On the Dynamics of a Higher-Order Rational Difference Equation

Abstract

The aim of this paper is to investigate the global asymptotic stability and the periodic character for the rational difference equation 𝑥_𝑛+1=𝛼𝑥_𝑛−1/(𝛽+𝛾Π^𝑘_𝑖=𝑙𝑥^𝑝𝑖_𝑛−2𝑖),𝑛=0,1,2,…, where the parameters 𝛼,𝛽,𝛾,𝑝_𝑙,𝑝_𝑙+1,…,𝑝_𝑘 are nonnegative real numbers, and 𝑙,𝑘 are nonnegative integers such that 𝑙≤𝑘.

1. Introduction

Difference equations have always played an important role in the construction and analysis of mathematical models of biology, ecology, physics, economic process, and so forth.

The study of nonlinear rational difference equations of higher order is of paramount importance, since we still know so little about such equations.

Amleh et al. [1] investigated the third-order rational difference equation 𝑥_𝑛+1=𝑎+𝑏𝑥_𝑛−1𝐴+𝐵𝑥_𝑛−2,𝑛=0,1,2,…, (1.1) where 𝑎,𝑏,𝐴,𝐵 are nonnegative real numbers and the initial conditions are nonnegative real numbers.

Ahmed [2] studied the global asymptotic behavior and the periodic character of solutions of the third-order rational difference equation 𝑥_𝑛+1=𝑏𝑥_𝑛−1𝐴+𝐵𝑥^𝑝_𝑛𝑥^𝑞_𝑛−2,𝑛=0,1,2,…, (1.2) where the parameters 𝑏,𝐴,𝐵,𝑝,𝑞 are nonnegative real numbers, and the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀ are arbitrary nonnegative real numbers.

For other related results, see [3] and also [4–15].

In this paper, the global asymptotic behavior and the periodic character of solutions of the rational difference equation 𝑥_𝑛+1=𝛼𝑥_𝑛−1𝛽+𝛾Π^𝑘_𝑖=𝑙𝑥^𝑝𝑖_𝑛−2𝑖,𝑛=0,1,2,…, (1.3) where the parameters 𝛼,𝛽,𝛾,𝑝_𝑙,𝑝_𝑙+1,…,𝑝_𝑘 are nonnegative real numbers, 𝑙,𝑘 are nonnegative integers such that 𝑙≤𝑘, and the initial conditions 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀ are arbitrary nonnegative real numbers such that 𝛽+𝛾^𝑘_𝑖=𝑙𝑥^𝑝𝑖_𝑛−2𝑖>0,∀𝑛≥0, (1.4) will be investigated.

Let 𝐼 be an interval of real numbers, and let 𝑓∶𝐼^2𝑘+1→𝐼 be a continuously differentiable function. Consider the difference equation 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−2𝑘,𝑛=0,1,2,…, (1.5) with 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀∈𝐼. Let 𝑥 be the equilibrium point of (1.5). The linearized equation of (1.5) about 𝑥 is 𝑦_𝑛+1=𝑐₁𝑦_𝑛+𝑐₂𝑦_𝑛−1+⋯+𝑐_2𝑘+1𝑦_𝑛−2𝑘,𝑛=0,1,2,…, (1.6) where 𝑐₁=𝜕𝑓𝜕𝑥_𝑛(𝑥,𝑥,…,𝑥),𝑐₂=𝜕𝑓𝜕𝑥_𝑛−1(𝑥,𝑥,…,𝑥),…,𝑐_2𝑘+1=𝜕𝑓𝜕𝑥_𝑛−2𝑘(𝑥,𝑥,…,𝑥). (1.7) The characteristic equation of (1.5) is 𝜆^2𝑘+1−𝑐₁𝜆^2𝑘−𝑐₂𝜆^2𝑘−1−⋯−𝑐_2𝑘+1=0. (1.8)

