Qualitative Study of Solutions of Some Difference Equations

Abstract

We obtain in this paper the solutions of the following recursive sequences 𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3/𝑥_𝑛−2(±1±𝑥_𝑛𝑥_𝑛−3), 𝑛=0,1,…, where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.

1. Introduction

In this paper, we obtain the solutions of the following recursive sequences:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3𝑥_𝑛−2±1±𝑥_𝑛𝑥_𝑛−3,𝑛=0,1,…, (1.1) where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solutions.

Recently, there has been a great interest in studying the qualitative properties of rational difference equations. For the systematical studies of rational and nonrational difference equations, one can refer to the papers [1–31] and references therein.

The study of rational difference equations of order greater than one is quite challenging and rewarding because 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. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Recently, Agarwal and Elsayed [4] investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation: 𝑥_𝑛+1=𝑎+𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘𝑏−𝑐𝑥_𝑛−𝑠. (1.2) Aloqeili [5] has obtained the solutions of the difference equation:𝑥_𝑛+1=𝑥_𝑛−1𝑎−𝑥_𝑛𝑥_𝑛−1. (1.3)Çinar [8, 9] investigated the solutions of the following difference equations:𝑥_𝑛+1=𝑥_𝑛−11+𝑎𝑥_𝑛𝑥_𝑛−1,𝑥_𝑛+1=𝑥_𝑛−1−1+𝑎𝑥_𝑛𝑥_𝑛−1. (1.4) Elabbasy et al. [10, 12] investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequences:𝑥_𝑛+1=𝑎𝑥_𝑛−𝑏𝑥_𝑛𝑐𝑥_𝑛−𝑑𝑥_𝑛−1,𝑥_𝑛+1=𝛼𝑥_𝑛−𝑘𝛽+𝛾∏^𝑘_𝑖=0𝑥_𝑛−𝑖. (1.5) Ibrahim [19] got the solutions of the rational difference equation:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−2𝑥_𝑛−1𝑎+𝑏𝑥_𝑛𝑥_𝑛−2. (1.6) Karatas et al. [20] got the form of the solution of the difference equation:𝑥_𝑛+1=𝑥_𝑛−51+𝑥_𝑛−2𝑥_𝑛−5. (1.7) Simsek et al. [26] obtained the solutions of the following difference equations:𝑥_𝑛+1=𝑥_𝑛−31+𝑥_𝑛−1. (1.8) Here, we recall some notations and results which will be useful in our investigation.

Let 𝐼 be some interval of real numbers and let𝑓∶𝐼^𝑘+1→𝐼 (1.9) 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,…, (1.10) has a unique solution {𝑥_𝑛}^∞_𝑛=−𝑘 [21].

Definition 1.1 (Equilibrium Point). A point 𝑥∈𝐼 is called an equilibrium point of (1.10) if𝑥=𝑓(𝑥,𝑥,…,𝑥). (1.11) That is, 𝑥_𝑛=𝑥, for 𝑛≥0,is a solution of (1.10), or equivalently, 𝑥 is a fixed point of 𝑓.

Definition 1.2 (Stability). (i) The equilibrium point 𝑥of (1.10) is locally stable if, for every 𝜖>0,there exists 𝛿>0such that for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼with 𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛿, (1.12)we have𝑥_𝑛−𝑥<𝜖∀𝑛≥−𝑘. (1.13)(ii) The equilibrium point 𝑥of (1.10) is locally asymptotically stable if 𝑥is locally stable solution of (1.10) and there exists 𝛾>0, such that for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼with𝑥_−𝑘−𝑥+𝑥_−𝑘+1−𝑥+⋯+𝑥₀−𝑥<𝛾, (1.14)we havelim_𝑛→∞𝑥_𝑛=𝑥. (1.15)(iii) The equilibrium point 𝑥of (1.10) is global attractor if, for all 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀∈𝐼,we havelim_𝑛→∞𝑥_𝑛=𝑥. (1.16)(iv) The equilibrium point 𝑥of (1.10) is globally asymptotically stable if 𝑥is locally stable, and 𝑥is also a global attractor of (1.10).(v) The equilibrium point 𝑥of (1.10) is unstable if 𝑥is not locally stable.The linearized equation of (1.10) about the equilibrium 𝑥is the linear difference equation𝑦_𝑛+1=^𝑘_𝑖=0𝜕𝑓(𝑥,𝑥,…,𝑥)𝜕𝑥_𝑛−𝑖𝑦_𝑛−𝑖. (1.17)

