Global Behavior of the Difference Equation 𝑥_𝑛+1=(𝑝+𝑥_𝑛−1)/(𝑞𝑥_𝑛+𝑥_𝑛−1)

Abstract

We study the following difference equation 𝑥_𝑛+1=(𝑝+𝑥_𝑛−1)/(𝑞𝑥_𝑛+𝑥_𝑛−1), 𝑛=0,1,…, where 𝑝,𝑞∈(0,+∞) and the initial conditions 𝑥₋₁,𝑥₀∈(0,+∞). We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.

1. Introduction

Kulenović and Ladas in [1] studied the following difference equation: 𝑥_𝑛+1=𝑝+𝑥_𝑛−1𝑞𝑥_𝑛+𝑥_𝑛−1,𝑛=0,1,…, (1.1) where 𝑝,𝑞∈(0,+∞) and the initial conditions 𝑥₋₁,𝑥₀∈(0,+∞), and they obtained the following theorems.

Theorem A (see [1, Theorem 6.6.2]). Equation (1.1) has a prime period-two solution …,𝜙,𝜓,𝜙,𝜓,… (1.2) if and only if 𝑞>1+4𝑝. Furthermore, when 𝑞>1+4𝑝, the prime period-two solution is unique and the values of 𝜙 and 𝜓 are the positive roots of the quadratic equation 𝑡²𝑝−𝑡+𝑞−1=0. (1.3)

Theorem B (see [1, Theorem 6.6.4]). Let {𝑥_𝑛}^+∞_𝑛=−1 be a solution of (1.1). Let 𝐼 be the closed interval with end points 1 and 𝑝/𝑞 and let 𝐽 and 𝐾 be the intervals which are disjoint from 𝐼 and such that 𝐼∪𝐽∪𝐾=(0,+∞). (1.4) Then either all the even terms of the solution lie in 𝐽 and all odd terms lie in 𝐾, or vice-versa, or for some 𝑁≥0, 𝑥_𝑛∈𝐼for𝑛≥𝑁, (E1) when (E1) holds, except for the length of the first semicycle of the solution, if 𝑝<𝑞, the length is one; if 𝑝>𝑞, the length is at most two.

Theorem C (see [1, Theorem 6.6.5]). (a) Assume 𝑞≤1+4𝑝. Then the equilibrium 𝑥=(1+1+4𝑝(1+𝑞))/(2(1+𝑞)) of (1.1) is global attractor.
(b) Assume 𝑞>1+4𝑝. Then every solution of (1.1) eventually enters and remains in the interval [𝑝/𝑞,1].

In [1], they proposed the following conjecture.

Conjecture 1 (see [1, Conjecture 6.10.4]). Assume that 𝑝,𝑞∈(0,+∞). Show that every positive solution of (1.1) either converges to a finite limit or to a two cycle.
Gibbons et al. in [2] trigged off the investigation of the second-order difference equations 𝑥_𝑛+1=𝑓(𝑥_𝑛,𝑥_𝑛−1) such that the function 𝑓(𝑥,𝑦) is increasing in 𝑦 and decreasing in 𝑥. Motivated by [2], Berg [3] and Stević [4] obtained some important results on the existence of monotone solutions of such equations which was later considerably developed in a series of papers [5–14] (for related papers see also [15–19]). The monotonous character of solutions of the equations was explained by Stević in [20]. For some other papers in the area, see also [1, 17–19, 21–26] and the references cited therein. In this paper, we shall confirm that the Conjecture 1 is true. The main idea used in this paper can be found in papers [24, 26].

2. Global behavior of (1.1)

Theorem 2.1. Let {𝑥_𝑛}^+∞_𝑛=−1 be a nonoscillatory solution of (1.1); then {𝑥_𝑛}^+∞_𝑛=−1 converges to the unique positive equilibrium 𝑥 of (1.1).

Proof. Since {𝑥_𝑛}^+∞_𝑛=−1 is a nonoscillatory solution of (1.1), we may assume without loss of generality that there exists 𝑁>0 such that 𝑥_𝑛≤𝑥 for any 𝑛≥𝑁. We claim 𝑥_𝑛+1≥𝑥_𝑛 for any 𝑛≥𝑁. Indeed, if 𝑥_𝑛+1<𝑥_𝑛 for some 𝑛≥𝑁, then 𝑝𝑞𝑥+𝑥+1=𝑞+1𝑝+𝑥𝑞𝑥+𝑥=𝑥≥𝑥_𝑛+2=𝑝+𝑥_𝑛𝑞𝑥_𝑛+1+𝑥_𝑛>𝑝+𝑥_𝑛𝑞𝑥_𝑛+𝑥_𝑛=𝑝𝑞𝑥_𝑛+𝑥_𝑛+1,𝑞+1 (2.1) which implies 𝑥_𝑛>𝑥; this is a contradiction. Let lim_𝑛→∞𝑥_𝑛=𝑎; then 𝑎=(𝑝+𝑎)/(𝑞𝑎+𝑎) and 𝑎=𝑥. The proof is complete.

