On the Dynamics of a Higher-Order Difference Equation

Abstract

This paper deals with the investigation of the following more general rational difference equation: 𝑦_𝑛+1=𝛼𝑦_𝑛/(𝛽+𝛾∑^𝑘_𝑖=0𝑦^𝑝_{𝑛−(2𝑖+1)}∏^𝑘_𝑖=0𝑦_{𝑛−(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.

1. Introduction

In this paper we investigate the global stability character and the oscillatory of the solutions of the following difference equation: 𝑦_𝑛+1=𝛼𝑦_𝑛𝛽+𝛾∑^𝑘_𝑖=0𝑦^𝑝_{𝑛−(2𝑖+1)}∏^𝑘_𝑖=0𝑦_{𝑛−(2𝑖+1)},𝑛=0,1,2,…, (1.1) 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 𝑥_𝑛+1=(𝛼+𝛽𝑥_𝑛+𝛾𝑥_𝑛−1)/(𝐴+𝐵𝑥_𝑛+𝐶𝑥_𝑛−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: 𝑥_𝑛+1=𝑎𝑥_𝑛−𝑏𝑥_𝑛𝑐𝑥_𝑛−𝑑𝑥_𝑛−1,𝑥_𝑛+1=𝛼𝑥_𝑛−𝑘𝛽+𝛾∏^𝑘_𝑖=0𝑥_𝑛−𝑖,𝑥_𝑛+1=𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘𝑐𝑥_𝑛−𝑠−𝑏+𝑎. (1.2) El-Metwally [6] gave some results about the global behavior of the solutions of the following more general rational difference equations 𝑥_𝑛+1=𝑎𝑥^𝑙0_𝑛−𝑘0𝑥^𝑙1_𝑛−𝑘1…𝑥^𝑙𝑖_{𝑛−𝑘𝑖}+𝑏𝑥^𝑠0_𝑛−𝑟0𝑥^𝑠1_𝑛−𝑟1…𝑥^𝑠𝑗_{𝑛−𝑟𝑗}𝑐𝑥^𝑙0_𝑛−𝑘0𝑥^𝑙1_𝑛−𝑘1…𝑥^𝑙𝑖_{𝑛−𝑘𝑖}+𝑑𝑥^𝑠0_𝑛−𝑟0𝑥^𝑠1_𝑛−𝑟1…𝑥^𝑠𝑗_{𝑛−𝑟𝑗},𝑦_𝑛+1=𝛼₀𝑦_𝑛+𝛼₁𝑦_𝑛−1+⋯+𝛼_𝑡𝑦_𝑛−𝑡𝛽₀𝑦_𝑛+𝛽₁𝑦_𝑛−1+⋯+𝛽_𝑡𝑦_𝑛−𝑡. (1.3) Çinar [7–9] obtained the solutions form of the difference equations 𝑥_𝑛+1=𝑥_𝑛−1/(1+𝑥_𝑛𝑥_𝑛−1), 𝑥_𝑛+1=𝑥_𝑛−1/(−1+𝑥_𝑛𝑥_𝑛−1)and𝑥_𝑛+1=𝑎𝑥_𝑛−1/(1+𝑏𝑥_𝑛𝑥_𝑛−1). Also, Cinar et al. [10] studied the existence and the convergence for the solutions of the difference equation 𝑥_𝑛+1=𝑥_𝑛−3/(−1+𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−2𝑥_𝑛−3). Simsek et al. [11] obtained the solution of the difference equation 𝑥_𝑛+1=𝑥_𝑛−3/(1+𝑥_𝑛−1). In [12] Yalcinkaya got the solution form of the difference equation 𝑥_𝑛+1=𝑥_{𝑛−(2𝑘+1)}/(1+𝑥_𝑛−𝑘𝑥_{𝑛−(2𝑘+1)}). In [13] Stević studied the difference equation 𝑥_𝑛+1=𝑥_𝑛−1/(1+𝑥_𝑛). Other related results on rational difference equations can be found in [14–19].

Let 𝐼 be some interval of real numbers and let 𝑓∶𝐼^𝑘+1⟶𝐼 (1.4) be a continuously differentiable function. Then for every set of initial conditions 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₀∈𝐼, the difference equation 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−𝑘,𝑛=0,1,… (1.5) has a unique solution {𝑥_𝑛}^∞_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≤𝑥_𝑛≤𝑀for all𝑛≥𝑁.

Definition 1.2 (periodicity). A sequence {𝑥_𝑛}^∞_𝑛=−𝑘is said to be periodic with period 𝑝 if 𝑥_𝑛+𝑝=𝑥_𝑛for all 𝑛≥−𝑘.

