Global Asymptotic Stability of a Family of Nonlinear Difference Equations

Abstract

In this note, we consider global asymptotic stability of the following nonlinear difference equation 𝑥_𝑛=(∏^𝑣_𝑖=1(𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1)+∏^𝑣_𝑖=1(𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1))/(∏^𝑣_𝑖=1(𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1)−∏^𝑣_𝑖=1(𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1)),𝑛=0,1,…, where 𝑘_𝑖∈ℕ(𝑖=1,2,…,𝑣),𝑣≥2, 𝛽₁∈[−1,1], 𝛽₂,𝛽₃,…,𝛽_𝑣∈(−∞,+∞), 𝑥_−𝑚,𝑥_−𝑚+1,…,𝑥₋₁∈(0,∞), and 𝑚=max_{1≤𝑖≤𝑣}{𝑘_𝑖}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007).

1. Introduction

The study of dynamical properties of nonlinear difference equations has been an area of intense interest in recent years (e.g., see [1–13]).

In [4], by analysis of semicycle structure, the authors discussed the global asymptotic stability of rational difference equation 𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−1+1𝑥_𝑛+𝑥_𝑛−1,𝑛=0,1,…, (1) where the initial values 𝑥₋₁,𝑥₀∈(0,+∞).

Li [5, 6] investigated the qualitative behavior of the rational difference equations 𝑥_𝑛=𝑥_𝑛−1+𝑥_𝑛−2+𝑥_𝑛−4+𝑥_𝑛−1𝑥_𝑛−2𝑥_𝑛−4+𝑎1+𝑥_𝑛−1𝑥_𝑛−2+𝑥_𝑛−2𝑥_𝑛−4+𝑥_𝑛−1𝑥_𝑛−4,𝑥+𝑎𝑛=0,1,2,…,_𝑛=𝑥_𝑛−2+𝑥_𝑛−3+𝑥_𝑛−4+𝑥_𝑛−2𝑥_𝑛−3𝑥_𝑛−4+𝑎1+𝑥_𝑛−2𝑥_𝑛−3+𝑥_𝑛−3𝑥_𝑛−4+𝑥_𝑛−2𝑥_𝑛−4,+𝑎𝑛=0,1,2,…, (2) with 𝑥₋₄,𝑥₋₃,…,𝑥₋₁∈(0,∞)  and  𝑎∈[0,∞) via analysis of semicycle structure and verified that every solution of (2) converges to equilibrium 1.

By using the transformation method, Berenhaut et al. [1] studied the behavior of positive solutions to the rational difference equation 𝑥_𝑛=𝑥_𝑛−𝑘+𝑥_𝑛−𝑚1+𝑥_𝑛−𝑘𝑥_𝑛−𝑚,𝑛=0,1,2,…, (3) with 𝑥_−𝑚,𝑥_−𝑚+1,…,𝑥₋₁∈(0,∞) and 1≤𝑘<𝑚  and proved that every solution of (3) converges to the unique equilibrium 1. Based on the above facts, Berenhaut et al. [1] put forward the following two conjectures.

Conjecture 1. Suppose that 1≤𝑘<𝑙<𝑚 and that {𝑥_𝑛} satisfies 𝑥_𝑛=x_𝑛−𝑘+𝑥_𝑛−𝑙+𝑥_𝑛−𝑚+𝑥_𝑛−𝑘𝑥_𝑛−𝑙𝑥_𝑛−𝑚1+𝑥_𝑛−𝑘𝑥_𝑛−𝑙+𝑥_𝑛−𝑙𝑥_𝑛−𝑚+𝑥_𝑛−𝑚𝑥_𝑛−𝑘,𝑛=0,1,2,… (4) with 𝑥_−𝑚,𝑥_−𝑚+1,…,𝑥₋₁∈(0,∞). Then, the sequence {𝑥_𝑛} converges to the unique equilibrium 1.

