Solution and Attractivity for a Rational Recursive Sequence

Abstract

This paper is concerned with the behavior of solution of the nonlinear difference equation 𝑥_𝑛+1=𝑎𝑥_𝑛−1+𝑏𝑥_𝑛𝑥_𝑛−1/(𝑐𝑥_𝑛+𝑑𝑥_𝑛−2), 𝑛=0,1,…, where the initial conditions 𝑥₋₂, 𝑥₋₁, 𝑥₀ are arbitrary positive real numbers and 𝑎,𝑏,𝑐,𝑑 are positive constants. Also, we give specific form of the solution of four special cases of this equation.

1. Introduction

In this paper we deal with the behavior of the solution of the following difference equation:𝑥_𝑛+1=𝑎𝑥_𝑛−1+𝑏𝑥_𝑛𝑥_𝑛−1𝑐𝑥_𝑛+𝑑𝑥_𝑛−2,𝑛=0,1,…, (1.1) where the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀ are arbitrary positive real numbers and 𝑎,𝑏,𝑐,𝑑 are positive constants. Also, we obtain the solution of some special cases of (1.1).

Let us introduce some basic definitions and some theorems that we need in the sequel.

Let 𝐼 be some interval of real numbers and let𝑓∶𝐼^𝑘+1⟶𝐼, (1.2) be a continuously differentiable function. Then for every set of initial conditions 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₀∈𝐼, the difference equation𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−𝑘,𝑛=0,1,…, (1.3) has a unique solution {𝑥_𝑛}^∞_𝑛=−𝑘 [1].

Definition 1.1 (equilibrium point). A point 𝑥∈𝐼 is called an equilibrium point of (1.3) if 𝑥=𝑓(𝑥,𝑥,…,𝑥). (1.4) That is, 𝑥_𝑛=𝑥 for 𝑛≥0,is a solution of (1.3), or equivalently, 𝑥 is a fixed point of 𝑓.

Definition 1.2 (stability). (i) The equilibrium point 𝑥of (1.3) is locally stable if for every 𝜖>0, there exists 𝛿>0such that for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼 with 𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛿, (1.5) we have 𝑥_𝑛−𝑥<𝜖∀𝑛≥−𝑘. (1.6)
(ii) The equilibrium point 𝑥of (1.3) is locally asymptotically stable if 𝑥is locally stable solution of (1.3) and there exists 𝛾>0, such that for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼 with 𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛾, (1.7) we have lim_𝑛→∞𝑥_𝑛=𝑥. (1.8)
(iii) The equilibrium point 𝑥 of (1.3) is global attractor if for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼, we have lim_𝑛→∞𝑥_𝑛=𝑥. (1.9)
(iv) The equilibrium point 𝑥of (1.3) is globally asymptotically stable if 𝑥is locally stable, and 𝑥is also a global attractor of (1.3).
(v) The equilibrium point 𝑥of (1.3) is unstable if 𝑥is not locally stable.The linearized equation of (1.3) about the equilibrium 𝑥is the linear difference equation 𝑦_𝑛+1=^𝑘_𝑖=0𝜕𝑓(𝑥,𝑥,…,𝑥)𝜕𝑥_𝑛−𝑖𝑦_𝑛−𝑖. (1.10)

Theorem A (see [2]). Assume that 𝑝,𝑞∈𝑅 and 𝑘∈{0,1,2,…}. Then 𝑝+𝑞<1, (1.11) is a sufficient condition for the asymptotic stability of the difference equation 𝑥_𝑛+1+𝑝𝑥_𝑛+𝑞𝑥_𝑛−𝑘=0,𝑛=0,1,…. (1.12)

Remark 1.3. Theorem A can be easily extended to a general linear equations of the form 𝑥_n+𝑘+𝑝₁𝑥_{𝑛+𝑘−1}+⋯+𝑝_𝑘𝑥_𝑛=0,𝑛=0,1,…, (1.13) where 𝑝₁,𝑝₂,…,𝑝_𝑘∈𝑅 and 𝑘∈{1,2,…}. Then (1.13) is asymptotically stable provided that ^𝑘_𝑖=1𝑝_𝑖<1. (1.14)

Consider the following equation 𝑥_𝑛+1=𝑔𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−2. (1.15) The following theorem will be useful for the proof of our results in this paper.