Theorem A (see [22]). Assume that 𝑝_𝑖∈𝑅,𝑖=1,2,…,𝑘,and𝑘∈{0,1,2,…}. Then, ^𝑘_𝑖=1𝑝_𝑖<1 (1.18) is a sufficient condition for the asymptotic stability of the difference equation: 𝑥_𝑛+𝑘+𝑝₁𝑥_{𝑛+𝑘−1}+⋯+𝑝_𝑘𝑥_𝑛=0,𝑛=0,1,…. (1.19)

Definition 1.3 (Periodicity). A sequence {𝑥_𝑛}^∞_𝑛=−𝑘is said to be periodic with period 𝑝 if 𝑥_𝑛+𝑝=𝑥_𝑛for all 𝑛≥−𝑘.

2. On the Equation𝑋_𝑛+1=𝑥_𝑛𝑥_𝑛−3/(𝑥_𝑛−2(1+𝑥_𝑛𝑥_𝑛−3))

In this section, we give a specific form of the solution of the first equation in the form:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3𝑥_𝑛−21+𝑥_𝑛𝑥_𝑛−3,𝑛=0,1,…, (2.1) where the initial values are arbitrary positive real numbers.

Theorem 2.1. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (2.1). Then, for 𝑛=0,1,…, 𝑥_6𝑛−3=𝑑^𝑛−1_𝑖=01+6𝑖𝑎𝑑1+(6𝑖+3)𝑎𝑑,𝑥_6𝑛−2=𝑐^𝑛−1_𝑖=01+(6𝑖+1)𝑎𝑑(),𝑥1+6𝑖+4𝑎𝑑_6𝑛−1=𝑏^𝑛−1_𝑖=01+(6𝑖+2)𝑎𝑑1+(6𝑖+5)𝑎𝑑,𝑥_6𝑛=𝑎^𝑛−1_𝑖=01+(6𝑖+3)𝑎𝑑(),𝑥1+6𝑖+6𝑎𝑑_6𝑛+1=𝑎𝑑𝑐(1+𝑎𝑑)^𝑛−1_𝑖=01+(6𝑖+4)𝑎𝑑1+(6𝑖+7)𝑎𝑑,𝑥_6𝑛+2=𝑎𝑑𝑏(1+2𝑎𝑑)^𝑛−1_𝑖=01+(6𝑖+5)𝑎𝑑(),1+6𝑖+8𝑎𝑑 (2.2) where𝑥₋₃=𝑑,𝑥₋₂=𝑐,𝑥₋₁=𝑏,𝑥₀=𝑎.