Definition 1.3 (semicycles). A positive semicycle of a sequence {𝑥_𝑛}^∞_𝑛=−𝑘consists of a “string” of terms {𝑥_𝑙,𝑥_𝑙+1,…,𝑥_𝑚} all greater than or equal to the equilibrium point 𝑥, with 𝑙≥−𝑘 and 𝑚≤∞ such that either 𝑙=−𝑘 or 𝑙>−𝑘 and 𝑥_𝑙−1<𝑥; and, either 𝑚=∞ or 𝑚<∞ and 𝑥_𝑚+1<𝑥. A negative semicycle of a sequence {𝑥_𝑛}^∞_𝑛=−𝑘consists of a "string" of terms {𝑥_𝑙,𝑥_𝑙+1,…,𝑥_𝑚} all less than the equilibrium point 𝑥, with 𝑙≥−𝑘 and 𝑚≤∞ such that: either 𝑙=−𝑘 or 𝑙>−𝑘 and 𝑥_𝑙−1≥𝑥; and, either 𝑚=∞ or 𝑚<∞ and 𝑥_𝑚+1≥𝑥.

Definition 1.4 (oscillation). A sequence {𝑥_𝑛}^∞_𝑛=−𝑘is called nonoscillatory about the point 𝑥 if there is exists 𝑁≥−𝑘 such that either 𝑥_𝑛>𝑥 for all 𝑛≥𝑁 or 𝑥_𝑛<𝑥 for all 𝑛≥𝑁. Otherwise {𝑥_𝑛}^∞_𝑛=−𝑘 is called oscillatory about 𝑥.

Recall that the linearized equation of (1.5) about the equilibrium 𝑥is the linear difference equation 𝑦_𝑛+1=^𝑘_𝑖=0𝜕𝑓(𝑥,𝑥,…,𝑥)𝜕𝑥_𝑛−𝑖𝑦_𝑛−𝑖. (1.6)

2. Dynamics of (1.1)

The change of variables 𝑦_𝑛=(𝛽/𝛾)^{1/(𝑝+𝑘+1)}𝑥_𝑛 reduces (1.1) to the following difference equation 𝑥_𝑛+1=𝑟𝑥_𝑛1+∑^𝑘_𝑖=0𝑥^𝑝_{𝑛−(2𝑖+1)}∏^𝑘_𝑖=0𝑥_{𝑛−(2𝑖+1)},𝑛=0,1,2,…, (2.1) 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 𝑟𝑥=𝑥1+(𝑘+1)𝑥^𝑝+𝑘+1. (2.2) 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 point 𝑥=0 of (2.1).

Theorem 2.1. The following statements are true:
(i) if 𝑟<1, then the equilibrium point 𝑥=0 of (2.1) is locally asymptotically stable,
(ii) if 𝑟>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 𝑢_𝑛+1−𝑟𝑢_𝑛=0. Then the associated eigenvalues are 𝜆=0 and 𝜆=𝑟. Then the proof is complete.

Theorem 2.2. Assume that 𝑟<1, then the equilibrium point 𝑥=0 of (2.1) is globally asymptotically stable.

Proof. Let {𝑥_𝑛}^∞_{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 lim𝑥_{𝑛𝑛→∞}=0. (2.3) Now it follows from (2.1) that 𝑥_𝑛+1=𝑟𝑥_𝑛1+𝑥^𝑝+1_𝑛−1𝑥_𝑛−3⋯𝑥_{𝑛−2𝑘+1}+𝑥_𝑛−1𝑥^𝑝+1_𝑛−3⋯𝑥_{𝑛−2𝑘+1}+⋯+𝑥_𝑛−1𝑥_𝑛−3⋯𝑥^𝑝+1_{𝑛−2𝑘+1}≤𝑟𝑥_𝑛<𝑥_𝑛. (2.4) Then the sequence {𝑥_𝑛}^∞_𝑛=0 is decreasing and this completes the proof.

