Global Attractivity and Periodic Character of Difference Equation of Order Four

Abstract

We investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence 𝑥_𝑛+1=𝑎𝑥_𝑛+((𝑏𝑥_𝑛−1+𝑐𝑥_𝑛−2+𝑑𝑥_𝑛−3)/(𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3)),𝑛=0,1,…, where the parameters 𝑎,𝑏,𝑐,𝑑,𝛼,𝛽, and𝛾are positive real numbers and the initial conditions 𝑥₋₃,𝑥₋₂,𝑥₋₁, and 𝑥₀are positive real numbers.

1. Introduction

Our goal in this paper is to investigate the global stability character, boundedness, and the periodicity of solutions of the recursive sequence 𝑥_𝑛+1=𝑎𝑥_𝑛+𝑏𝑥_𝑛−1+𝑐𝑥_𝑛−2+𝑑𝑥_𝑛−3𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3, (1.1) where the parameters 𝑎,𝑏,𝑐,𝑑,𝛼,𝛽,and𝛾are positive real numbers and the initial conditions 𝑥₋₃,𝑥₋₂,𝑥₋₁, and 𝑥₀are positive real numbers.

Recently there has been a lot of interest in studying the global attractivity, the boundedness character and the periodicity nature of nonlinear difference equations, see for example [1–15].

The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order, recently many researchers have investigated the behavior of the solution of difference equations—for example, in [3] Elabbasy et al. investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence: 𝑥_𝑛+1=𝑎𝑥_𝑛−𝑏𝑥_𝑛𝑐𝑥_𝑛−𝑑𝑥_𝑛−1. (1.2) In [5] Elabbasy and Elsayed investigated the global stability character and boundedness of solutions of the recursive sequence 𝑥_𝑛+1=𝑎𝑥^𝑝_𝑛+𝑏∏^𝑝_𝑟=1𝑥_𝑛−𝑟𝑐𝑥^𝑝_𝑛+𝑑∏^𝑝_𝑟=1𝑥_𝑛−𝑟. (1.3) Elsayed [11] investigated the global character of solutions of the nonlinear, fourth-order, rational difference equation 𝑥_𝑛+1=𝑎𝑥_𝑛−2+𝑏𝑥_𝑛𝑥_𝑛−2𝑐𝑥_𝑛+𝑑𝑥_𝑛−3. (1.4) Saleh and Aloqeili [16] investigated the difference equation 𝑦_𝑛+1𝑦=𝐴+^𝑛𝑦_𝑛−𝑘. (1.5) Yang et al. [17] investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequence 𝑥_𝑛+1=𝑎𝑥_𝑛−1+𝑏𝑥_𝑛−2𝑐+𝑑𝑥_𝑛−1𝑥_𝑛−2. (1.6) For some related work see [16–26].

Here, we recall some basic definitions and some theorems that we need in the sequel.

Let 𝐼 be some interval of real numbers and let 𝐹∶𝐼^𝑘+1→𝐼 (1.7) 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,…, (1.8) has a unique solution {𝑥_𝑛}^∞_𝑛=−𝑘.

Definition 1.1   (Equilibrium Point). A point 𝑥∈𝐼 is called an equilibrium point of (1.8) if 𝑥=𝐹(𝑥,𝑥,…,𝑥). (1.9) That is, 𝑥_𝑛=𝑥 for 𝑛≥0is a solution of (1.8), or equivalently, 𝑥 is a fixed point of 𝑓.

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   (Stability). (i) The equilibrium point 𝑥of (1.8) is locally stable if for every  𝜖>0,  there exists 𝛿>0  such that for all 𝑥_−𝑘,  𝑥_−𝑘+1,…,  𝑥₋₁,𝑥₀∈𝐼with 𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛿, (1.10) we have 𝑥_𝑛−𝑥<𝜖∀𝑛≥−𝑘. (1.11)
(ii) The equilibrium point 𝑥of (1.8) is locally asymptotically stable if 𝑥is locally stable solution of (1.8) and there exists 𝛾>0, such that for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁, 𝑥₀∈𝐼with 𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛾, (1.12) we have lim_𝑛→∞𝑥_𝑛=𝑥. (1.13)
(iii) The equilibrium point 𝑥of (1.8) is a global attractor if for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼,we have lim_𝑛→∞𝑥_𝑛=𝑥. (1.14)
(iv) The equilibrium point 𝑥of (1.8) is globally asymptotically stable if 𝑥is locally stable, and 𝑥is also a global attractor of (1.8).
(v) The equilibrium point 𝑥of (1.8) is unstable if 𝑥is not locally stable.
The linearized equation of (1.8) about the equilibrium 𝑥is the linear difference equation 𝑦_𝑛+1=^𝑘_𝑖=0𝜕𝐹(𝑥,𝑥,…,𝑥)𝜕𝑥_𝑛−𝑖𝑦_𝑛−𝑖. (1.15)