Proof. For 𝑛=0, the result holds. Now, suppose that 𝑛>0 and that our assumption holds for 𝑛−1. That is, 𝑥_6𝑛−9=𝑑^𝑛−2_𝑖=01+6𝑖𝑎𝑑1+(6𝑖+3)𝑎𝑑,𝑥_6𝑛−8=𝑐^𝑛−2_𝑖=01+(6𝑖+1)𝑎𝑑(),𝑥1+6𝑖+4𝑎𝑑_6𝑛−7=𝑏^𝑛−2_𝑖=01+(6𝑖+2)𝑎𝑑1+(6𝑖+5)𝑎𝑑,𝑥_6𝑛−6=𝑎^𝑛−2_𝑖=01+(6𝑖+3)𝑎𝑑(),𝑥1+6𝑖+6𝑎𝑑_6𝑛−5=𝑎𝑑𝑐(1+𝑎𝑑)^𝑛−2_𝑖=01+(6𝑖+4)𝑎𝑑1+(6𝑖+7)𝑎𝑑,𝑥_6𝑛−4=𝑎𝑑𝑏(1+2𝑎𝑑)^𝑛−2_𝑖=01+(6𝑖+5)𝑎𝑑().1+6𝑖+8𝑎𝑑 (2.3) Now, it follows from (2.1) that 𝑥_6𝑛−3=𝑥_6𝑛−4𝑥_6𝑛−7𝑥_6𝑛−61+𝑥_6𝑛−4𝑥_6𝑛−7=𝑎𝑑𝑏()∏1+2𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+5)𝑎𝑑()𝑏∏1+6𝑖+8𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+2)𝑎𝑑1+(6𝑖+5)𝑎𝑑𝑎∏^𝑛−2_𝑖=01+(6𝑖+3)𝑎𝑑1+(6𝑖+6)𝑎𝑑1+𝑎𝑑𝑏()∏1+2𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+5)𝑎𝑑1+(6𝑖+8)𝑎𝑑𝑏∏^𝑛−2_𝑖=01+(6𝑖+2)𝑎𝑑()=1+6𝑖+5𝑎𝑑𝑎𝑑∏1+2𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+5)𝑎𝑑()∏1+6𝑖+8𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+2)𝑎𝑑1+(6𝑖+5)𝑎𝑑𝑎∏^𝑛−2_𝑖=01+(6𝑖+3)𝑎𝑑1+(6𝑖+6)𝑎𝑑1+𝑎𝑑()∏1+2𝑎𝑑^{𝑛−2𝑖=0}1+(6𝑖+2)𝑎𝑑()=1+6𝑖+8𝑎𝑑𝑎𝑑(1+(6𝑛+2)𝑎𝑑)𝑎∏^𝑛−2_𝑖=01+(6𝑖+3)𝑎𝑑1+(6𝑖+6)𝑎𝑑1+𝑎𝑑(())=𝑑1+6𝑛+2𝑎𝑑∏^𝑛−2_𝑖=01+(6𝑖+3)𝑎𝑑()()))1+6𝑖+6𝑎𝑑((1+6𝑛+2𝑎𝑑+𝑎𝑑=𝑑∏^𝑛−2_𝑖=01+(6𝑖+6)𝑎𝑑()11+6𝑖+3𝑎𝑑(()).1+6𝑛+3𝑎𝑑 (2.4) Hence, we have 𝑥_6𝑛−3=𝑑^𝑛−1_𝑖=01+6𝑖𝑎𝑑().1+6𝑖+3𝑎𝑑 (2.5) Similarly, 𝑥_6𝑛+1=𝑥_6𝑛𝑥_6𝑛−3𝑥_6𝑛−21+𝑥_6𝑛𝑥_6𝑛−3=𝑎𝑑∏^𝑛−1_𝑖=01+(6𝑖+3)𝑎𝑑1+(6𝑖+6)𝑎𝑑𝑑∏^𝑛−1_𝑖=01+6𝑖𝑎𝑑1+(6𝑖+3)𝑎𝑑𝑐∏^𝑛−1_𝑖=01+(6𝑖+1)𝑎𝑑1+(6𝑖+4)𝑎𝑑1+𝑎∏^𝑛−1_𝑖=01+(6𝑖+3)𝑎𝑑1+(6𝑖+6)𝑎𝑑𝑑∏^𝑛−1_𝑖=01+6𝑖𝑎𝑑()=1+6𝑖+3𝑎𝑑𝑎𝑑∏^𝑛−1_𝑖=01+6𝑖𝑎𝑑1+(6𝑖+6)𝑎𝑑𝑐∏^𝑛−1_𝑖=01+(6𝑖+1)𝑎𝑑1+(6𝑖+4)𝑎𝑑1+𝑎𝑑∏^𝑛−1_𝑖=01+6𝑖𝑎𝑑()=1+6𝑖+6𝑎𝑑𝑎𝑑(1+6𝑛𝑎𝑑)𝑐∏^𝑛−1_𝑖=01+(6𝑖+1)𝑎𝑑1+(6𝑖+4)𝑎𝑑1+𝑎𝑑=1+6𝑛𝑎𝑑𝑎𝑑𝑐∏^𝑛−1_𝑖=01+(6𝑖+1)𝑎𝑑()()=1+6𝑖+4𝑎𝑑1+6𝑛𝑎𝑑+𝑎𝑑_{𝑛−1𝑖=0}1+(6𝑖+4)𝑎𝑑1+(6𝑖+1)𝑎𝑑𝑎𝑑𝑐().1+7𝑛𝑎𝑑 (2.6) Hence, we have 𝑥_6𝑛+1=𝑎𝑑𝑐(1+𝑎𝑑)^𝑛−1_𝑖=01+(6𝑖+4)𝑎𝑑().1+6𝑖+7𝑎𝑑 (2.7) Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Theorem 2.2. Equation (2.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.

