The Dynamics of the Solutions of Some Difference Equations

Abstract

This paper is devoted to investigate the global behavior of the following rational difference equation: 𝑦_𝑛+1=𝛼𝑦_𝑛−𝑡/(𝛽+𝛾∑^𝑘_𝑖=0𝑦^𝑝_{𝑛−(2𝑖+1)}∏^𝑘_𝑖=0𝑦^𝑞_{𝑛−(2𝑖+1)}),𝑛=0,1,2,…, where 𝛼,𝛽,𝛾,𝑝,𝑞∈(0,∞) and 𝑘,𝑡∈{0,1,2,…} with the initial conditions 𝑥₀,𝑥₋₁,…,𝑥_−2𝑘,𝑥_{−2max{𝑘,𝑡}−1}∈(0,∞). We will find and classify the equilibrium points of the equations under studying and then investigate their local and global stability. Also, we will study the oscillation and the permanence of the considered equations.

1. Introduction

The aim of this paper is to study the dynamics of the solutions of the following recursive sequence: 𝑦_𝑛+1=𝛼𝑦_𝑛−𝑡𝛽+𝛾∑^𝑘_𝑖=0𝑦^𝑝_{()𝑛−2𝑖+1}∏^𝑘_𝑖=0𝑦^𝑞_{()𝑛−2𝑖+1},𝑛=0,1,2,…, (1.1) where 𝛼,𝛽,𝛾,𝑝,𝑞∈(0,∞) and 𝐾∈{0,1,2,…}, where 𝐾=max{𝑘,𝑡}, with the initial conditions 𝑥₀,  𝑥₋₁,…,  𝑥_−2𝑘,𝑥_−2𝐾−1∈(0,∞). We deal with the classification of the equilibrium points of (1.1) in terms of being stable or unstable, where we investigate the global attractor of the solutions of (1.1) as well as the permanence of the equation. Also, we establish some appropriate conditions, which grantee the oscillation of the solutions of (1.1). For more results in the direction of this study, see, for example, [1–23] and the papers therein.

In the sequel, we present some well-known results and definition that will be useful in our investigation of (1.1). Let 𝐼 be some interval of real numbers and let 𝑓 : 𝐼^𝑘+1→𝐼 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,…, (1.2) has a unique solution {𝑥_𝑛}^∞_𝑛=−𝑘.

Definition 1.1 (permanence). The difference equation (1.2) 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≤𝑀 for all 𝑛≥𝑁.

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

Definition 1.3 (semicycles). A positive semicycle of a sequence {𝑥_𝑛}^∞_𝑛=−𝑘 consists 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; either 𝑚=∞ or 𝑚<∞ and 𝑥_𝑚+1<𝑥. A negative semicycle of a sequence {𝑥_𝑛}^∞_𝑛=−𝑘 consists 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; either 𝑚=∞ or 𝑚<∞ and 𝑥_𝑚+1≥𝑥.

Definition 1.4 (oscillation). A sequence {𝑥_𝑛}^∞_𝑛=−𝑘 is called nonoscillatory about the point 𝑥 if there 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.

2. Dynamics of (1.1)

The change of variables 𝑦_𝑛=(𝛽/𝛾)^{1/[𝑝+(𝑘+1)𝑞]}𝑥_𝑛 reduces (1.1) to the following difference equation 𝑥_𝑛+1=𝑟𝑥_𝑛−𝑡1+∑^𝑘_𝑖=0𝑥^𝑝_{()𝑛−2𝑖+1}∏^𝑘_𝑖=0𝑥^𝑞_{()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 points of (2.1) are given by 𝑟𝑥=𝑥1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}. (2.2) Then, whenever 𝑟≤1, (2.1) has the only equilibrium point 𝑥=0, and, while at 𝑟>1, (2.1) possesses the unique positive equilibrium point 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)}.