Theorem A (see [18]). Assume that 𝑝_𝑖∈𝑅,𝑖=1,2,…,𝑘and𝑘∈{0,1,2,…}. Then ^𝑘_𝑖=1𝑝_𝑖<1 (1.16) is a sufficient condition for the asymptotic stability of the difference equation 𝑥_𝑛+𝑘+𝑝₁𝑥_{𝑛+𝑘−1}+⋯+𝑝_𝑘𝑥_𝑛=0𝑛=0,1,…. (1.17)

Theorem B (see [19]). Let 𝑔∶[𝑎,𝑏]^𝑘+1→[𝑎,𝑏] be a continuous function, where 𝑘 is a positive integer, and where [𝑎,𝑏] is an interval of real numbers. Consider the difference equation 𝑥_𝑛+1=𝑔𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−𝑘,𝑛=0,1,…. (1.18) Suppose that 𝑔 satisfies the following conditions.(1)For each integer 𝑖 with 1≤𝑖≤𝑘+1; the function 𝑔(𝑧₁,𝑧₂,…,𝑧_𝑘+1) is weakly monotonic in 𝑧_𝑖 for fixed 𝑧₁,𝑧₂,…,𝑧_𝑖−1,𝑧_𝑖+1,…,𝑧_𝑘+1.(2)If (𝑚,𝑀) is a solution of thmhje system 𝑚=𝑔𝑚₁,𝑚₂,…,𝑚_𝑘+1,𝑀=𝑔𝑀₁,𝑀₂,…,𝑀_𝑘+1, (1.19)then 𝑚=𝑀, where for each 𝑖=1,2,…,𝑘+1, we set 𝑚_𝑖=𝑚𝑖𝑓𝑔𝑖𝑠𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑧_𝑖,𝑀𝑖𝑓𝑔𝑖𝑠𝑛𝑜𝑛𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑧_𝑖,𝑀_𝑖=𝑀𝑖𝑓𝑔𝑖𝑠𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑧_𝑖,𝑚𝑖𝑓𝑔𝑖𝑠𝑛𝑜𝑛𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑧_𝑖. (1.20) Then there exists exactly one equilibrium 𝑥 of (1.18), and every solution of (1.18) converges to 𝑥.
The paper proceeds as follows. In Section 2, we show that when ()(),(),()𝛼+𝛽+𝛾𝑏+𝑐+𝑑>max2𝛼𝑐+𝑑−2𝑏𝛽+𝛾2𝛾𝑏+𝑐−2𝑑𝛼+𝛽2𝛽𝑏+𝑑−2𝑐𝛼+𝛾 (1.21) then the equilibrium point of (1.1) is locally asymptotically stable. In Section 3 we prove that the solution is bounded and persists when 𝑎<1 and the solution of (1.1) is unbounded if 𝑎>1. In Section 4 we prove that there exists a period two solution of (1.1). In Section 5 we prove that the equilibrium point of (1.1) is global attractor. Finally we give a numerical examples of some special cases of (1.1) and draw it by using Matlab.

2. Local Stability of the Equilibrium Point of (1.1)

This section deals with study of the local stability character of the equilibrium point of  (1.1).

Equation (1.1) has equilibrium point and is given by 𝑥=𝑎𝑥+𝑏+𝑐+𝑑.𝛼+𝛽+𝛾 (2.1) If 𝑎<1, then the only positive equilibrium point of (1.1) is given by 𝑥=𝑏+𝑐+𝑑().𝛼+𝛽+𝛾1−𝑎 (2.2) Let 𝑓∶(0,∞)⁴→(0,∞) be a continuously differentiable function defined by 𝑓(𝑢,𝑣,𝑤,𝑡)=𝑎𝑢+𝑏𝑣+𝑐𝑤+𝑑𝑡.𝛼𝑣+𝛽𝑤+𝛾𝑡 (2.3) Therefore, it follows that 𝜕𝑓(𝑢,𝑣,𝑤,𝑡)()𝜕𝑢=𝑎,𝜕𝑓𝑢,𝑣,𝑤,𝑡=𝑡𝜕𝑣𝑏𝛽−𝑐𝛼𝑤+𝑏𝛾−𝑑𝛼𝛼𝑣+𝛽𝑤+𝛾𝑡²,𝜕𝑓(𝑢,𝑣,𝑤,𝑡)=−𝑡𝜕𝑤𝑏𝛽−𝑐𝛼𝑣+𝑐𝛾−𝑑𝛽𝛼𝑣+𝛽𝑤+𝛾𝑡²,𝜕𝑓(𝑢,𝑣,𝑤,𝑡)=−𝑤𝜕𝑡𝑏𝛾−𝑑𝛼𝑣−𝑐𝛾−𝑑𝛽𝛼𝑣+𝛽𝑤+𝛾𝑡². (2.4) Then we see that 𝜕𝑓(𝑥,𝑥,𝑥,𝑥)𝜕𝑢=𝑎=−𝑎₃,𝜕𝑓(𝑥,𝑥,𝑥,𝑥)=+𝜕𝑣𝑏𝛽−𝑐𝛼𝑏𝛽−𝑑𝛼𝛼+𝛽+𝛾²𝑥=𝑏𝛽−𝑐𝛼+𝑏𝛾−𝑑𝛼(1−𝑎)𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)=−𝑎₂,𝜕𝑓(𝑥,𝑥,𝑥,𝑥)=−+𝜕𝑤𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾²𝑥=−𝑏𝛽−𝑐𝛼+𝑐𝛾−𝑑𝛽(1−𝑎)𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)=−𝑎₁,𝜕𝑓(𝑥,𝑥,𝑥,𝑥)=−−𝜕𝑡𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾²𝑥=−𝑏𝛾−𝑑𝛼−𝑐𝛾−𝑑𝛽(1−𝑎)𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)=−𝑎₀. (2.5) Then the linearized equation of (1.1) about 𝑥 is 𝑦_𝑛+1+𝑎₃𝑦_𝑛+𝑎₂𝑦_𝑛−1+𝑎₁𝑦_𝑛−2+𝑎₀𝑦_𝑛−3=0, (2.6) whose characteristic equation is 𝜆⁴+𝑎₃𝜆³+𝑎₂𝜆²+𝑎₁𝜆+𝑎₀=0. (2.7)