Definition 1.1. Let 𝑥 be an equilibrium point of (1.5).(i)The equilibrium point 𝑥 of (1.5) is called locally stable if, for every 𝜖>0, there exists 𝛿>0 such that, for all 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀∈𝐼 with |𝑥_−2𝑘−𝑥|+|𝑥_−2𝑘+1−𝑥|+⋯+|𝑥₀−𝑥|<𝛿, we have |𝑥_𝑛−𝑥|<𝜖 for all 𝑛≥−2𝑘.(ii)The equilibrium point 𝑥 of (1.5) is called locally asymptotically stable if it is locally stable, and if there exists 𝛾>0 such that for all 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀∈𝐼 with |𝑥_−2𝑘−𝑥|+|𝑥_−2𝑘+1−𝑥|+⋯+|𝑥₀−𝑥|<𝛾, we have lim_𝑛→∞𝑥_𝑛=𝑥.(iii)The equilibrium point 𝑥 of (1.5) is called global attractor if, for every 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀∈𝐼, we have lim_𝑛→∞𝑥_𝑛=𝑥.(iv)The equilibrium point 𝑥 of (1.5) is called globally asymptotically stable if it is locally stable and global attractor.(v)The equilibrium point 𝑥 of (1.5) is called unstable if it is not stable.(vi)The equilibrium point 𝑥 of (1.5) is called source or repeller if there exists 𝑟>0 such that, for all 𝑥_−2𝑘,𝑥_−2𝑘+1,…,𝑥₀∈𝐼 with 0<|𝑥_−2𝑘−𝑥|+|𝑥_−2𝑘+1−𝑥|+⋯+|𝑥₀−𝑥|<𝑟, there exists 𝑁≥1 such that |𝑥_𝑁−𝑥|≥𝑟. Clearly, a repeller is an unstable equilibrium.

Theorem A (linearized stability theorem). The following statements are true.(1)If all roots of (1.8) have modulus less than one, then the equilibrium point 𝑥 of (1.5) is locally asymptotically stable.(2)If at least one of the roots of (1.8) has modulus greater than one, then the equilibrium point 𝑥 of (1.5) is unstable.
The equilibrium point 𝑥 of (1.5) is called a “saddle point” if (1.8) has roots both inside and outside the unit disk.

2. The Special Cases 𝛼𝛽𝛾Σ^𝑘_𝑖=𝑙𝑝_𝑖=0

In this section, we examine the character of solutions of (1.3) when one or more of the parameters in (1.3) are zero.

There are four such equations; namely, 𝑥_𝑛+1𝑥=0,𝑛=0,1,2,…,_𝑛+1=𝛼𝛽𝑥_𝑛−1𝑥,𝑛=0,1,2,…,_𝑛+1=𝛼𝑥𝛽+𝛾^𝑛−1𝑥,𝑛=0,1,2,…,_𝑛+1=𝛼𝑥_𝑛−1𝛾Π^𝑘_𝑖=𝑙𝑥^𝑝𝑖_𝑛−2𝑖,𝑛=0,1,2,…. (2.1)(2.2)(2.3)(2.4)

Equation (2.1) is trivial, (2.2) and (2.3) are linear, and (2.4) is a non-linear difference equation; the change of variables 𝑥_𝑛=𝑒^𝑦𝑛 reduces it to a linear difference equation.

3. A General Oscillation Result

The change of variables 𝑥_𝑛=(𝛽/𝛾)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}𝑦_𝑛 reduces (1.3) to the difference equation 𝑦_𝑛+1=𝑟𝑦_𝑛−11+Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_𝑛−2𝑖,𝑛=0,1,2,…, (3.1) where 𝑟=𝛼/𝛽>0.

Note that 𝑦₁=0 is always an equilibrium point. When 𝑟>1, (3.1) also possesses the unique positive equilibrium 𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}.