Theorem 2.3. Assume that 𝑟>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 {𝑥_𝑛}^∞_{𝑛=−2𝑘+1} be a solution of (2.1). Without loss of generality assume that {𝑥_𝑛}^∞_{𝑛=−2𝑘+1} is a nonoscillatory solution of (2.1), then it suffices to show that lim_𝑛→∞𝑥_𝑛=𝑥. Assume that 𝑥_𝑛≥𝑥 for 𝑛≥𝑛₀ (the case where 𝑥_𝑛≤𝑥 for 𝑛≥𝑛₀ is similar and will be omitted). It follows from (2.1) that 𝑥_𝑛+1=𝑟𝑥_𝑛1+𝑥^𝑝+1_𝑛−1𝑥_𝑛−3⋯𝑥_{𝑛−2𝑘−1}+𝑥_𝑛−1𝑥^𝑝+1_𝑛−3⋯𝑥_{𝑛−2𝑘−1}+⋯+𝑥_𝑛−1𝑥_𝑛−3⋯𝑥^𝑝+1_{𝑛−2𝑘−1}≤𝑥_𝑛𝑟1+(𝑘+1)𝑥^𝑝+𝑘+1=𝑥_𝑛. (2.5) Hence {𝑥_𝑛} is monotonic for 𝑛≥𝑛₀+2𝑘+1, therefore it has a limit. Let lim_𝑛→∞𝑥_𝑛=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)𝜇^𝑝+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 {𝑥_𝑛}^∞_{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 {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} is a strictly oscillatory solution of (2.1). Let 𝑁≥2𝑘+2 and let {𝑥_𝑁,𝑥_𝑁+1,…,𝑥_𝑀} be a positive semicycle followed by the negative semicycle {𝑥_𝑀,𝑥_𝑀+1,…,𝑥_𝑀}. Now it follows from (2.1) that 𝑥_𝑁+2𝑘+2−𝑥_𝑁=𝑟𝑥_𝑁+2𝑘+11+𝑥^𝑝+1_𝑁+2𝑘𝑥_{𝑁+2𝑘−2}⋯𝑥_𝑁+𝑥_𝑁+2𝑘𝑥^𝑝+1_{𝑁+2𝑘−2}⋯𝑥_𝑁+⋯+𝑥_𝑁+2𝑘⋯𝑥_𝑁^𝑝+1−𝑥_𝑁≤𝑥_𝑁+2𝑘+1𝑟1+(𝑘+1)𝑥^𝑝+𝑘+1−𝑥_𝑁=𝑥_𝑁+2𝑘+1−𝑥_𝑁=𝑟𝑥_𝑁+2𝑘1+𝑥^𝑝+1_{𝑁+2𝑘−1}𝑥_{𝑁+2𝑘−3}⋯𝑥_𝑁−1+⋯+𝑥_{𝑁+2𝑘−1}⋯𝑥^𝑝+1_𝑁−1−𝑥_𝑁≤𝑥_𝑁+2𝑘𝑟1+(𝑘+1)𝑥^𝑝+𝑘+1−𝑥_𝑁=𝑥_𝑁+2𝑘−𝑥_𝑁≤𝑥_{𝑁+2𝑘−1}−𝑥_𝑁≤⋯≤𝑥_𝑁+1−𝑥_𝑁≤𝑥_𝑁−𝑥_𝑁=0. (2.6)
Then 𝑥_𝑁≥𝑥_{𝑁+2(𝑘+1)} for all 𝑁≥2(𝑘+1).
Similarly, we see from (2.1) that 𝑥_𝑀+2𝑘+2−𝑥_𝑀=𝑟𝑥_𝑀+2𝑘+11+𝑥^𝑝+1_𝑀+2𝑘𝑥_{𝑀+2𝑘−2}⋯𝑥_𝑀+𝑥_𝑀+2𝑘𝑥^𝑝+1_{𝑀+2𝑘−2}⋯𝑥_𝑀+⋯+𝑥_𝑀+2𝑘⋯𝑥_𝑀^𝑝+1−𝑥_𝑀≥𝑥_𝑀+2𝑘+1𝑟1+(𝑘+1)𝑥^𝑝+𝑘+1−𝑥_𝑀=𝑥_𝑀+2𝑘+1−𝑥_𝑀=𝑟𝑥_𝑀+2𝑘1+𝑥^𝑝+1_{𝑀+2𝑘−1}𝑥_{𝑀+2𝑘−3}⋯𝑥_𝑀−1+⋯+𝑥_{𝑀+2𝑘−1}⋯𝑥^𝑝+1_𝑀−1−𝑥_𝑀≥𝑥_𝑀+2𝑘𝑟1+(𝑘+1)𝑥^𝑝+𝑘+1−𝑥_𝑀=𝑥_𝑀+2𝑘−𝑥_𝑀≤𝑥_{𝑀+2𝑘−1}−𝑥_𝑀≥⋯≥𝑥_𝑀+1−𝑥_𝑀≥𝑥_𝑀−𝑥_𝑀=0. (2.7)
Therefore 𝑥_{𝑀+2(𝑘+1)}≥𝑥_𝑀 for all 𝑀≥2(𝑘+1). The proof is so complete.