Theorem 2.1. Assume that ()(),(),()𝛼+𝛽+𝛾𝑏+𝑐+𝑑>max2𝛼𝑐+𝑑−2𝑏𝛽+𝛾2𝛾𝑏+𝑐−2𝑑𝛼+𝛽2𝛽𝑏+𝑑−2𝑐𝛼+𝛾. (2.8) Then the positive equilibrium point of (1.1) is locally asymptotically stable.

Proof. It is follows by Theorem A that (2.6) is asymptotically stable if all roots of (2.7) lie in the open disc |𝜆|<1, that is, if 𝑎₃+𝑎₂+𝑎₁+𝑎₀|𝑎|++)<1.𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼(1−𝑎()+−+)𝛼+𝛽+𝛾𝑏+𝑐+𝑑𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽(1−𝑎()+−−)𝛼+𝛽+𝛾𝑏+𝑐+𝑑𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽(1−𝑎𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)<1, (2.9) and so ++−++−−<).𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼(1−𝑎)𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽(1−𝑎)𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽(1−𝑎)𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)(1−𝑎 (2.10) Dividing the denominator and numerator by (1−𝑎) gives ++−++−−<).𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.11) Suppose that 𝐵₁=+,𝐵𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼₂+,𝐵=−𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽₃=−𝑏𝛾−𝑑𝛼−𝑐𝛾−𝑑𝛽. (2.12) We consider the following cases.
(1)𝐵₁>0,  𝐵₂>0,  and  𝐵₃>0. In this case we see from (2.11) that +−+−−<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.13) if and only if 𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)>0, (2.14) which is always true.
(2)𝐵₁>0,  𝐵₂>0, and  𝐵₃<0.It follows from (2.11) that +−+++<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.15) if and only if 2𝛾(𝑏+𝑐)−2𝑑𝛼+𝛽<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.16) which is satisfied by (2.8).
(3)𝐵₁>0,  𝐵₂<0,  and  𝐵₃>0.  We see from (2.11) that ++−−−<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.17) if and only if 2𝛽(𝑏+𝑑)−2𝑐𝛼+𝛾<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.18) which is satisfied by (2.8).
(4)𝐵₁>0,  𝐵₂<0,  and  𝐵₃<0. It follows from (2.11) that ++−++<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.19) if and only if 2𝑏𝛽+𝛾−2𝛼(𝑐+𝑑)<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.20) which is satisfied by (2.8).
(5)𝐵₁<0,  𝐵₂>0,  and  𝐵₃>0.  We see from (2.11) that −−−+−−<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.21) if and only if 2𝛼(𝑐+𝑑)−2𝑏𝛽+𝛾<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.22) which is satisfied by (2.8).
(6)𝐵₁<0,  𝐵₂>0, and  𝐵₃<0.  It follows from (2.11) that −−−+++<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.23) if and only if 2𝑐𝛼+𝛾−2𝛽(𝑏+𝑑)<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.24) which is satisfied by (2.8).
(7)𝐵₁<0,  𝐵₂<0, and  𝐵₃>0.  We see from (2.11) that −−+−−−<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.25) if and only if 2𝑑𝛼+𝛽−2𝛾(𝑏+𝑐)<𝛼+𝛽+𝛾(𝑏+𝑐+𝑑), (2.26) which is satisfied by (2.8).
(8)𝐵₁<0,  𝐵₂<0,  and  𝐵₃<0.  It follows from (2.11) that −−+−++<),𝑏𝛽−𝑐𝛼𝑏𝛾−𝑑𝛼𝑏𝛽−𝑐𝛼𝑐𝛾−𝑑𝛽𝑏𝛾−𝑑𝛼𝑐𝛾−𝑑𝛽𝛼+𝛽+𝛾(𝑏+𝑐+𝑑 (2.27) if and only if 𝛼+𝛽+𝛾(𝑏+𝑐+𝑑)>0, (2.28) which is always true. The proof is complete.