Theorem B (see [1]). Let [𝑎,𝑏] be an interval of real numbers and assume that 𝑔∶[𝑎,𝑏]³⟶[𝑎,𝑏], (1.16) is a continuous function satisfying the following properties:(a)𝑔(𝑥,𝑦,𝑧) is nondecreasing in 𝑥 and 𝑦 in [𝑎,𝑏] for each 𝑧∈[𝑎,𝑏], and is nonincreasing in 𝑧∈[𝑎,𝑏] for each 𝑥 and 𝑦 in [𝑎,𝑏];(b)if (𝑚,𝑀)∈[𝑎,𝑏]×[𝑎,𝑏] is a solution of the system𝑀=𝑔(𝑀,𝑀,𝑚),𝑚=𝑔(𝑚,𝑚,𝑀), (1.17) then 𝑚=𝑀. (1.18) Then (1.15) has a unique equilibrium 𝑥∈[𝑎,𝑏] and every solution of (1.15) converges to 𝑥.

Definition 1.4 (periodicity). A sequence {𝑥_𝑛}^∞_𝑛=−𝑘is said to be periodic with period 𝑝 if 𝑥_𝑛+𝑝=𝑥_𝑛for all 𝑛≥−𝑘.

Definition 1.5 (Fibonacci sequence). The sequence {𝐹_𝑚}^∞_𝑚=0={1,2,3,5,8,13,…}, that is, 𝐹_𝑚=𝐹_𝑚−1+𝐹_𝑚−2,𝑚≥0,𝐹₋₂=0,𝐹₋₁=1 is called Fibonacci sequence.

Recently there has been a great interest in studying the qualitative properties of rational difference equations. Some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results for rational difference equations.

However, there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one so far. From the known work, one can see that it is extremely difficult to understand thoroughly the global behaviors of solutions of rational difference equations although they have simple forms (or expressions). One can refer to [3–23] for examples to illustrate this. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Many researchers have investigated the behavior of the solution of difference equations, for example, Aloqeili [24] has obtained the solutions of the difference equation𝑥_𝑛+1=𝑥_𝑛−1𝑎−𝑥_𝑛𝑥_𝑛−1. (1.19) Amleh et al. [25] studied the dynamics of the difference equation𝑥_𝑛+1=𝑎+𝑏𝑥_𝑛−1𝐴+𝐵𝑥_𝑛−2. (1.20)Çinar [26, 27] got the solutions of the following difference equation𝑥_𝑛+1=𝑥_𝑛−1±1+𝑎𝑥_𝑛𝑥_𝑛−1. (1.21) In [28], Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence𝑥_𝑛+1=𝑎𝑥_𝑛−𝑏𝑥_𝑛𝑐𝑥_𝑛−𝑑𝑥_𝑛−1. (1.22) Elabbasy et al. [29] investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the difference equation𝑥_𝑛+1=𝛼𝑥_𝑛−𝑘𝛽+𝛾∏^𝑘_𝑖=0𝑥_𝑛−𝑖. (1.23) In [30], Ibrahim got the form of the solution of the rational difference equation 𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−2𝑥_𝑛−1𝑎+𝑏𝑥_𝑛𝑥_𝑛−2. (1.24) Karatas et al. [31] got the solution of the difference equation𝑥_𝑛+1=𝑥_𝑛−51+𝑥_𝑛−2𝑥_𝑛−5. (1.25) Yalçinkaya and Çinar [32] considered the dynamics of the difference equation𝑥_𝑛+1=𝑎𝑥_𝑛−𝑘𝑏+𝑐𝑥^𝑝_𝑛. (1.26) Yang [33] investigated the global asymptotic stability of the difference equation 𝑥_𝑛+1=𝑥_𝑛−1𝑥_𝑛−2+𝑥_𝑛−3+𝑎𝑥_𝑛−1+𝑥_𝑛−2𝑥_𝑛−3+𝑎. (1.27) See also [1, 2, 30, 31, 34–40]. Other related results on rational difference equations can be found in [32, 33, 41–48].

2. Local Stability of (1.1)

In this section we investigate the local stability character of the solutions of (1.1). Equation (1.1) has a unique equilibrium point and is given by𝑥=𝑎𝑏𝑥+𝑥²𝑐𝑥+𝑑𝑥, (2.1) or𝑥²(1−𝑎)(𝑐+𝑑)=𝑏𝑥², (2.2) if (𝑐+𝑑)(1−𝑎)≠𝑏, then the unique equilibrium point is 𝑥=0.