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.(iii)If 𝑟=1, then the equilibrium point 𝑥=0 of (2.1) is nonhyperbolic with 𝜆=0<1 and 𝜆=1.

Proof. The linearized equation of (2.1) about 𝑥=0 is 𝑢_𝑛+1−𝑟𝑢_𝑛−𝑡=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 in Theorem 2.1 that the equilibrium point  𝑥=0 of (2.1) is locally asymptotically stable. So it suffices to show that lim_𝑛→∞𝑥_𝑛=0. (2.3) Now, it follows from (2.1) that 𝑥_𝑛+1=𝑟𝑥_𝑛−𝑡1+𝑥^𝑝+𝑞_𝑛−1𝑥^𝑞_𝑛−3⋯𝑥^𝑞_{()𝑛−2𝑘+1}+𝑥^𝑞_𝑛−1𝑥^𝑝+𝑞_𝑛−3⋯𝑥^𝑞_{()𝑛−2𝑘+1}+⋯+𝑥^𝑞_𝑛−1𝑥^𝑞_𝑛−3⋯𝑥₍₎^𝑝+𝑞_{𝑛−2𝑘+1}≤𝑟𝑥_𝑛−2𝑡. (2.4) Now, assume that 𝑋=max{𝑥_−2𝑡,𝑥_−2𝑡+1,…,𝑥₋₁,𝑥₀/𝑟}. Then, it follows from (1.1) and after some simple computations are achieved that 𝑥_{2(𝑛−1)𝑡+𝑖}≤𝑟^𝑛𝑋, 𝑖=0,1,…,2𝑡. Therefore 𝑥_𝑛→0 as 𝑛→∞, 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)𝑞)}.

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 is 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_𝑛−𝑡1+𝑥^𝑝+𝑞_𝑛−1𝑥^𝑞_𝑛−3⋯𝑥^𝑞_{()𝑛−2𝑘+1}+𝑥^𝑞_𝑛−1𝑥^𝑝+𝑞_𝑛−3⋯𝑥^𝑞_{()𝑛−2𝑘+1}+⋯+𝑥^𝑞_𝑛−1𝑥^𝑞_𝑛−3⋯𝑥₍₎^𝑝+𝑞_{𝑛−2𝑘+1}≤𝑥_𝑛−𝑡𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}=𝑥_𝑛−𝑡. (2.5) Hence, each subsequence {𝑥_{(𝑡+1)𝑛+𝑖}},  𝑗=0,1,…,𝑡, of {𝑥_𝑛}^∞_𝑛=0 is decreasing sequence and therefore it has a limit. Let for some 𝑗=0,1,…,𝑡, lim_𝑛→∞𝑥_{(𝑡+1)𝑛+𝑗}=𝜇, and, for the sake of contradiction, assume that 𝜇>𝑥. Then, by taking the limit of both sides of 𝑥_{(𝑡+1)𝑛+𝑗+1}=𝑟𝑥_{(𝑡+1)𝑛+𝑗−𝑡}1+∑^𝑘_𝑠=0𝑥^𝑝_{()()𝑡+1𝑛+𝑗−2𝑠+1}∏^𝑘_𝑠=0𝑥^𝑞_{()()𝑡+1𝑛+𝑗−2𝑠+1},𝑛=0,1,2,…, (2.6) we obtain 𝜇=𝑟𝜇/(1+(𝑘+1)𝜇^{𝑝+(𝑘+1)𝑞}), which contradicts the hypothesis that 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)} is the only positive solution of (2.2). Therefore, lim_𝑛→∞𝑥_{(𝑡+1)𝑛+𝑗}=𝜇, for all 𝑗=0,1,…,𝑡. This means that all the subsequences {𝑥_{(𝑡+1)𝑛+𝑖}},  𝑗=0,1,…,𝑡, of {𝑥_𝑛}^∞_𝑛=0 have the same limit, 𝑥, and therefore lim_𝑛→∞𝑥_𝑛=𝑥, which completes the proof.