3. Boundedness of Solutions of (1.1)

Here we study the boundedness nature and persistence of solutions of (1.1).

Theorem 3.1. Every solution of (1.1) is bounded and persists if 𝑎<1.

Proof. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (1.1). It follows from (1.1) that 𝑥_𝑛+1=𝑎𝑥_𝑛+𝑏𝑥_𝑛−1+𝑐𝑥_𝑛−2+𝑑𝑥_𝑛−3𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3=𝑎𝑥_𝑛+𝑏𝑥_𝑛−1𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3+𝑐𝑥_𝑛−2𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3+𝑑𝑥_𝑛−3𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3. (3.1) Then 𝑥_𝑛+1≤𝑎𝑥_𝑛+𝑏𝑥_𝑛−1𝛼𝑥_𝑛−1+𝑐𝑥_𝑛−2𝛽𝑥_𝑛−2+𝑑𝑥_𝑛−3𝛾𝑥_𝑛−3=𝑎𝑥_𝑛+𝑏𝛼+𝑐𝛽+𝑑𝛾∀𝑛≥1. (3.2) By using a comparison, we see that limsup_𝑛→∞𝑥_𝑛≤𝑏𝛽𝛾+𝑐𝛼𝛾+𝑑𝛼𝛽𝛼𝛽𝛾(1−𝑎)=𝑀. (3.3) Thus the solution is bounded.
Now we wish to show that there exists 𝑚>0 such that 𝑥_𝑛≥𝑚∀𝑛≥1. (3.4) The transformation 𝑥_𝑛=1𝑦_𝑛 (3.5) will reduce (1.1) to the equivalent form 1𝑦_𝑛+1=𝑎𝛼𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2+𝑦_𝑛𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2𝑦_𝑛𝛼𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2, (3.6) or 𝑦_𝑛+1=𝑦_𝑛𝛼𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2𝑎𝛼𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2+𝑦_𝑛𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2. (3.7) It follows that 𝑦_𝑛+1≤𝑦_𝑛𝛼𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2𝑦_𝑛𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2=𝛼𝑦_𝑛−2𝑦_𝑛−3𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2+𝛽𝑦_𝑛−1𝑦_𝑛−3𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2+𝛾𝑦_𝑛−1𝑦_𝑛−2𝑏𝑦_𝑛−2𝑦_𝑛−3+𝑐𝑦_𝑛−1𝑦_𝑛−3+𝑑𝑦_𝑛−1𝑦_𝑛−2≤𝛼𝑦_𝑛−2𝑦_𝑛−3𝑏𝑦_𝑛−2𝑦_𝑛−3+𝛽𝑦_𝑛−1𝑦_𝑛−3𝑐𝑦_𝑛−1𝑦_𝑛−3+𝛾𝑦_𝑛−1𝑦_𝑛−2𝑑𝑦_𝑛−1𝑦_𝑛−2=𝛼𝑏+𝛽𝑐+𝛾𝑑=𝛼𝑐𝑑+𝛽𝑏𝑑+𝛾𝑏𝑐𝑏𝑐𝑑=𝐻∀𝑛≥1. (3.8) Thus we obtain 𝑥_𝑛=1𝑦_𝑛≥1𝐻=𝑏𝑐𝑑𝛼𝑐𝑑+𝛽𝑏𝑑+𝛾𝑏𝑐=𝑚∀𝑛≥1. (3.9) From (3.3) and (3.9), we see that 𝑚≤𝑥_𝑛≤𝑀∀𝑛≥1. (3.10) Therefore, every solution of (1.1) is bounded and persists.

Theorem 3.2. Every solution of (1.1) is unbounded if 𝑎>1.

Proof. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (1.1). Then from (1.1) we see that 𝑥_𝑛+1=𝑎𝑥_𝑛+𝑏𝑥_𝑛−1+𝑐𝑥_𝑛−2+𝑑𝑥_𝑛−3𝛼𝑥_𝑛−1+𝛽𝑥_𝑛−2+𝛾𝑥_𝑛−3>𝑎𝑥_𝑛∀𝑛≥1. (3.11) We see that the right-hand side can be written as follows: 𝑦_𝑛+1=𝑎𝑦_𝑛⇒𝑦_𝑛=𝑎^𝑛𝑦₀, (3.12) and this equation is unstable because 𝑎>1, and lim_𝑛→∞𝑦_𝑛=∞. Then by using ratio test {𝑥_𝑛}^∞_𝑛=−3 is unbounded from above.