Let 𝑓∶(0,∞)³→(0,∞) be a function defined by𝑓(𝑢,𝑣,𝑤)=𝑎𝑣+𝑏𝑢𝑣𝑐𝑢+𝑑𝑤. (2.3) Therefore it follows that 𝑓_𝑢(𝑢,𝑣,𝑤)=𝑏𝑑𝑣𝑤(𝑐𝑢+𝑑𝑤)²,𝑓_𝑣(𝑢,𝑣,𝑤)=𝑎+𝑏𝑢𝑐𝑢+𝑑𝑤,𝑓_𝑤(𝑢,𝑣,𝑤)=−𝑏𝑑𝑢𝑣(𝑐𝑢+𝑑𝑤)², (2.4) we see that𝑓_𝑢(𝑥,𝑥,𝑥)=𝑏𝑑(𝑐+𝑑)²,𝑓_𝑣(𝑥,𝑥,𝑥)𝑏=𝑎+𝑐+𝑑,𝑓_𝑤(𝑥,𝑥,𝑥)=−𝑏𝑑(𝑐+𝑑)². (2.5) The linearized equation of (1.1) about 𝑥 is𝑦_𝑛+1−𝑏𝑑(𝑐+𝑑)²𝑦_𝑛−𝑏𝑎+𝑦𝑐+𝑑^𝑛−1+𝑏𝑑(𝑐+𝑑)²𝑦_𝑛−1=0. (2.6)

Theorem 2.1. Assume that 𝑏(𝑐+3𝑑)<(1−𝑎)(𝑐+𝑑)². (2.7) Then the equilibrium point of (1.1) is locally asymptotically stable.

Proof. It follows from Theorem A that (2.6) is asymptotically stable if 𝑏𝑑(𝑐+𝑑)²+𝑏𝑎++𝑐+𝑑𝑏𝑑(𝑐+𝑑)²<1, (2.8) or 𝑏𝑎++𝑐+𝑑2𝑏𝑑(𝑐+𝑑)²<1, (2.9) and so, 𝑏(𝑐+3𝑑)(𝑐+𝑑)²<(1−𝑎). (2.10) The proof is complete.

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

In this section we investigate the global attractivity character of solutions of (1.1).

Theorem 3.1. The equilibrium point 𝑥 of (1.1) is global attractor if 𝑐(1−𝑎)≠𝑏.

Proof. Let 𝑝,𝑞 be real numbers and assume that𝑔∶[𝑝,𝑞]³→[𝑝,𝑞] is a function defined by 𝑔(𝑢,𝑣,𝑤)=𝑎𝑣+𝑏𝑢𝑣/(𝑐𝑢+𝑑𝑤), then we can easily see that the function 𝑔(𝑢,𝑣,𝑤) is increasing in 𝑢,𝑣 and decreasing in 𝑤.Suppose that (𝑚,𝑀)is a solution of the system 𝑀=𝑔(𝑀,𝑀,𝑚),𝑚=𝑔(𝑚,𝑚,𝑀). (3.1) Then from (1.1), we see that 𝑀=𝑎𝑀+𝑏𝑀²𝑐𝑀+𝑑𝑚,𝑚=𝑎𝑚+𝑏𝑚²𝑐𝑚+𝑑𝑀, (3.2) or 𝑀(1−𝑎)=𝑏𝑀²()=𝑐𝑀+𝑑𝑚,𝑚1−𝑎𝑏𝑚²𝑐𝑚+𝑑𝑀, (3.3) then 𝑑(1−𝑎)𝑀𝑚+𝑐(1−𝑎)𝑀²=𝑏𝑀²,𝑑(1−𝑎)𝑀𝑚+𝑐(1−𝑎)𝑚²=𝑏𝑚², (3.4) subtracting, we obtain 𝑐(1−𝑎)𝑀²−𝑚²=𝑏𝑀²−𝑚²,𝑐(1−𝑎)≠𝑏. (3.5) Thus 𝑀=𝑚. (3.6) It follows from Theorem B that 𝑥 is a global attractor of (1.1), and then the proof is complete.

4. Boundedness of Solutions of (1.1)

In this section we study the boundedness of solutions of (1.1).