Conjecture 2. Suppose that 𝑚 is odd and 1≤𝑘₁<𝑘₂<⋯<𝑘_𝑚,  and define 𝑆={1,2,…,𝑚}. If {𝑥_𝑛} satisfies 𝑥_𝑛=𝑓₁𝑥_𝑛−𝑘1,𝑥_𝑛−𝑘2,…,𝑥_{𝑛−𝑘𝑚}𝑓₂𝑥_𝑛−𝑘1,𝑥_𝑛−𝑘2,…,𝑥_{𝑛−𝑘𝑚},𝑛=0,1,2,…, (5) with 𝑥_−𝑘𝑚,𝑥_{−𝑘𝑚+1},…,𝑥₋₁∈(0,∞), where 𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑚=_{{}𝑗∈1,3,…,𝑚𝑡1,𝑡2,…,𝑡𝑗𝑡⊂𝑆;1<𝑡2<⋯<𝑡𝑗}𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝑗,𝑓₂𝑥₁,𝑥₂,…,𝑥_𝑚=1+_{{}𝑗∈2,4,…,𝑚−1𝑡1,𝑡2,…,𝑡𝑗𝑡⊂𝑆;1<𝑡2<⋯<𝑡𝑗}𝑥_𝑡1,𝑥_𝑡2,…,𝑥_𝑡𝑗, (6) then the sequence {𝑥_𝑛} converges to the unique equilibrium 1.

Recently, by method used in [4–6], the authors of [12] studied the global asymptotic stability of the following nonlinear difference equation. 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−2,𝑥_𝑛−3𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−2,𝑥_𝑛−3,𝑛=0,1,…, (7) where 𝐹𝑥,𝑦,𝑧,𝑤=𝑥^𝛼1𝑦^𝛼2+𝑥^𝛼1𝑧^𝛼3+𝑥^𝛼1𝑤^𝛼4+𝑦^𝛼2𝑧^𝛼3+𝑦^𝛼2𝑤^𝛼4+𝑧^𝛼3𝑤^𝛼4+𝑥^𝛼1𝑦^𝛼2𝑧^𝛼3𝑤^𝛼4𝐺+1,𝑥,𝑦,𝑧,𝑤=𝑥^𝛼1+𝑦^𝛼2+𝑧^𝛼3+𝑤^𝛼4+𝑥^𝛼1𝑦^𝛼2𝑧^𝛼3+𝑥^𝛼1𝑦^𝛼2𝑤^𝛼4+𝑥^𝛼1𝑧^𝛼3𝑤^𝛼4+𝑦^𝛼2𝑧^𝛼3𝑤^𝛼4, (8) the parameter 𝛼₁∈(0,1],𝛼₂,𝛼₃,𝛼₄∈(0,+∞),  and the initial values 𝑥₋₃,𝑥₋₂,𝑥₋₁,𝑥₀∈(0,+∞).

Motivated by the above studies, in this note, we propose and consider the following nonlinear difference equation. 𝑥_𝑛=∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1+∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖},−1𝑛=0,1,…, (9) where 𝑘_𝑖∈ℕ(𝑖=1,2,…,𝑣),𝑣≥2, 𝛽₁∈[−1,1], 𝛽₂,𝛽₃,…,𝛽_𝑣∈(−∞,+∞), 𝑥_−𝑚,𝑥_−𝑚+1,…,𝑥₋₁∈(0,∞), and 𝑚=max_{1≤𝑖≤𝑣}{𝑘_𝑖}.

It is noticed that, letting 𝑣=2, 𝛽₁=𝛽₂=1, 𝑘₁=1,  and  𝑘₂=2, (9) reduces to (1); letting 𝑣=3,  𝛽₁=𝛽₂=𝛽₃=1,  𝑘₁=1,𝑘₂=2,  and  𝑘₃=4 and 𝑣=3,   𝛽₁=𝛽₂=𝛽₃=1,   𝑘₁=2,   𝑘₂=3,   and  𝑘₃=4, (9) reduces to (2); letting 𝑣=2,  𝛽₁=𝛽₂=1, 𝑘₁=𝑘,  and  𝑘₂=𝑚, (9) reduces to (3); letting 𝑣=3, 𝛽₁=𝑘,   𝛽₂=𝑙,  and  𝛽₃=𝑚, (9) reduces to (4); letting 𝑣=4,  𝛽₁∈(0,1],  𝛽_𝑖=𝛼_𝑖(𝑖=1,2,3,4),  𝑘₁=1,  𝑘₂=2, 𝑘₃=3, and 𝑘₄=4, (9) reduces to (7); letting 𝑣=𝑚 be odd, 1≤𝑘₁<𝑘₂<⋯<𝑘_𝑚, and 𝛽₁=𝛽₂=⋯=𝛽_𝑚=1, (9) reduces to (5). Clearly, (5) is a special example of (9).