4. Existence of Periodic Solutions

In this section we study the existence of periodic solutions of (1.1). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two.

Theorem 4.1. Equation (1.1) has positive prime period two solutions if and only if (i)(𝑏+𝑑−𝑐)(𝛼+𝛾−𝛽)(1+𝑎)+4(𝑎𝛽(𝑏+𝑑)+𝑐(𝛼+𝛾))>0,𝛼+𝛾>𝛽,𝑏+𝑑>𝑐.

Proof. First suppose that there exists a prime period two solution …,𝑝,𝑞,𝑝,𝑞,…, (4.1) of (1.1). We will prove that condition (i) holds.
We see from (1.1) that 𝑝=𝑎𝑞+𝑏𝑝+𝑐𝑞+𝑑𝑝𝛼𝑝+𝛽𝑞+𝛾𝑝=𝑎𝑞+𝑒𝑝+𝑐𝑞𝑓𝑝+𝛽𝑞,where𝑒=𝑏+𝑑,𝑓=𝛼+𝛾,𝑞=𝑎𝑝+𝑏𝑞+𝑐𝑝+𝑑𝑞𝛼𝑞+𝛽𝑝+𝛾𝑞=𝑎𝑝+𝑒𝑞+𝑐𝑝.𝑓𝑞+𝛽𝑝 (4.2) Then 𝑓𝑝²+𝛽𝑝𝑞=𝑎𝑓𝑝𝑞+𝑎𝛽𝑞²+𝑒𝑝+𝑐𝑞,𝑓𝑞²+𝛽𝑝𝑞=𝑎𝑓𝑝𝑞+𝑎𝛽𝑝²+𝑒𝑞+𝑐𝑝. (4.3)(4.4) Subtracting (4.3) from (4.4) gives 𝑓𝑝²−𝑞²=−𝑎𝛽𝑝²−𝑞²+(𝑒−𝑐)𝑝−𝑞. (4.5) Since 𝑝≠𝑞, it follows that ()𝑝+𝑞=𝑒−𝑐.𝑓+𝑎𝛽 (4.6) Again, adding (4.3) and (4.4) yields 𝑓𝑝²+𝑞²+2𝛽𝑝𝑞=2𝑎𝑓𝑝𝑞+𝑎𝛽𝑝²+𝑞²+(𝑒+𝑐)𝑝+𝑞 (4.7) or 𝑓−𝑎𝛽𝑝²+𝑞²+2𝛽−𝑎𝑓𝑝𝑞=(𝑒+𝑐)𝑝+𝑞. (4.8) It follows by (4.6), (4.8) and the relation 𝑝²+𝑞²=𝑝+𝑞²−2𝑝𝑞∀𝑝,𝑞∈𝑅 (4.9) that 2()2()𝛽−𝑓1+𝑎𝑝𝑞=𝑒𝑎𝛽+𝑐𝑓𝑒−𝑐𝑓+𝑎𝛽². (4.10) Thus ()𝑝𝑞=𝑒𝑎𝛽+𝑐𝑓𝑒−𝑐𝑓+𝑎𝛽²().𝛽−𝑓1+𝑎 (4.11) Now it is clear from (4.6) and (4.11) that 𝑝 and 𝑞 are the two distinct roots of the quadratic equation 𝑡²−(𝑒−𝑐)()𝑓+𝑎𝛽𝑡+𝑒𝑎𝛽+𝑐𝑓𝑒−𝑐𝑓+𝑎𝛽²()𝑡𝛽−𝑓1+𝑎=0,𝑓+𝑎𝛽²−()()𝑒−𝑐𝑡+𝑒𝑎𝛽+𝑐𝑓𝑒−𝑐𝑓+𝑎𝛽𝛽−𝑓(1+𝑎)=0, (4.12)(4.13) and so [𝑒−𝑐]²−4𝑒𝑎𝛽+𝑐𝑓(𝑒−𝑐)𝛽−𝑓(1+𝑎)>0, (4.14) or [𝑒−𝑐]²+4𝑒𝑎𝛽+𝑐𝑓(𝑒−𝑐)()())𝑓−𝛽1+𝑎>0,𝑒−𝑐𝑓−𝛽(1+𝑎+4𝑒𝑎𝛽+𝑐𝑓>0. (4.15) Therefore, inequality (i) holds.
Second, suppose that inequality (i) is true. We will show that (1.1) has a prime period two solution.
Assume that 𝑝=𝑒−𝑐+𝜁2,𝑓+𝑎𝛽𝑞=𝑒−𝑐−𝜁2,𝑓+𝑎𝛽 (4.16) where 𝜁=[𝑒−𝑐]²−4(𝑒𝑎𝛽+𝑐𝑓)(𝑒−𝑐)/(𝛽−𝑓)(1+𝑎).
We see from inequality (i) that (𝑒−𝑐)𝑓−𝛽(1+𝑎)+4𝑒𝑎𝛽+𝑐𝑓>0,𝑒>𝑐,𝑓>𝛽, (4.17) which is equivalent to (𝑒−𝑐)²>4𝑒𝑎𝛽+𝑐𝑓(𝑒−𝑐)𝛽−𝑓(1+𝑎). (4.18) Therefore,  𝑝  and 𝑞 are distinct real numbers.