Theorem 2.4. Assume that 𝑡=0, 𝑟>1, and let {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)} 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 𝐿≥𝑀≥𝑁≥2(𝑘+1), and let {𝑥_𝑁,𝑥_𝑁+1,…,𝑥_𝑀} be a positive semicycle followed by the negative semicycle {𝑥_𝑀,𝑥_𝑀+1,…,𝑥_𝐿}. Now, it follows from (2.1) that 𝑥_{𝑁+2(𝑘+1)}−𝑥_𝑁=𝑟𝑥_𝑁+2𝑘+11+𝑥^𝑝+𝑞_𝑁+2𝑘𝑥^𝑞_{𝑁+2𝑘−2}⋯𝑥^𝑞_𝑁+𝑥^𝑞_𝑁+2𝑘𝑥^𝑝+𝑞_{𝑁+2𝑘−2}⋯𝑥^𝑞_𝑁+⋯+𝑥^𝑞_𝑁+2𝑘⋯𝑥_𝑁^𝑝+𝑞−𝑥_𝑁≤𝑥_𝑁+2𝑘+1𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑁=𝑥_𝑁+2𝑘+1−𝑥_𝑁=𝑟𝑥_{𝑁+2(𝑘+𝑡)}1+𝑥^𝑝+𝑞_{𝑁+2𝑘−1}𝑥^𝑞_{𝑁+2𝑘−3}⋯𝑥^𝑞_𝑁−1+⋯+𝑥^𝑞_{𝑁+2𝑘−1}⋯𝑥^𝑝+𝑞_𝑁−1−𝑥_𝑁≤𝑥_𝑁+2𝑘𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑁=𝑥_𝑁+2𝑘−𝑥_𝑁≤𝑥_{𝑁+2𝑘−1}−𝑥_𝑁≤⋯≤𝑥_𝑁+1−𝑥_𝑁≤𝑥_𝑁−𝑥_𝑁=0. (2.7) Then, 𝑥_𝑁≥𝑥_{𝑁+2(𝑘+1)} for all 𝑁≥2(𝑘+1).
Similarly, we see from (2.1) that 𝑥_{𝑀+2(𝑘+1)}−𝑥_𝑀=𝑟𝑥_𝑀+2𝑘+11+𝑥^𝑝+𝑞_𝑀+2𝑘𝑥^𝑞_{𝑀+2𝑘−2}⋯𝑥^𝑞_𝑀+𝑥^𝑞_𝑀+2𝑘𝑥^𝑝+𝑞_{𝑀+2𝑘−2}⋯𝑥^𝑞_𝑀+⋯+𝑥^𝑞_𝑀+2𝑘⋯𝑥_𝑀^𝑝+𝑞−𝑥_𝑀≥𝑥_𝑀+2𝑘+1𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑀=𝑥_𝑀+2𝑘+1−𝑥_𝑀=𝑟𝑥_𝑀+2𝑘1+𝑥^𝑝+𝑞_{𝑀+2𝑘−1}𝑥^𝑞_{𝑀+2𝑘−3}⋯𝑥^𝑞_𝑀−1+⋯+𝑥^𝑞_{𝑀+2𝑘−1}⋯𝑥^𝑝+𝑞_𝑀−1−𝑥_𝑀≥𝑥_𝑀+2𝑘𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑀=𝑥_𝑀+2𝑘−𝑥_𝑀≤𝑥_{𝑀+2𝑘−1}−𝑥_𝑀≥⋯≥𝑥_𝑀+1−𝑥_𝑀≥𝑥_𝑀−𝑥_𝑀=0. (2.8)
Therefore, 𝑥_{𝑀+2(𝑘+1)}≥𝑥_𝑀 for all 𝑀≥2(𝑘+1). The proof is complete.