Theorem B (see [8]). Assume that 𝐹∈𝐶([0,∞)^2𝑘+1→[0,∞)) is nonincreasing in the odd arguments, and nondecreasing in the even arguments. Let 𝑥 be an equilibrium point of the difference equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−2𝑘,𝑛=0,1,2,…, (3.2) and let {𝑥_𝑛}^∞_{𝑛=−2𝑘} be a solution of (3.2) such that either 𝑥_−2𝑘,𝑥_−2𝑘+2,…,𝑥₀≥𝑥,𝑥_−2𝑘+1,𝑥_−2𝑘+3,…,𝑥₋₁<𝑥, (3.3) or 𝑥_−2𝑘,𝑥_−2𝑘+2,…,𝑥₀<𝑥,𝑥_−2𝑘+1,𝑥_−2𝑘+3,…,𝑥₋₁≥𝑥. (3.4) Then {𝑥_𝑛}^∞_{𝑛=−2𝑘} oscillates about 𝑥 with semicycles of length one.

Corollary 3.1. Assume that 𝑟>1; let {𝑦_𝑛}^∞_{𝑛=−2𝑘} be a solution of (3.1) such that either 𝑦_−2𝑘,𝑦_−2𝑘+2,…,𝑦₀≥𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖},𝑦_−2𝑘+1,𝑦_−2𝑘+3,…,𝑦₋₁<𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}, (3.5) or 𝑦_−2𝑘,𝑦_−2𝑘+2,…,𝑦₀<𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}𝑦_−2𝑘+1,𝑦_−2𝑘+3,…,𝑦₋₁≥𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}. (3.6) Then {𝑦_𝑛}^∞_{𝑛=−2𝑘} oscillates about the positive equilibrium point 𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖} with semicycles of length one.

Proof. The proof follows immediately from Theorem B.

4. The Dynamics of (3.1)

In this section, we investigate the dynamics of (3.1) with nonnegative initial conditions.

Theorem 4.1. For (3.1), we have the following results. (i)Assume that 𝑟<1, then the zero equilibrium point is locally asymptotically stable. (ii)Assume that 𝑟>1, then the zero equilibrium point is saddle point. (iii)The positive equilibrium point 𝑦₂ is unstable.

Proof. The linearized equation associated with (3.1) about 𝑦₁=0 has the form 𝑧_𝑛+1−𝑟𝑧_𝑛−1=0,𝑛=0,1,2,…, (4.1) so, the characteristic equation of (3.1) about 𝑦₁=0 is 𝜆^2𝑘+1−𝑟𝜆^2𝑘−1=0, (4.2) then the proof of (i) and (ii) follows immediately from Theorem A.
The linearized equation of (3.1) about 𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖} is 𝑧_𝑛+1−𝑧_𝑛−1+^𝑘_𝑖=𝑙𝑝_𝑖11−𝑟𝑧_𝑛−2𝑖=0,𝑛=0,1,2,…, (4.3) so, the characteristic equation of (3.1) about 𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖} is 𝜆^2𝑘+1−𝜆^2𝑘−1+^𝑘_𝑖=𝑙𝑝_𝑖11−𝑟𝜆^{2𝑘−2𝑖}=0. (4.4) Set 𝑓(𝜆)=𝜆^2𝑘+1−𝜆^2𝑘−1+^𝑘_𝑖=𝑙𝑝_𝑖11−𝑟𝜆^{2𝑘−2𝑖}, (4.5) then 𝑓(−1)=((Σ^𝑘_𝑖=𝑙𝑝_𝑖)(𝑟−1))/𝑟>0, and lim_{𝜆→−∞}𝑓(𝜆)=−∞ so 𝑓(𝜆) has at least a root in (−∞,−1). Then the proof of (iii) follows.

Theorem 4.2. Assume 𝑟<1. Then the zero equilibrium point of (3.1) is globally asymptotically stable.

Proof. We know by Theorem 4.1 that the equilibrium point 𝑦₁=0 is locally asymptotically stable of (3.1), and so it suffices to show that 𝑦₁=0 is a global atractor of (3.1) as follows: 0≤𝑦_𝑛+1=𝑟𝑦_𝑛−11+Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_𝑛−2𝑖≤𝑟𝑦_𝑛−1, (4.6) since 𝑟<1, then lim_𝑛→∞𝑦_𝑛=0. (4.7)

The next theorem shows that (3.1) has a prime-period two solutions when 𝑟=1.