Set 𝑥₋₃=𝑝,𝑥₋₂=𝑞,𝑥₋₁=𝑝,𝑥₀=𝑞. (4.19) We wish to show that 𝑥₁=𝑥₋₁=𝑝,𝑥₂=𝑥₀=𝑞. (4.20) It follows from (1.1) that 𝑥₁=𝑎𝑞+𝑏𝑝+𝑐𝑞+𝑑𝑝𝛼𝑝+𝛽𝑞+𝛾𝑝=𝑎𝑞+𝑒𝑝+𝑐𝑞𝑓𝑝+𝛽𝑞=𝑎𝑒−𝑐−𝜁2+𝑒()()𝑓+𝑎𝛽𝑒−𝑐+𝜁/2𝑓+𝑎𝛽+𝑐𝑒−𝑐−𝜁/2𝑓+𝑎𝛽𝑓)).(𝑒−𝑐+𝜁/2𝑓+𝑎𝛽+𝛽(𝑒−𝑐−𝜁/2𝑓+𝑎𝛽 (4.21) Dividing the denominator and numerator by 2(𝑓+𝑎𝛽)gives 𝑥₁=𝑎𝑒−𝑎𝑐−𝑎𝜁2+𝑒()()𝑓+𝑎𝛽𝑒−𝑐+𝜁+𝑐𝑒−𝑐−𝜁𝑓()()=𝑒−𝑐+𝜁+𝛽𝑒−𝑐−𝜁𝑎𝑒−𝑎𝑐−𝑎𝜁2+()𝑓+𝑎𝛽𝑒−𝑐)(𝑒+𝑐+𝜁()+𝜁.𝑓+𝛽𝑒−𝑐𝑓−𝛽 (4.22) Multiplying the denominator and numerator of the right side by (𝑓+𝛽)(𝑒−𝑐)−(𝑓−𝛽)𝜁 gives 𝑥₁=𝑎𝑒−𝑎𝑐−𝑎𝜁2+()()−𝜁𝑓+𝑎𝛽𝑒−𝑐)(𝑒+𝑐+𝜁𝑓+𝛽𝑒−𝑐𝑓−𝛽()+𝜁()−𝜁=𝑓+𝛽𝑒−c𝑓−𝛽𝑓+𝛽𝑒−𝑐𝑓−𝛽𝑎𝑒−𝑎𝑐−𝑎𝜁2+(𝑒𝑓+𝑎𝛽𝑒−𝑐)𝑓+𝛽²−𝑐²+𝜁𝑓+𝛽(𝑒−𝑐)−𝑓−𝛽(𝑒+𝑐)−𝑓−𝛽𝜁²𝑓+𝛽²(𝑒−𝑐)²−𝑓−𝛽²𝜁²=𝑎𝑒−𝑎𝑐−𝑎𝜁2+(𝑒𝑓+𝑎𝛽𝑒−𝑐)𝑓+𝛽²−𝑐²+2𝜁𝛽𝑒−𝑐𝑓−𝑓−𝛽𝒜𝑓+𝛽²(𝑒−𝑐)²−𝑓−𝛽²𝒜=𝑎𝑒−𝑎𝑐−𝑎𝜁2+(𝑒𝑓+𝑎𝛽𝑒−𝑐)𝑓+𝛽²−𝑐²+2𝜁𝛽𝑒−𝑐𝑓−ℬ𝑓+𝛽²(𝑒−𝑐)²−𝑓−𝛽²𝒜=𝑎𝑒−𝑎𝑐−𝑎𝜁2+(2(/(𝑓+𝑎𝛽𝑒−𝑐)𝑒−𝑐)𝑓𝑐+𝛽𝑒−2𝑒𝑎𝛽+𝑐𝑓1+𝑎)+2𝜁𝛽𝑒−𝑐𝑓4(()+/(,𝑒−𝑐)𝛽𝑓𝑒−𝑐𝛽−𝑓𝑒𝑎𝛽+𝑐𝑓1+𝑎) (4.23) where 𝒜 denotes to [𝑒−𝑐]²−4(𝑒𝑎𝛽+𝑐𝑓)(𝑒−𝑐)/(𝛽−𝑓)(1+𝑎) and ℬ  denotes to (𝑓−𝛽)(𝑒−𝑐)²+4(𝑒𝑎𝛽+𝑐𝑓)(𝑒−𝑐)/(1+𝑎).
Multiplying the denominator and numerator of the right side by (1+𝑎), we obtain𝑥₁=𝑎𝑒−𝑎𝑐−𝑎𝜁2+(()()𝑓+𝑎𝛽𝑒−𝑐)𝑓𝑐+𝛽𝑒1+𝑎−2𝑒𝑎𝛽+𝑐𝑓+𝜁1+𝑎𝛽𝑒−𝑐𝑓2()+=𝛽𝑓𝑒−𝑐)(1+𝑎𝛽−𝑓𝑒𝑎𝛽+𝑐𝑓𝑎𝑒−𝑎𝑐−𝑎𝜁2+())()𝑓+𝑎𝛽𝑒−𝑐𝛽𝑒−𝑓𝑐(1−𝑎+𝜁1+𝑎𝛽𝑒−𝑐𝑓2()+=𝛽𝑓𝑒−𝑐)(1+𝑎𝛽−𝑓𝑒𝑎𝛽+𝑐𝑓𝑎𝑒−𝑎𝑐−𝑎𝜁2+)(𝑓+𝑎𝛽𝛽𝑒−𝑓𝑐(𝑒−𝑐)(1−𝑎+𝜁1+𝑎)2=𝛽𝑒−𝑐𝑓𝑓+𝑎𝛽𝑎𝑒−𝑎𝑐−𝑎𝜁2+()()𝑓+𝑎𝛽𝑒−𝑐)(1−𝑎+𝜁1+𝑎2=()()𝑓+𝑎𝛽𝑎𝑒−𝑎𝑐−𝑎𝜁+𝑒−𝑐)(1−𝑎+𝜁1+𝑎2=𝑓+𝑎𝛽𝑒−𝑐+𝜁2𝑓+𝑎𝛽=𝑝. (4.24) Similarly as before one can easily show that 𝑥₂=𝑞. (4.25) Then it follows by induction that 𝑥_2𝑛=𝑞,𝑥_2𝑛+1=𝑝∀𝑛≥−1. (4.26) Thus (1.1) has the prime period two solution …,𝑝,𝑞,𝑝,𝑞,…, (4.27) where 𝑝 and 𝑞 are the distinct roots of the quadratic equation (4.13) and the proof is complete.