Theorem 4.1. Every solution of (1.1) is bounded if (𝑎+𝑏/𝑐)<1.

Proof. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (1.1). It follows from (1.1) that 𝑥_𝑛+1=𝑎𝑥_𝑛−1+𝑏𝑥_𝑛𝑥_𝑛−1𝑐𝑥_𝑛+𝑑𝑥_𝑛−2≤𝑎𝑥_𝑛−1+𝑏𝑥_𝑛𝑥_𝑛−1𝑐𝑥_𝑛=𝑏𝑎+𝑐𝑥_𝑛−1. (4.1) Then 𝑥_𝑛+1≤𝑥_𝑛−1∀𝑛≥0. (4.2) Then the subsequences {𝑥_2𝑛−1}^∞_𝑛=0, {𝑥_2𝑛}^∞_𝑛=0 are decreasing and so are bounded from above by 𝑀=max{𝑥₋₂,𝑥₋₁,𝑥₀}.

5. Special Cases of (1.1)

Our goal in this section is to find a specific form of the solutions of some special cases of (1.1) when 𝑎,𝑏,𝑐, and 𝑑 are integers and give numerical examples of each case and draw it by using MATLAB 6.5.

5.1. On the Difference Equation 𝑥_𝑛+1=𝑥_𝑛−1+𝑥_𝑛𝑥_𝑛−1/(𝑥_𝑛+𝑥_𝑛−2)

In this subsection we study the following special case of (1.1): 𝑥_𝑛+1=𝑥_𝑛−1+𝑥_𝑛𝑥_𝑛−1𝑥_𝑛+𝑥_𝑛−2, (5.1) where the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀are arbitrary positive real numbers.

Theorem 5.1. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (5.1). Then for 𝑛=0,1,2,…𝑥_2𝑛−1=𝑘^𝑛−1_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟,𝑥_2𝑛=ℎ^𝑛−1_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟, (5.2) where𝑥₋₂=𝑟,𝑥₋₁=𝑘,𝑥₀=ℎ,{𝐹_𝑚}^∞_𝑚=0={0,1,1,2,3,5,8,13,…}.

Proof. For 𝑛=0 the result holds. Now suppose that 𝑛>0 and that our assumption holds for 𝑛−1,𝑛−2. That is, 𝑥_2𝑛−3=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟,𝑥_2𝑛−2=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟,𝑥_2𝑛−4=ℎ^𝑛−3_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟. (5.3) Now, it follows from (5.1) that 𝑥_2𝑛−1=𝑥_2𝑛−3+𝑥_2𝑛−2𝑥_2𝑛−3𝑥_2𝑛−2+𝑥_2𝑛−4=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟+ℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝑘∏^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟+ℎ∏^𝑛−3_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟+𝐹_4𝑛−3ℎ+𝐹_4𝑛−4𝑟/𝐹_4𝑛−4ℎ+𝐹_4𝑛−5𝑟𝑘∏^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟𝐹_4𝑛−3ℎ+𝐹_4𝑛−4𝑟/𝐹_4n−4ℎ+𝐹_4𝑛−5𝑟+1=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟+𝐹_4𝑛−3ℎ+𝐹_4𝑛−4𝑟𝑘∏^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟𝐹_4𝑛−3ℎ+𝐹_4𝑛−4𝑟+𝐹_4𝑛−4ℎ+𝐹_4𝑛−5𝑟=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟+𝐹_4𝑛−3ℎ+𝐹_4𝑛−4𝑟𝑘∏^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟𝐹1+^4𝑛−3ℎ+𝐹_4𝑛−4𝑟𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟=𝑘^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟𝐹_4𝑛−1ℎ+𝐹_4𝑛−2𝑟𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟. (5.4) Therefore 𝑥_2𝑛−1=𝑘^𝑛−1_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟. (5.5) Also, we see from (5.1) that 𝑥_2𝑛=𝑥_2𝑛−2+𝑥_2𝑛−1𝑥_2𝑛−2𝑥_2𝑛−1+𝑥_2𝑛−3=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟+𝑘∏^𝑛−1_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝑘∏^𝑛−1_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟+𝑘∏^𝑛−2_𝑖=0𝐹_4𝑖+3ℎ+𝐹_4𝑖+2𝑟/𝐹_4𝑖+2ℎ+𝐹_4𝑖+1𝑟=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟+𝐹_4𝑛−1ℎ+𝐹_4𝑛−2𝑟/𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟ℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝐹_4𝑛−1ℎ+𝐹_4𝑛−2𝑟/𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟+1=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟+𝐹_4𝑛−1ℎ+𝐹_4𝑛−2rℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝐹_4𝑛−1ℎ+𝐹_4𝑛−2𝑟+𝐹_4𝑛−2ℎ+𝐹_4𝑛−3𝑟=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟+𝐹_4𝑛−1ℎ+𝐹_4𝑛−2𝑟ℎ∏^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟/𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝐹_4𝑛ℎ+𝐹_4𝑛−1𝑟=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝐹1+^4𝑛−1ℎ+𝐹_4𝑛−2𝑟𝐹_4𝑛ℎ+𝐹_4𝑛−1𝑟=ℎ^𝑛−2_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟𝐹_4𝑛+1ℎ+𝐹_4𝑛𝑟𝐹_4𝑛ℎ+𝐹_4𝑛−1𝑟. (5.6) Thus 𝑥_2𝑛=ℎ^𝑛−1_𝑖=0𝐹_4𝑖+5ℎ+𝐹_4𝑖+4𝑟𝐹_4𝑖+4ℎ+𝐹_4𝑖+3𝑟. (5.7) Hence, the proof is completed.