Theorem 2.5. Equation (2.1) is permanent.

Proof. Let {𝑥_𝑛}^∞_{𝑛=−2𝑘+1} be a solution of (2.1). There are two cases to consider:
(i){𝑥_𝑛}^∞_{𝑛=−2𝑘+1} is a nonoscillatory solution of (2.1). Then it follows from Theorem 2.3 that lim𝑥_{𝑛𝑛→∞}=𝑥, (2.8) that is there is a sufficiently large positive integer 𝑁 such that |𝑥_𝑛−𝑥|<𝜀 for all 𝑛≥𝑁 and for some 𝜀>0. So, 𝑥−𝜀<𝑥_𝑛<𝑥+𝜀, this means that there are two positive real numbers, say 𝐶 and 𝐷, such that 𝐶≤𝑥_𝑛≤𝐷, (2.9)(ii){𝑥_𝑛}^∞_{𝑛=−2𝑘+1} is strictly oscillatory about 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+𝑘+1)}.
Now let {𝑥_𝑠+1,𝑥_𝑠+2,…,𝑥_𝑡} be a positive semicycle followed by the negative semicycle {𝑥_𝑡+1,𝑥_𝑡+2,…,𝑥_𝑢}. If 𝑥_𝑉 and 𝑥_𝑊 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 𝑥_𝐿=𝑥_𝐿−1𝑟1+𝑥^𝑝+1_𝐿−2𝑥_𝐿−4⋯𝑥_{𝐿−2𝑘−2}+𝑥_𝐿−2𝑥^𝑝+1_𝐿−4⋯𝑥_{𝐿−2𝑘−2}+⋯+𝑥_𝐿−2𝑥_𝐿−4⋯𝑥^𝑝+1_{𝐿−2𝑘−2}=𝑟²𝑥_𝐿−21+𝑥^𝑝+1_𝐿−3⋯𝑥_{𝐿−2𝑘−3}+⋯+𝑥_𝐿−3⋯𝑥^𝑝+1_{𝐿−2𝑘−3}1+𝑥^𝑝+1_𝐿−2⋯𝑥_{𝐿−2𝑘−2}+⋯+𝑥_𝐿−2⋯𝑥^𝑝+1_{𝐿−2𝑘−2}⋮=𝑥_𝐿−𝜁𝑟^𝜁𝜁_𝜂=111+∑^𝑘_𝑖=0𝑥^𝑝_{𝐿−(2𝑖+1)−𝜂}∏^𝑘_𝑖=0𝑥_{𝐿−(2𝑖+1)−𝜂}=𝑥_𝜇𝑟^{𝐿−𝜇𝐿−1}_𝜂=𝜇11+∑^𝑘_𝑖=0𝑥^𝑝_{𝜂−(2𝑖+1)}∏^𝑘_𝑖=0𝑥_{𝜂−(2𝑖+1)}. (2.10) Therefor for 𝑉=𝐿 and 𝑠=𝜇 we obtain 𝑥_𝑉=𝑥_𝑠𝑟^{𝑉−𝑠𝑉−1}_𝜂=𝑠11+∑^𝑘_𝑖=0𝑥^𝑝_{𝜂−(2𝑖+1)}∏^𝑘_𝑖=0𝑥_{𝜂−(2𝑖+1)}≤𝑥𝑟^2𝑘+1=𝐻. (2.11)
Again whenever 𝑊=𝐿 and 𝜇=𝑡, we see that 𝑥_𝑊=𝑥_𝑡𝑟^{𝑊−𝑡𝑊−1}_𝜂=𝑡11+∑^𝑘_𝑖=0𝑥^𝑝_{𝜂−(2𝑖+1)}∏^𝑘_𝑖=0𝑥_{𝜂−(2𝑖+1)}≥𝑥𝑟^{𝑊−𝑡𝑊−1}_𝜂=𝑡11+(𝑘+1)𝐻^𝑝+𝑘+1=𝑥𝑟^𝑊−𝑡11+(𝑘+1)𝐻^{𝑝+𝑘+1𝑊−𝑡−1}≥𝑥11+(𝑘+1)𝐻^{𝑝+𝑘+12𝑘+1}=𝐺. (2.12) That is, 𝐺≤𝑥_𝑛≤𝐻. It follows from (i) and (ii) that min{𝐶,𝐺}≤𝑥_𝑛≤max{𝐷,𝐻}. (2.13) Then the proof is complete.