In the sequel, let 𝑞>1+4𝑝 and …,𝜙,𝜓,𝜙,𝜓,… the unique prime period-two solution of (1.1) with 𝜙<𝜓. Define 𝑓∈𝐶([𝜙,𝜓]×[𝜙,𝜓],[𝜙,𝜓]) by 𝑓(𝑥,𝑦)=𝑝+𝑦𝑞𝑥+𝑦 (2.2) for any 𝑥,𝑦∈[𝜙,𝜓] and 𝑔∈𝐶([𝜙,𝜓],[𝜙,𝜓]) by 𝑦^∗=𝑔(𝑦)=𝑝+𝑦−𝑦²𝑞𝑦 (2.3) for any 𝑦∈[𝜙,𝜓]. Then 𝑓(𝑦^∗,𝑦)=𝑦. (2.4)

Lemma 2.2. Let 𝑞>1+4𝑝, then the following statements are true.
(i)𝑓(𝑥,𝑦)>𝑦 if and only if 𝑥<𝑦^∗.(ii)𝑥>𝑦 if and only if 𝑥^∗<𝑦^∗.(iii)If 𝑥<𝑦<𝜓, then 𝑓(𝑦,𝑦^∗)<y^∗ and 𝑦>𝑦^∗∗. If 𝜙<𝑦<𝑥, then 𝑓(𝑦,𝑦^∗)>𝑦^∗ and 𝑦^∗∗>𝑦.

Proof. (i) Since 𝑓 is decreasing in 𝑥 and 𝑓(𝑦^∗,𝑦)=𝑦, 𝑥<𝑦^∗ if and only if 𝑓(𝑥,𝑦)>𝑓(𝑦^∗,𝑦)=𝑦.
(ii) Since 𝑦^∗=𝑔(𝑦) is a decreasing function for 𝑦, 𝑥>𝑦 if and only if 𝑥^∗<𝑦^∗.
(iii) Since 𝑓(𝑦,𝑦^∗)−𝑦^∗=𝑝+((𝑝+𝑦−𝑦²)/𝑞𝑦)𝑞𝑦+((𝑝+𝑦−𝑦²−)/𝑞𝑦)𝑝+𝑦−𝑦²=𝑞𝑦(𝑞²−1)[𝑦−(1−1+4𝑝+4𝑝𝑞)/2(𝑞+1)](𝑦−𝜙)(𝑦−𝑥)(𝑦−𝜓)𝑞𝑦[(𝑞²−1)𝑦²,+𝑝+𝑦] (2.5) it follows that 𝑥<𝑦<𝜓⇒𝑓(𝑦,𝑦^∗)<𝑦^∗,𝜙<𝑦<𝑥⇒𝑓(𝑦,𝑦^∗)>𝑦^∗. (2.6) By (i), we obtain 𝑦>𝑦^∗∗ if 𝑥<𝑦<𝜓 and 𝑦^∗∗>𝑦 if 𝜙<𝑦<𝑥. The proof is complete.

Lemma 2.3. Let 𝑞>1+4𝑝 and {𝑥_𝑛}^+∞_𝑛=−1 is a positive solution of (1.1); then {𝑥_2𝑛}^∞_𝑛=0 and {𝑥_2𝑛−1}^∞_𝑛=0 do exactly one of the following.
(i)Eventually, they are both monotonically increasing.(ii)Eventually, they are both monotonically decreasing.(iii)Eventually, one of them is monotonically increasing and the other is monotonically decreasing.

Proof. See [20] (also see [27]).

Remark 2.4. Stević in [20] noticed the relationship between the monotonicity of the subsequences 𝑥_2𝑛 and 𝑥_2𝑛−1 of solution {𝑥_𝑛}^+∞_𝑛=−1 of a second-order difference equation 𝑥_𝑛+1=𝑓(𝑥_𝑛,𝑥_𝑛−1) and the monotonicity of the function 𝑓(𝑥,𝑦) in variables 𝑥 and 𝑦. A simple observation shows that Stević's proof works in the general case if the function 𝑦/𝑥 is replaced by 𝑓(𝑥,𝑦). The result was later used for many times by Stević and his collaborators (see, e.g., [21, 23–26]).