For confirming the results of this section, we consider numerical example for 𝑥₋₂=7,𝑥₋₁=6,𝑥₀=3.(See Figure 1).

Figure 1 

5.2. On the Difference Equation 𝑥_𝑛+1=𝑥_𝑛−1+𝑥_𝑛𝑥_𝑛−1/(𝑥_𝑛−𝑥_𝑛−2)

In this subsection we give a specific form of the solutions of the difference equation 𝑥_𝑛+1=𝑥_𝑛−1+𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−2, (5.8) where the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀are arbitrary positive real numbers with 𝑥₋₂≠𝑥₀.

Theorem 5.2. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (5.8). Then for 𝑛=0,1,2,…𝑥_2𝑛−1=𝑘^𝑛−1_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟,𝑥_2𝑛=ℎ^𝑛−1_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟, (5.9) where 𝑥₋₂=𝑟,𝑥₋₁=𝑘,𝑥₀=ℎ,{𝐹_𝑚}^∞_𝑚=−1={1,0,1,1,2,3,5,8,13,…}.

Proof. For 𝑛=0 the result holds. Now suppose that 𝑛>0 and that our assumption holds for 𝑛−1,𝑛−2. That is, 𝑥_2𝑛−3=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟,𝑥_2𝑛−2=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟,𝑥_2𝑛−4=ℎ^𝑛−3_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟. (5.10) Now, it follows from (5.8) that 𝑥_2𝑛−1=𝑥_2𝑛−3+𝑥_2𝑛−2𝑥_2𝑛−3𝑥_2𝑛−2−𝑥_2𝑛−4=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟+ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−F_2𝑖𝑟𝑘∏^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟−ℎ∏^𝑛−3_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟+𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟/𝐹_2𝑛−2ℎ−𝐹_2𝑛−4𝑟𝑘∏^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟/𝐹_2𝑛−2ℎ−𝐹_2𝑛−4𝑟−1=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟+𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟𝑘∏^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟𝐹1+^2𝑛ℎ−𝐹_2𝑛−2𝑟𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟+𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟=𝑘^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟. (5.11) Therefore 𝑥_2𝑛−1=𝑘^𝑛−1_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟. (5.12) Also, we see from (5.8) that 𝑥_2𝑛=𝑥_2𝑛−2+𝑥_2𝑛−1𝑥_2𝑛−2𝑥_2𝑛−1−𝑥_2𝑛−3=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟+𝑘∏^𝑛−1_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝑘∏^𝑛−1_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟−𝑘∏^𝑛−2_𝑖=0𝐹_2𝑖+3ℎ−𝐹_2𝑖+1𝑟/𝐹_2𝑖+1ℎ−𝐹_2𝑖−1𝑟=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟+𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟/𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟/𝐹_2𝑛−1ℎ−𝐹_2𝑛−3𝑟−1=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟+𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟−𝐹_2𝑛−1ℎ+𝐹_2𝑛−3𝑟=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟+𝐹_2𝑛+1ℎ−𝐹_2𝑛−1𝑟ℎ∏^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟/𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝐹_2𝑛ℎ−𝐹_2𝑛−2r=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝐹1+^2𝑛+1ℎ−𝐹_2𝑛−1𝑟𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟=ℎ^𝑛−2_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟𝐹_2𝑛+2ℎ−𝐹_2𝑛𝑟𝐹_2𝑛ℎ−𝐹_2𝑛−2𝑟. (5.13) Thus 𝑥_2𝑛=ℎ^𝑛−1_𝑖=0𝐹_2𝑖+4ℎ−𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ−𝐹_2𝑖𝑟. (5.14) Hence, the proof is completed.