Proof. For the equilibrium points of (2.1), we can write 𝑥=𝑥²𝑥1+𝑥³. (2.8) Then, we have 𝑥²1+𝑥²=𝑥²,𝑥²1+𝑥²−1=0, (2.9) or 𝑥⁴=0. (2.10) Thus the equilibrium point of (2.1) is 𝑥=0.
Let 𝑓∶(0,∞)³→(0,∞) be a function defined by 𝑓(𝑢,𝑣,𝑤)=𝑢𝑤𝑣().1+𝑢𝑤 (2.11) Therefore, it follows that 𝑓_𝑢()=𝑤𝑢,𝑣,𝑤𝑣(1+𝑢𝑤)²,𝑓_𝑣(𝑢,𝑣,𝑤)=−𝑢𝑤𝑣²(),𝑓1+𝑢𝑤_𝑤()=𝑢𝑢,𝑣,𝑤𝑣(1+𝑢𝑤)², (2.12) we see that 𝑓_𝑢(𝑥,𝑥,𝑥)=1,𝑓_𝑣(𝑥,𝑥,𝑥)=1,𝑓_𝑤(𝑥,𝑥,𝑥)=1. (2.13) The proof follows by using Theorem A.

Numerical Examples
For confirming the results of this section, we consider numerical examples which represent different types of solutions to (2.1).
Example 2.3. We assume 𝑥₋₃=11,𝑥₋₂=7,𝑥₋₁=13,𝑥₀=3, (see Figure 1).Example 2.4. See Figure 2, since 𝑥₋₃=2,𝑥₋₂=9,𝑥₋₁=3,𝑥₀=5.

Figure 1 

Figure 2 

3. On the Equation 𝑋_𝑛+1=𝑥_𝑛𝑥_𝑛−3/(𝑥_𝑛−2(−1+𝑥_𝑛𝑥_𝑛−3))

In this section, we obtain the solution of the second equation in the form:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3𝑥_𝑛−2−1+𝑥_𝑛𝑥_𝑛−3,𝑛=0,1,…, (3.1) where the initial values are arbitrary nonzero real numbers with 𝑥₀𝑥₋₃≠1.

Theorem 3.1. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (3.1). Then, (3.1) has unboundedness solution and, for 𝑛=0,1,…,𝑥_6𝑛−3=𝑑(−1+𝑎𝑑)^𝑛,𝑥_6𝑛−2=𝑐(−1+𝑎𝑑)^𝑛,𝑥_6𝑛−1=𝑏(−1+𝑎𝑑)^𝑛,𝑥_6𝑛=𝑎(−1+𝑎𝑑)^𝑛,𝑥_6𝑛+1=𝑎𝑑𝑐(−1+𝑎𝑑)^𝑛+1,𝑥_6𝑛+2=𝑎𝑑𝑏(−1+𝑎𝑑)^𝑛, (3.2) where𝑥₋₃=𝑑,𝑥₋₂=𝑐,𝑥₋₁=𝑏,𝑥₀=𝑎.

Proof. For 𝑛=0 the result holds. Now, suppose that 𝑛>0 and that our assumption holds for 𝑛−1. That is, 𝑥_6𝑛−9=𝑑(−1+𝑎𝑑)^𝑛−1,𝑥_6𝑛−8=𝑐(−1+𝑎𝑑)^𝑛−1,𝑥_6𝑛−7=𝑏(−1+𝑎𝑑)^𝑛−1,𝑥_6𝑛−6=𝑎(−1+𝑎𝑑)^𝑛−1,𝑥_6𝑛−5=𝑎𝑑𝑐(−1+𝑎𝑑)^𝑛,𝑥_6𝑛−4=𝑎𝑑𝑏(−1+𝑎𝑑)^𝑛−1. (3.3) Now, it follows from (3.1) that 𝑥_6𝑛−3=𝑥_6𝑛−4𝑥_6𝑛−7𝑥_6𝑛−6−1+𝑥_6𝑛−4𝑥_6𝑛−7=(𝑎𝑑/𝑏)(−1+𝑎𝑑)^𝑛−1𝑏/(−1+𝑎𝑑)^𝑛−1𝑎(−1+𝑎𝑑)^𝑛−1−1+(𝑎𝑑/𝑏)(−1+𝑎𝑑)^𝑛−1𝑏/(−1+𝑎𝑑)^𝑛−1=𝑑(−1+𝑎𝑑)^𝑛−1()=𝑑−1+𝑎𝑑(−1+𝑎𝑑)^𝑛,𝑥_6𝑛−2=𝑥_6𝑛−3𝑥_6𝑛−6𝑥_6𝑛−5−1+𝑥_6𝑛−3𝑥_6𝑛−6=𝑑/(−1+𝑎𝑑)^𝑛𝑎(−1+𝑎𝑑)^𝑛−1𝑎𝑑/𝑐(−1+𝑎𝑑)^𝑛−1+(𝑑/(−1+𝑎𝑑)^𝑛)𝑎(−1+𝑎𝑑)^𝑛−1=𝑐(−1+𝑎𝑑)^𝑛−1(((()−1+𝑎𝑑/−1+𝑎𝑑)))−1+𝑎𝑑()()−1+𝑎𝑑=𝑐−1+𝑎𝑑^𝑛,𝑥_6𝑛−1=𝑥_6𝑛−2𝑥_6𝑛−5𝑥_6𝑛−4−1+𝑥_6𝑛−2𝑥_6𝑛−5=𝑐(−1+𝑎𝑑)^𝑛𝑎𝑑/𝑐(−1+𝑎𝑑)^𝑛(𝑎𝑑/𝑏)(−1+𝑎𝑑)^𝑛−1(−1+𝑐(−1+𝑎𝑑)^𝑛(𝑎𝑑/𝑐(−1+𝑎𝑑)^𝑛=𝑏))(−1+𝑎𝑑)^𝑛−1()=𝑏−1+𝑎𝑑(−1+𝑎𝑑)^𝑛. (3.4) Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Theorem 3.2. Equation (3.1) has a periodic solution of period six iff 𝑎𝑑=2 and will take the form:   {𝑑,𝑐,𝑏,𝑎,𝑎𝑑/𝑐,𝑎𝑑/𝑏,𝑑,𝑐,𝑏,𝑎,𝑎𝑑/𝑐,𝑎𝑑/𝑏,…}.