Lemma 2.5. Let 𝑞>1+4𝑝. Assume that there exists some 𝑖 such that 𝜓≥𝑥_𝑖≥𝑥_𝑖+2>𝑥>𝑥_𝑖+1≥𝜙; then 𝑥_𝑖+1≥𝑥_𝑖+3.

Proof. Since 𝑥_𝑖+2=𝑓(𝑥_𝑖+1,𝑥_𝑖)≤𝑥_𝑖=𝑓(𝑥^∗_𝑖,𝑥_𝑖), it follows that 𝑥_𝑖+1≥𝑥^∗_𝑖. By Lemma 2.2(ii), we get 𝑥_𝑖^∗∗≥𝑥^∗_𝑖+1, which with Lemma 2.2(iii) implies 𝑥_𝑖≥𝑥_𝑖^∗∗≥𝑥^∗_𝑖+1. Since 𝑓(𝑥,𝑦) is increasing in 𝑦 (𝑥,𝑦∈[𝜙,𝜓]) and 𝑥_𝑖≥𝑥^∗_𝑖+1, it follows that 𝑥_𝑖+2=𝑓(𝑥_𝑖+1,𝑥_𝑖)≥𝑓(𝑥_𝑖+1,𝑥^∗_𝑖+1). (2.7) By Lemma 2.2(iii), we have 𝑥_𝑖+2≥𝑓(𝑥_𝑖+1,𝑥^∗_𝑖+1)≥𝑥^∗_𝑖+1 as 𝑥≥𝑥_𝑖+1≥𝜙. Thus 𝑥_𝑖+1=𝑓(𝑥^∗_𝑖+1,𝑥_𝑖+1)≥𝑓(𝑥_𝑖+2,𝑥_𝑖+1)=𝑥_𝑖+3. The proof is complete.

Theorem 2.6. Let 𝑞>1+4𝑝 and {𝑥_𝑛}^+∞_𝑛=−1 be an oscillatory solution of (1.1); then {𝑥_𝑛}^+∞_𝑛=−1 converges to the unique prime period-two solution of (1.1).

Proof. It follows from Theorem C(b) that there exists 𝑁>0 such that for any 𝑛≥𝑁, 𝑥_𝑛𝑝∈[𝑞,1], (2.8) and 𝑥_𝑁≥𝑥 and 𝑥_𝑁+1<𝑥. We assume without loss of generality that 𝑥_𝑛𝑝∈[𝑞,1]forany𝑛≥−1, (2.9) and 𝑥₋₁≥𝑥 and 𝑥₀<𝑥. Since ℎ(𝑥,𝑦)=𝑝+𝑦𝑝𝑞𝑥+𝑦(𝑥,𝑦∈[𝑞,1]) (2.10) is decreasing in 𝑥 and increasing in 𝑦, it follows that 𝑥_2𝑛−1>𝑥 and 𝑥_2𝑛<𝑥 for any 𝑛≥1.
If 𝑥_2𝑛−1>𝑥 is eventually increasing or 𝑥_2𝑛<𝑥 is eventually decreasing, then it follows from Theorem A that lim_𝑛→∞𝑥_2𝑛−1=𝜓 and lim_𝑛→∞𝑥_2𝑛=𝜙.
If 𝑥_2𝑛−1>𝑥 is eventually decreasing and 𝑥_2𝑛<𝑥 is eventually increasing, we may assume without loss of generality that 𝑥_2𝑛≤𝑥_2𝑛+2<𝑥<𝑥_2𝑛+1≤𝑥_2𝑛−1 for any 𝑛≥0. It follows from Lemma 2.5 that 𝑥_2𝑛≤𝑥_2𝑛+2≤𝜙<𝑥<𝜓≤𝑥_2𝑛+1≤𝑥_2𝑛−1 for any 𝑛≥0. By Theorem A, we obtain lim_𝑛→∞𝑥_2𝑛−1=𝜓 and lim_𝑛→∞𝑥_2𝑛=𝜙. The proof is complete.

We confirm from Theorems 2.1, 2.6, and C(a) that the Conjecture 1 is true.

Acknowledgment

The project is supported by NNSF of China(10861002) and NSF of Guangxi (2010GXNSFA013106) and SF of Education Department of Guangxi (200911MS212).