Theorem 2.5. Assume that 𝑡=2𝑘,𝑟>1, and let {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)} of (2.1). Then, the extreme point in any semicycle occurs in one of the first 2𝑘 terms of the semicycle.

Proof. Assume that {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} is a strictly oscillatory solution of (2.1). Let 𝐿≥𝑀≥𝑁≥2𝑘+1, and  let {𝑥_𝑁,𝑥_𝑁+1,…,𝑥_𝑀} be a positive semicycle followed by the negative semicycle {𝑥_𝑀,𝑥_𝑀+1,…,𝑥_𝐿}. Now, it follows from (2.1) that 𝑥_𝑁+2𝑘+1−𝑥_𝑁=𝑟𝑥_𝑁1+𝑥^𝑝+𝑞_𝑁+2𝑘𝑥^𝑞_{𝑁+2𝑘−2}⋯𝑥^𝑞_𝑁+𝑥^𝑞_𝑁+2𝑘𝑥^𝑝+𝑞_{𝑁+2𝑘−2}⋯𝑥^𝑞_𝑁+⋯+𝑥^𝑞_𝑁+2𝑘⋯𝑥_𝑁^𝑝+𝑞−𝑥_𝑁≤𝑥_𝑁𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑁=𝑥_𝑁−𝑥_𝑁=0. (2.9) Then, 𝑥_𝑁≥𝑥_𝑁+2𝑘 for all 𝑁≥2𝑘+1.
Similarly, we see from (2.1) that 𝑥_𝑀+2𝑘+1−𝑥_𝑀=𝑟𝑥_𝑀1+𝑥^𝑝+𝑞_𝑀+2𝑘𝑥^𝑞_{𝑀+2𝑘−2}⋯𝑥^𝑞_𝑀+𝑥^𝑞_𝑀+2𝑘𝑥^𝑝+𝑞_{𝑀+2𝑘−2}⋯𝑥^𝑞_𝑀+⋯+𝑥^𝑞_𝑀+2𝑘⋯𝑥_𝑀^𝑝+𝑞−𝑥_𝑀≥𝑥_𝑀𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑀=𝑥_𝑀−𝑥_𝑀=0. (2.10) Therefore, 𝑥_𝑀+2𝑘≥𝑥_𝑀 for all 𝑀≥2𝑘+1. The proof is complete.

Theorem 2.6. Assume that 𝑡=2𝑘+1, 𝑟>1, and let {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)} of (2.1). Then, the extreme point in any semicycle occurs in one of the first 2𝑘+1 terms of the semicycle.

Proof. Assume that {𝑥_𝑛}^∞_{𝑛=−2𝑘−1} be a strictly oscillatory solution of (2.1). Let 𝐿≥𝑀≥𝑁≥2𝑘+1 and let {𝑥_𝑁,𝑥_𝑁+1,…,𝑥_𝑀} be a positive semicycle followed by the negative semicycle {𝑥_𝑀,𝑥_𝑀+1,…,𝑥_𝐿}. Now it follows from (2.1) that 𝑥_{𝑁+2(𝑘+1)}−𝑥_𝑁=𝑟𝑥_𝑁1+𝑥^𝑝+𝑞_𝑁+2𝑘+1𝑥^𝑞_{𝑁+2𝑘−1}⋯𝑥^𝑞_𝑁+𝑥^𝑞_𝑁+2𝑘+1𝑥^𝑝+𝑞_{𝑁+2𝑘−1}⋯𝑥^𝑞_𝑁+⋯+𝑥^𝑞_𝑁+2𝑘+1⋯𝑥_𝑁^𝑝+𝑞−𝑥_𝑁≤𝑥_𝑁𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑁=𝑥_𝑁−𝑥_𝑁=0. (2.11) Then, 𝑥_𝑁≥𝑥_𝑁+2𝑘 for all 𝑁≥2𝑘+1.
Similarly, we see from (2.1) that 𝑥_{𝑀+2(𝑘+1)}−𝑥_𝑀=𝑟𝑥_𝑀1+𝑥^𝑝+𝑞_𝑀+2𝑘+1𝑥^𝑞_{𝑀+2𝑘−1}⋯𝑥^𝑞_𝑀+𝑥^𝑞_𝑀+2𝑘+1𝑥^𝑝+𝑞_{𝑀+2𝑘−1}⋯𝑥^𝑞_𝑀+⋯+𝑥^𝑞_𝑀+2𝑘+1⋯𝑥_𝑀^𝑝+𝑞−𝑥_𝑀≥𝑥_𝑀𝑟1+(𝑘+1)𝑥^{𝑝+(𝑘+1)𝑞}−𝑥_𝑀=𝑥_𝑀−𝑥_𝑀=0. (2.12) Therefore 𝑥_𝑀+2𝑘≥𝑥_𝑀 for all 𝑀≥2𝑘+1. The proof is complete.