In 2007, Berenhaut and Stević [2] had proved Conjecture 1. In this paper, by making full use of analytical techniques, we mainly prove that the unique positive equilibrium point of (9) is globally asymptotically stable. It is clear that our result generalizes the corresponding works in [1, 2, 4–9, 12] and simultaneously conforms to Conjecture 2.

2. Existence of a Unique Positive Equilibrium

In this section, we mainly show the existence of a unique positive equilibrium of (9).

Theorem 3. In (9) there exists a unique positive equilibrium point 𝑥=1.

Proof. A positive equilibrium point 𝑥 of (9) satisfies the next equation: ∏𝑥=^𝑣𝑖=1𝑥^𝛽𝑖+1+∏^𝑣_𝑖=1𝑥^𝛽𝑖−1∏^𝑣_𝑖=1𝑥^𝛽𝑖+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖,−1 (10) from which we may get (𝑥−1)^𝑣_𝑖=1𝑥^𝛽𝑖+1=(𝑥+1)^𝑣_𝑖=1𝑥^𝛽i−1; (11) that is, (𝑥−1)𝑥^𝛽1+1^𝑣_𝑖=2𝑥^𝛽𝑖=(+1𝑥+1)𝑥^𝛽1−1^𝑣_𝑖=2𝑥^𝛽𝑖−1. (12) From the above equation, we can get 𝑥^𝛽1+1−1^𝑣_𝑖=2𝑥^𝛽𝑖+1−^𝑣_𝑖=2𝑥^𝛽𝑖+−1𝑥−𝑥^𝛽1𝑣_𝑖=2𝑥^𝛽𝑖+1+^𝑣_𝑖=2𝑥^𝛽𝑖−1=0. (13) One can see that for any 𝑥>0  and  𝑣≥2, ^𝑣_𝑖=2𝑥^𝛽𝑖+1−^𝑣_𝑖=2𝑥^𝛽𝑖−1>0,_𝑣𝑖=2𝑥^𝛽𝑖+1+^𝑣_𝑖=2𝑥^𝛽𝑖−1>0. (14)(i)If 𝛽₁=−1,0,1, from (13) and (14), we can get that (9) has a unique positive equilibrium 𝑥=1.(ii)If  −1<𝛽₁<0 or 0<𝛽₁<1 and 0<𝑥<1, we have 𝑥<𝑥^𝛽1,𝑥^𝛽1+1<1. (15) Further, we have 𝑥^𝛽1+1−1^𝑣_𝑖=2𝑥^𝛽𝑖+1−^𝑣_𝑖=2𝑥^𝛽𝑖+−1𝑥−𝑥^𝛽1𝑣_𝑖=2𝑥^𝛽𝑖+1+^𝑣_𝑖=2𝑥^𝛽𝑖−1<0. (16)(iii)If −1<𝛽₁<0 or 0<𝛽₁<1 and  𝑥>1, we have 𝑥^𝛽1+1−1^𝑣_𝑖=2𝑥^𝛽𝑖+1−^𝑣_𝑖=2𝑥^𝛽𝑖+−1𝑥−𝑥^𝛽1𝑣_𝑖=2𝑥^𝛽𝑖+1+^𝑣_𝑖=2𝑥^𝛽𝑖−1>0. (17)It is clear that (9) has a unique positive equilibrium 𝑥=1. The proof is complete.

3. Global Asymptotic Stability for the Unique Positive Equilibrium Point

In this section, we give our main result.

Theorem 4. The unique positive equilibrium point 𝑥=1 of (9) is globally asymptotically stable.

In order to prove Theorem 4, we introduce the following lemma by Kruse and Nesemann [3] and make full use of analytical techniques.

Lemma 5. Consider the difference equation 𝑥_𝑛+𝑘=𝑓𝑥_{𝑛+𝑘−1},…,𝑥_𝑛,𝑛=0,1,2,…, (18) where 𝑘∈ℕ and 𝑓∶(0,∞)^𝑘→(0,∞) is a continuous function with some unique equilibrium 𝑥. Suppose that there is a 𝑝∈ℕ such that for all solutions {𝑥_𝑛} of (18) 𝑥_𝑛−𝑥_𝑛+𝑝𝑥²𝑥_𝑛−𝑥_𝑛+𝑝≤0, (19) where equality holds if and only if 𝑥_𝑛=𝑥. Then 𝑥 is globally asymptotically stable.