Assume that𝑥₋₂=0.7,𝑥₋₁=1.6,𝑥₀=13. (See Figure 2), and for 𝑥₋₂=9,𝑥₋₁=5,𝑥₀=2. (See Figure 3).

Figure 2 

Figure 3 

The following cases can be treated similarly.

5.3. On the Difference Equation 𝑥_𝑛+1=𝑥_𝑛−1−𝑥_𝑛𝑥_𝑛−1/(𝑥_𝑛+𝑥_𝑛−2)

In this subsection we obtain the solution of the following difference equation𝑥_𝑛+1=𝑥_𝑛−1−𝑥_𝑛𝑥_𝑛−1𝑥_𝑛+𝑥_𝑛−2, (5.15) where the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀ are arbitrary positive real numbers.

Theorem 5.3. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (5.15). Then for 𝑛=0,1,2,…𝑥_2𝑛−1=𝑘^𝑛−1_𝑖=0𝐹_2𝑖ℎ+𝐹_2𝑖+1𝑟𝐹_2𝑖+1ℎ+𝐹_2𝑖+2𝑟,𝑥_2𝑛=ℎ^𝑛−1_𝑖=0𝐹_2𝑖+1ℎ+𝐹_2𝑖+2𝑟𝐹_2𝑖+2ℎ+𝐹_2𝑖+3𝑟, (5.16) where 𝑥₋₂=𝑟,𝑥₋₁=𝑘,𝑥₀=ℎ,{𝐹_𝑚}^∞_𝑚=0={0,1,1,2,3,5,8,13,…}.
Figure 4 shows the solution when 𝑥₋₂=3,𝑥₋₁=7,𝑥₀=12.

Figure 4 

5.4. On the Difference Equation 𝑥_𝑛+1=𝑥_𝑛−1−𝑥_𝑛𝑥_𝑛−1/(𝑥_𝑛−𝑥_𝑛−2)

In this subsection we give the solution of the following special case of (1.1) 𝑥_𝑛+1=𝑥_𝑛−1−𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−2 (5.17) where the initial conditions 𝑥₋₂,𝑥₋₁,𝑥₀ are arbitrary real numbers. with 𝑥₋₂≠𝑥₀,𝑥₋₂,𝑥₋₁,𝑥₀≠0.

Theorem 5.4. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (5.17). Then every solution of (5.17) is periodic with period 12. Moreover {𝑥_𝑛}^∞_𝑛=−2 takes the form 𝑟,𝑘,ℎ,𝑘𝑟𝑟−ℎ,ℎ−𝑟,ℎ𝑘𝑟−ℎ,−𝑟,−𝑘,−ℎ,𝑘𝑟(),ℎ−𝑟,−ℎ−𝑟ℎ𝑘ℎ−𝑟,𝑟,𝑘,ℎ,…, (5.18) where 𝑥₋₂=𝑟,𝑥₋₁=𝑘,𝑥₀=ℎ.
Figure 5 shows the solution when 𝑥₋₂=3,𝑥₋₁=7,𝑥₀=2.

Figure 5 

6. Conclusion

This paper discussed global stability, boundedness, and the solutions of some special cases of (1.1). In Section 2 we proved when𝑏(𝑐+3𝑑)<(1−𝑎)(𝑐+𝑑)², (1.1) local stability.In Section 3 we showed that the unique equilibrium of (1.1) is globally asymptotically stable if 𝑐(1−𝑎)≠𝑏. In Section 4 we proved that the solution of (1.1) is bounded if (𝑎+𝑏/𝑐)<1. In Section 5 we gave the form of the solution of four special cases of (1.1) and gave numerical examples of each case and drew them by using Matlab 6.5.