zero. Moreover, for each s∈{0,1,…,2pq-1}, the subsequence (x2pqm+s)m∈ℕ0 converges monotonically to zero as m→∞.Theorem 2.4.
Assume thatc>1, gcd(p,q)=1, and p is odd, then every positive solution of (1.3) tends to infinity. Moreover, for each s∈{0,1,…,2pq-1}, the subsequence (x2pqm+s)m∈ℕ0 converges increasingly to infinity as m→∞.Finally, there are two concluding interesting remarks.Remark 2.5.
Note that, since the functionscos((2k+1)πn/q) and sin((2k+1)πn/q) are periodic with period 2q and the functions cos(2lπn/p) and sin(2lπn/p) are periodic with period p, from the representation (2.9) we also obtain Theorem 2.1.Remark 2.6.
The results in papers [16, 17, 39], which are obtained in much complicated ways, are particular cases of our results. Namely, in [16] Özban studied a system which is transformed into (1.3) with p=1,q=m+k+1 and c=1, in [17] he studied a system which is transformed into (1.3) with p=3, and c=b/a, while in [39] the authors considered a system which is transformed into (1.3) with c=b/a, but they only considered the case when p≤q.



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