Proof of Theorem 4. Let {𝑥_𝑛}^∞_𝑛=−𝑚 be any solution of (9). We have 𝑥_𝑛−𝑥^𝛽1_𝑛−𝑘1=∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1+∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1−𝑥^𝛽1_𝑛−𝑘1=1−𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1+1+𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}=−11−𝑥^𝛽1_𝑛−𝑘11+𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖},1−1𝑛=0,1,…,𝑥^𝛽1_𝑛−𝑘1−𝑥_𝑛=1𝑥^𝛽1_𝑛−𝑘1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1+∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}=−11−𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−1+𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖},=−11−𝑥^𝛽1_𝑛−𝑘11+𝑥^𝛽1_𝑛−𝑘1∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1+∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛−𝑘𝑖},−1𝑛=0,1,…. (20)
It follows from (20) that 𝑥_𝑛+𝑘1−𝑥^𝛽1_𝑛=1−𝑥^𝛽1_𝑛1+𝑥^𝛽1_𝑛∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖}+1−∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖},−1𝑛=−𝑘₁,−𝑘₁1+1,…,𝑥^𝛽1_𝑛−𝑥_𝑛+𝑘1=1−𝑥^𝛽1_𝑛1+𝑥^𝛽1_𝑛∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖}+1+∏^𝑣_𝑖=2𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖}−1∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛+k1−𝑘𝑖}+1−∏^𝑣_𝑖=1𝑥^𝛽𝑖_{𝑛+𝑘1−𝑘𝑖},−1𝑛=−𝑘₁,−𝑘₁+1,…. (21) Clearly, from (21), we have 𝑥^𝛽1_𝑛−𝑥_𝑛+𝑘11𝑥^𝛽1_𝑛−𝑥_𝑛+𝑘1≤0,𝑛=−𝑘₁,−𝑘₁+1,…. (22) From (22), we have 1−𝑥_𝑛+𝑘11𝑥^𝛽1_𝑛+𝑥^𝛽1_𝑛+𝑥²_𝑛+𝑘1≤0,𝑛=−𝑘₁,−𝑘₁+1,…. (23) If 𝛽₁=±1, it is clear that 1𝑥^𝛽1_𝑛+𝑥^𝛽1_𝑛=1𝑥_𝑛+𝑥_𝑛. (24) If 0<𝑥_𝑛<1 and −1<𝛽₁<1, we have 𝑥_𝑛<𝑥^𝛽1_𝑛 and 0<𝑥^𝛽1_𝑛⁺¹<1, so that 𝑥_𝑛−𝑥^𝛽1_𝑛11−𝑥_𝑛𝑥^𝛽1_𝑛>0. (25) Similarly, if 𝑥_𝑛>1 and −1<𝛽₁<1, we have 𝑥_𝑛>𝑥^𝛽1_𝑛 and 𝑥^𝛽1_𝑛⁺¹>1, so that 𝑥_𝑛−𝑥^𝛽1_𝑛11−𝑥_𝑛𝑥^𝛽1_𝑛>0. (26) Hence, for −1≤𝛽₁≤1, we always have 1𝑥^𝛽1_𝑛+𝑥^𝛽1_𝑛≤1𝑥_𝑛+𝑥_𝑛. (27) Further, from (23) and (27), we have 1−𝑥_𝑛+𝑘11𝑥_𝑛+𝑥_𝑛+𝑥²_𝑛+𝑘1≤1−𝑥_𝑛+𝑘11𝑥^𝛽1_𝑛+𝑥^𝛽1_𝑛+𝑥²_𝑛+𝑘1≤0,𝑛=−𝑘₁,−𝑘₁+1,…. (28) Therefore, 𝑥_𝑛−𝑥_𝑛+𝑘11𝑥_𝑛−𝑥_𝑛+𝑘1≤0,𝑛=−𝑘₁,−𝑘₁+1,…, (29) where equality holds if and only if 𝑥_𝑛=𝑥=1. By Lemma 5 and (29), with 𝑝=𝑘₁∈ℕ, it follows that the unique positive equilibrium point 𝑥=1 of (9) is globally asymptotically stable. The proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the referees for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this work. This paper is supported partly by Hunan Provincial Natural Science Foundation of China (no. 13JJ3075), Soft Science Fund of Science and Technology Department of Hunan Province (no. 2011ZK3066), Start-up Fund of University of South China (no. 2011XQD49), and the construct program in USC.