Proof. First suppose that there exists a prime period six solution: 𝑑,𝑐,𝑏,𝑎,𝑎𝑑𝑐,𝑎𝑑𝑏,𝑑,𝑐,𝑏,𝑎,𝑎𝑑𝑐,𝑎𝑑𝑏,…, (3.5) of (3.1), we see from the form of the solution of (3.1) that 𝑑𝑑=(−1+𝑎𝑑)^𝑛,𝑐=𝑐(−1+𝑎𝑑)^𝑛𝑏,𝑏=(−1+𝑎𝑑)^𝑛,𝑎=𝑎(−1+𝑎𝑑)^𝑛,𝑎𝑑𝑐=𝑎𝑑𝑐(−1+𝑎𝑑)^𝑛+1,𝑎𝑑𝑏=𝑎𝑑𝑏(−1+𝑎𝑑)^𝑛, (3.6) or (−1+𝑎𝑑)^𝑛=1. (3.7) Then, 𝑎𝑑=2. (3.8) Second, assume that 𝑎𝑑=2. Then, we see from the form of the solution of (3.1) that 𝑥_6𝑛−3=𝑑,𝑥_6𝑛−2=𝑐,𝑥_6𝑛−1=𝑏,𝑥_6𝑛=𝑎,𝑥_6𝑛+1=𝑎𝑑𝑐,𝑥_6𝑛+2=𝑎𝑑𝑏. (3.9) Thus, we have a periodic solution of period six and the proof is complete.

Theorem 3.3. Equation (3.1) has two equilibrium points which are 0,√³2 and these equilibrium points are not locally asymptotically stable.

Proof. For the equilibrium points of (3.1), we can write 𝑥=𝑥²𝑥−1+𝑥². (3.10) Then, we have 𝑥²−1+𝑥²=𝑥², (3.11) or 𝑥²𝑥²−2=0, (3.12) Thus, the equilibrium points of (3.1) are 0,±√2.
Let 𝑓∶(0,∞)³→(0,∞) be a function defined by 𝑓(𝑢,𝑣,𝑤)=𝑢𝑤𝑣().−1+𝑢𝑤 (3.13) Therefore, it follows that 𝑓_𝑢()𝑤𝑢,𝑣,𝑤=−𝑣(−1+𝑢𝑤)²,𝑓_𝑣(𝑢,𝑣,𝑤)=−𝑢𝑤𝑣²(),𝑓−1+𝑢𝑤_𝑤()𝑢𝑢,𝑣,𝑤=−𝑣(−1+𝑢𝑤)², (3.14) we see that 𝑓_𝑢(𝑥,𝑥,𝑥)=−1,𝑓_𝑣(𝑥,𝑥,𝑥)=±1,𝑓_𝑤(𝑥,𝑥,𝑥)=−1. (3.15) The proof follows by using Theorem A.

Numerical Examples
Here, we will represent different types of solutions of (3.1).
Example 3.4. We consider 𝑥₋₃=2,𝑥₋₂=9,𝑥₋₁=3,𝑥₀=5, (see Figure 3).Example 3.5. See Figure 4 since 𝑥₋₃=7,𝑥₋₂=2,𝑥₋₁=8,𝑥₀=2/7.