5. Global Attractor of the Equilibrium Point of (1.1)

In this section, we investigate the global asymptotic stability of (1.1).

Lemma 5.1. For any values of the quotient 𝑏/𝛼,𝑐/𝛽,  and𝑑/𝛾,the function 𝑓(𝑢,𝑣,𝑤,𝑡)defined by (2.3) has the monotonicity behavior in its three arguments.

Proof. The proof follows by some computations and it will be omitted.

Remark 5.2. It follows from (1.1), when𝑏/𝛼=𝑐/𝛽=𝑑/𝛾, that 𝑥_𝑛+1=𝑎𝑥_𝑛+𝜆∀𝑛≥−3andforsomeconstant𝜆. (5.1)
Whenever the quotients 𝛼/𝐴,  𝛽/𝐵,and𝛾/𝐶 are not equal, we get the following result.

Theorem 5.3. The equilibrium point 𝑥 is a global attractor of (1.1) if one of the following statements holds: (1)𝑏𝛼≥𝑐𝛽≥𝑑𝛾(2)𝑏,𝑑≥𝑏+𝑐,𝛼≥𝑑𝛾≥𝑐𝛽(3)𝑐,𝑐≥𝑏+𝑑,𝛽≥𝑏𝛼≥𝑑𝛾(4)𝑐,𝑑≥𝑏+𝑐,𝛽≥𝑑𝛾≥𝑏𝛼(5)𝑑,𝑏≥𝑐+𝑑,𝛾≥𝑐𝛽≥𝑏𝛼(6)𝑑,𝑏≥𝑐+𝑑,𝛾≥𝑏𝛼≥𝑐𝛽,𝑐≥𝑏+𝑑. (5.2)

Proof. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (1.1) and again let 𝑓be a function defined by (2.3).
We will prove the theorem when case (1.1) is true, and the proof of the other cases is similar and so will be omitted.

Assume that (5.2) is true, then it is easy from the equations after (2.3) to see that the function 𝑓(𝑢,𝑣,𝑤,𝑡)is nondecreasing in 𝑢,𝑣 and nonincreasing in 𝑡 and it is not clear what is going on with 𝑤. So we consider the following two cases.