Theorem 2.7. 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 lim_𝑛→∞𝑥_𝑛=𝑥, (2.13) 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.(ii){𝑥_𝑛}^∞_{𝑛=−2𝑘−1} is strictly oscillatory about 𝑥=((𝑟−1)/(𝑘+1))^{1/(𝑝+(𝑘+1)𝑞)}.
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 semicycles, 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+𝑞_𝐿−2𝑥^𝑞_𝐿−4⋯𝑥^𝑞_{𝐿−2𝑘−2}+𝑥^𝑞_𝐿−2𝑥^𝑝+𝑞_𝐿−4⋯𝑥^𝑞_{𝐿−2𝑘−2}+⋯+𝑥^𝑞_𝐿−2𝑥^𝑞_𝐿−4⋯𝑥^𝑝+𝑞_{𝐿−2𝑘−2}=𝑟²𝑥_𝐿−21+𝑥^𝑝+𝑞_𝐿−3⋯𝑥^𝑞_{𝐿−2𝑘−3}+⋯+𝑥^𝑞_𝐿−3⋯𝑥^𝑝+𝑞_{𝐿−2𝑘−3}1+𝑥^𝑝+𝑞_𝐿−2⋯𝑥^𝑞_{𝐿−2𝑘−2}+⋯+𝑥^𝑞_𝐿−2⋯𝑥^𝑝+𝑞_{𝐿−2𝑘−2}⋮=𝑥_𝐿−𝜁𝑟^𝜁𝜁_𝜂=111+∑^𝑘_𝑖=0𝑥^𝑝_{()𝐿−2𝑖+1−𝜂}∏^𝑘_𝑖=0𝑥^𝑞_{()𝐿−2𝑖+1−𝜂}=𝑥_𝜇𝑟^{𝐿−𝜇𝐿−1}_𝜂=𝜇11+∑^𝑘_𝑖=0𝑥^𝑝_{()𝜂−2𝑖+1}∏^𝑘_𝑖=0𝑥^𝑞_{()𝜂−2𝑖+1}. (2.14) Therefore, 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^𝑞_{()𝜂−2𝑖+1}≤𝑥𝑟^2𝑘+1=𝐻. (2.15) 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^𝑞_{()𝜂−2𝑖+1}≥𝑥𝑟^{𝑊−𝑡𝑊−1}_𝜂=𝑡11+(𝑘+1)𝐻^{𝑝+(𝑘+1)𝑞}=𝑥𝑟^𝑊−𝑡11+(𝑘+1)𝐻^{𝑝+(𝑘+1)𝑞𝑊−𝑡−1}≥𝑥11+(𝑘+1)𝐻^{𝑝+(𝑘+1)𝑞2𝑘+1}=𝐺. (2.16) That is 𝐺≤𝑥_𝑛≤𝐻. It follows from (i) and (ii) that min{𝐶,𝐺}≤𝑥_𝑛≤max{𝐷,𝐻}. Then, the proof is complete.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support.