Figure 3 

Figure 4 

The following cases can be proved similarly.

4. On the Equation 𝑋_𝑛+1=𝑥_𝑛𝑥_𝑛−3/(𝑥_𝑛−2(1−𝑥_𝑛𝑥_𝑛−3))

In this section, we get the solution of the third following equation:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3𝑥_𝑛−21−𝑥_𝑛𝑥_𝑛−3,𝑛=0,1,…, (4.1) where the initial values are arbitrary positive real numbers.

Theorem 4.1. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (4.1). Then, for 𝑛=0,1,…, 𝑥_6𝑛−3=𝑑^𝑛−1_𝑖=01−6𝑖𝑎𝑑1−(6𝑖+3)𝑎𝑑,𝑥_6𝑛−2=𝑐^𝑛−1_𝑖=01−(6𝑖+1)𝑎𝑑(),𝑥1−6𝑖+4𝑎𝑑_6𝑛−1=𝑏^𝑛−1_𝑖=01−(6𝑖+2)𝑎𝑑1−(6𝑖+5)𝑎𝑑,𝑥_6𝑛=𝑎^𝑛−1_𝑖=01−(6𝑖+3)𝑎𝑑(),𝑥1−6𝑖+6𝑎𝑑_6𝑛+1=𝑎𝑑𝑐(1−𝑎𝑑)^𝑛−1_𝑖=01−(6𝑖+4)𝑎𝑑1−(6𝑖+7)𝑎𝑑,𝑥_6𝑛+2=𝑎𝑑𝑏(1−2𝑎𝑑)^𝑛−1_𝑖=01−(6𝑖+5)𝑎𝑑().1−6𝑖+8𝑎𝑑 (4.2)

Theorem 4.2. Equation (4.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.

Example 4.3. Assume that 𝑥₋₃=1,𝑥₋₂=8,𝑥₋₁=3,𝑥₀=9,(see Figure 5).

Figure 5 

Example 4.4. See Figure 6 since 𝑥₋₃=11,𝑥₋₂=8,𝑥₋₁=18,𝑥₀=9.

Figure 6 

5. On the Equation 𝑋_𝑛+1=𝑥_𝑛𝑥_𝑛−3/(𝑥_𝑛−2(−1−𝑥_𝑛𝑥_𝑛−3))

Here, we obtain a form of the solutions of the equation𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−3𝑥_𝑛−2−1−𝑥_𝑛𝑥_𝑛−3,𝑛=0,1,…, (5.1) where the initial values are arbitrary nonzero real numbers with 𝑥₋₃𝑥₀≠−1.

Theorem 5.1. Let {𝑥_𝑛}^∞_𝑛=−3 be a solution of (5.1). Then, (5.1) has unboundedness solution and, for 𝑛=0,1,…, 𝑥_6𝑛−3=𝑑(−1−𝑎𝑑)^𝑛,𝑥_6𝑛−2=𝑐(−1−𝑎𝑑)^𝑛,𝑥_6𝑛−1=𝑏(−1−𝑎𝑑)^𝑛,𝑥_6𝑛=𝑎(−1−𝑎𝑑)^𝑛,𝑥_6𝑛+1=𝑎𝑑𝑐(−1−𝑎𝑑)^𝑛+1,𝑥_6𝑛+2=𝑎𝑑𝑏(−1−𝑎𝑑)^𝑛. (5.2)

Theorem 5.2. Equation (5.1) has a periodic solution of period six if and only if 𝑎𝑑=−2 and will take the form:{𝑑,𝑐,𝑏,𝑎,𝑎𝑑/𝑐,𝑎𝑑/𝑏,𝑑,𝑐,𝑏,𝑎,𝑎𝑑/𝑐,𝑎𝑑/𝑏,…}.

Theorem 5.3. Equation (5.1) has a unique equilibrium point which is 0,and this equilibrium point is not locally asymptotically stable.

Example 5.4. Consider 𝑥₋₃=6,𝑥₋₂=8,𝑥₋₁=12,𝑥₀=4,(see Figure 7).

Figure 7 

Example 5.5. Figure 8 shows the solutions when 𝑥₋₃=−6,𝑥₋₂=11,𝑥₋₁=3,𝑥₀=2/6.

Figure 8 

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under Grant no. (1431/662/415). The authors, therefore, acknowledge with thanks DSR technical and financial support. Last, but not least, sincere appreciations are dedicated to all our colleagues at the Faculty of Science, Rabigh branch, for their nice wishes.