Case 1. Assume that the function 𝑓(𝑢,𝑣,𝑤,𝑡)is nondecreasing in 𝑤.
Suppose that (𝑚,𝑀)is a solution of the system 𝑀=𝑓(𝑀,𝑀,𝑀,𝑚)  and  𝑚=𝑔(𝑚,𝑚,𝑚,𝑀). Then from (1.1), we see that 𝑀=𝑎𝑀+𝑏𝑀+𝑐𝑀+𝑑𝑚𝛼𝑀+𝛽𝑀+𝛾𝑚,𝑚=𝑎𝑚+𝑏𝑚+𝑐𝑚+𝑑𝑀,𝛼𝑚+𝛽𝑚+𝛾𝑀 (5.3) or 𝑀(1−𝑎)=𝑏𝑀+𝑐𝑀+𝑑𝑚()=𝛼𝑀+𝛽𝑀+𝛾𝑚,𝑚1−𝑎𝑏𝑚+𝑐𝑚+𝑑𝑀.𝛼𝑚+𝛽𝑚+𝛾𝑀 (5.4) Then 𝛼+𝛽(1−𝑎)𝑀²()()()𝑚+𝛾1−𝑎𝑀𝑚=𝑏+𝑐𝑀+𝑑𝑚,𝛼+𝛽1−𝑎²+𝛾(1−𝑎)𝑀𝑚=(𝑏+𝑐)𝑚+𝑑𝑀. (5.5) Subtracting this two equations, we obtain (𝑀−𝑚)𝛼+𝛽(1−𝑎)(𝑀+𝑚)+(𝑑−𝑏−𝑐)=0, (5.6) under the conditions 𝑑≥𝑏+𝑐,  𝑎<1,we see that 𝑀=𝑚. (5.7) It follows by Theorem B that 𝑥 is a global attractor of (1.1) and then the proof is complete.

Case 2. Assumes that the function 𝑓(𝑢,𝑣,𝑤,𝑡)is nonincreasing in 𝑤.
Suppose that (𝑚,𝑀)is a solution of the system 𝑀=𝑓(𝑀,𝑀,𝑚,𝑚)  and  𝑚=𝑔(𝑚,𝑚,𝑀,𝑀). Then from (1.1), we see that 𝑀=𝑎𝑀+𝑏𝑀+𝑐𝑀+𝑑𝑚𝛼𝑀+𝛽𝑀+𝛾𝑚,𝑚=𝑎𝑚+𝑏𝑚+𝑐𝑚+𝑑𝑀,𝛼𝑚+𝛽𝑚+𝛾𝑀 (5.8) or 𝑀(1−𝑎)=𝑏𝑀+𝑐𝑚+𝑑𝑚()=𝛼𝑀+𝛽𝑚+𝛾𝑚,𝑚1−𝑎𝑏𝑚+𝑐𝑀+𝑑𝑀.𝛼𝑚+𝛽𝑀+𝛾𝑀 (5.9) Then 𝛼(1−𝑎)𝑀²+()()𝛼()𝑚𝛽+𝛾1−𝑎𝑀𝑚=𝑏𝑀+𝑐+𝑑𝑚,1−𝑎²+𝛽+𝛾(1−𝑎)𝑀𝑚=𝑏𝑚+(𝑐+𝑑)𝑀. (5.10) Subtracting these two equations we obtain (𝑀−𝑚){𝛼(1−𝑎)(𝑀+𝑚)+(𝑐+𝑑−𝑏)}=0. (5.11) Under the conditions 𝑑≥𝑏+𝑐,  𝑎<1,we see that 𝑀=𝑚. (5.12) It follows by Theorem B that 𝑥 is a global attractor of (1.1) and then the proof is complete.

6. Numerical Examples

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to (1.1).

Example 6.1. We assume 𝑥₋₃=2,𝑥₋₂=3,𝑥₋₁=8,𝑥₀=5,𝑎=0.6,𝑏=7,𝑐=3,𝑑=9,  𝛼=3.8,𝛽=0.2,  and  𝛾=1.2. See Figure 1.

Figure 1 

Example 6.2 (see [Figure 2]). Since 𝑥₋₃=5,𝑥₋₂=6,𝑥₋₁=3,𝑥₀=4,𝑎=1.3,𝑏=5,𝑐=3,𝑑=8,  𝛼=6,𝛽=4,  and   𝛾=7.

Figure 2 

Example 6.3. We consider 𝑥₋₃=5,𝑥₋₂=7,𝑥₋₁=11,𝑥₀=3,  𝑎=5,𝑏=2,  𝑐=3,  𝑑=4,𝛼=3,  𝛽=2,  and  𝛾=7. See Figure 3.

Figure 3 

Example 6.4 (see [Figure 4]). Since 𝑥₋₃=7,𝑥₋₂=2,𝑥₋₁=3,𝑥₀=5,𝑎=0.1,𝑏=2,𝑐=5,𝑑=1.4,𝛼=1.3,𝛽=2,and𝛾=3.

Figure 4 

Example 6.5 (see [Figure 5]). It shows the solutions when𝑥₋₃=𝑥₋₁=𝑝,𝑥₋₂=𝑥₀=𝑞,𝑎=0.6,𝑏=7,𝑐=3,𝑑=9,𝛼=3.8,𝛽=0.2,𝛾=1.2()±Since𝑝,𝑞=𝑒−𝑐[𝑒−𝑐]²−4𝑒𝑎𝛽+𝑐𝑓(𝑒−𝑐)/𝛽−𝑓(1+𝑎)2𝑓+𝑎𝛽. (6.1)

Figure 5