Theorem 4.3. For (3.1), we have the following results.
 (a) Equation (3.1) possesses the prime-period two solutions …,𝜙,0,𝜙,0,𝜙,… (4.8) with 𝜙>0, when 𝑟=1.
 (b) Assume that 𝑟=1, then every solution of (3.1) converges to a period (not necessarily prime) two solutions (4.8) with 𝜙≥0.

Proof. (a) Let …,𝜙,𝜓,𝜙,𝜓,… (4.9) be period two solutions of (3.1). Then 𝜙=𝑟𝜙1+𝜓^{Σ𝑘𝑖=𝑙𝑝𝑖},𝜓=𝑟𝜓1+𝜙^{Σ𝑘𝑖=𝑙𝑝𝑖}. (4.10) If 𝜙≠0 and 𝜓≠0, then 𝜙=𝜓=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖}, which is impossible. Hence, 𝜓=0 which implies that (𝑟−1)𝜙=0, so 𝑟=1.
(b) Assume that 𝑟=1, and let {𝑦_𝑛}^∞_{𝑛=−2𝑘} be a solution of (3.1), then 𝑦_𝑛+1−𝑦_𝑛−1=−𝑦_𝑛−1Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_𝑛−2𝑖1+Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_𝑛−2𝑖≤0. (4.11) So the even terms of this solution decrease to a limit (say Φ≥0), and the odd terms decrease to a limit (say Ψ≥0). Thus, ΦΦ=1+Ψ^{Σ𝑘𝑖=𝑙𝑝𝑖}Ψ,Ψ=1+Φ^{Σ𝑘𝑖=𝑙𝑝𝑖}, (4.12) which implies that ΦΨ^{Σ𝑘𝑖=𝑙𝑝𝑖}=0,ΨΦ^{Σ𝑘𝑖=𝑙𝑝𝑖}=0. (4.13) This completes the proof.

The next theorem shows that when 𝑟>1, (3.1) possesses unbounded solutions.

Theorem 4.4. Assume 𝑟>1. Then (3.1) possesses unbounded solutions. In particular, every solution of (3.1) which oscillates about the equilibrium 𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖} with semicycles of length one is unbounded.

Proof. we will prove that every solution {𝑦_𝑛}^∞_{𝑛=−2𝑘} of (3.1) which oscillates with semicycles of length one is unbounded (see corollary 3.1).
Assume that {𝑦_𝑛}^∞_{𝑛=−2𝑘} is a solution of (3.1) such that 𝑦_2𝑛+1<𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖},𝑦_2𝑛>𝑦₂=(𝑟−1)^{1/Σ𝑘𝑖=𝑙𝑝𝑖},𝑛≥−𝑘. (4.14) Then 𝑦_2𝑛+2=𝑟𝑦_2𝑛1+Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_{2𝑛+1−2𝑖}>𝑟𝑦_2𝑛1+𝑦^{Σ𝑘𝑖=𝑙𝑝𝑖}₂=𝑦_2𝑛,𝑦_2𝑛+3=𝑟𝑦_2𝑛+11+Π^𝑘_𝑖=𝑙𝑦^𝑝𝑖_{2𝑛+2−2𝑖}<𝑟𝑦_2𝑛+11+𝑦^{Σ𝑘𝑖=𝑙𝑝𝑖}₂=𝑦_2𝑛+1. (4.15) From which it follows that lim_𝑛→∞𝑦_2𝑛=∞,lim_𝑛→∞𝑦_2𝑛+1=0, (4.16) which completes the proof.

Acknowledgments

This paper was supported by King Saud University, Deanship of Scientific Research. The author would like to thank Deanship of Scientific Research, King Saud University, Riyadh, Saudi Arabia, for funding and supporting this research.