Quantitative Bounds for Positive Solutions of a Stević Difference Equation

Abstract

This paper studies the behavior of positive solutions to the following particular case of a difference equation by Stević 𝑥_𝑛+1=𝐴+𝑥^𝑝_𝑛/𝑥^𝑝𝑘+1_𝑛−𝑘, 𝑛∈ℕ₀, where 𝐴,𝑝∈(0,+∞), 𝑘∈ℕ, and presents theoretically computable explicit lower and upper bounds for the positive solutions to this equation. Besides, a concrete example is given to show the computing approaches which are effective for small parameters. Some analogous results are also established for the corresponding Stević max-type difference equation.

1. Introduction

The study regarding the behavior of positive solutions to the difference equation𝑥_𝑛𝑥=𝐴+^{𝑝𝑛−𝑘}𝑥^𝑞_𝑛−𝑚,𝑛∈ℕ₀, (1.1) where 𝐴,𝑝,𝑞∈(0,+∞) and 𝑘,𝑚∈ℕ,𝑘≠𝑚, was put forward by Stević at many conferences (see, e.g., [1–3]). For numerous papers in this area and some closely related results, see [1–39] and the references cited therein.

In [4, 24], the authors proved some conditions for the global asymptotic stability of the positive equilibrium to the difference equation given by𝑦_𝑛+1𝑦=𝐴+^𝑛𝑦_𝑛−𝑘,𝑛∈ℕ₀, (1.2) with 𝐴>0,𝑘∈ℕ.

Motivated by these papers, the authors of [8] studied the quantitative bounds for the recursive equation (1.2) where 𝑦_−𝑘,…,𝑦₋₁,𝑦₀,𝐴>0, and 𝑘∈ℕ⧵{1}, and quantitative bounds of the form 𝑅_𝑖≤𝑦_𝑖≤𝑆_𝑖,𝑖≥𝑘+1 were provided. Exponential convergence was shown to persist for all solutions. The authors also took 𝐴=𝑘=2 as an example, and eventually obtained the concrete bounds as follows:2+^𝑛−2_{𝑖=𝑛−3}(1−(12)17^{[(𝑖+2)/10]})≤𝑦_𝑛≤2+^𝑛−2_{𝑖=𝑛−3}2(1+(3)^{2[(𝑖−3)/10]+1}),𝑛>6. (1.3) In [20], Stević investigated positive solutions of the following difference equation:𝑥_𝑛+1𝑥=𝐴+^𝑝𝑛𝑥^𝑟_𝑛−1,𝑛∈ℕ₀, (1.4) where 𝐴,𝑝,𝑟∈(0,+∞), and gave a complete picture concerning the boundedness character of the positive solutions to (1.4) as well as of positive solutions of the following counterpart in the class of max-type difference equations:𝑦_𝑛+1𝑦=max{𝐴,^𝑝𝑛𝑦^𝑟_𝑛−1},𝑛∈ℕ₀, (1.5) where 𝐴,𝑝,𝑟 are positive real numbers.

Motivated by the above work and works in [6, 9, 10, 12, 17, 21, 22], our aim in this paper is to discuss the quantitative bounds of the solutions to the following higher-order difference equation:𝑥_𝑛+1𝑥=𝐴+^𝑝𝑛𝑥^𝑝𝑘+1_𝑛−𝑘,𝑛∈ℕ₀, (1.6) where 𝐴,𝑝∈(0,+∞),𝑘∈ℕ, and the initial values are positive. Following the methods and ideas from [8], we obtain theoretically computable explicit bounds of the form𝐴+^𝑛−2_{𝑗=𝑛−𝑘−1}𝑎(2[𝑗+𝑘−14𝑘+2])^{𝑝𝑛−𝑗−1}≤𝑥_𝑛≤𝐴+^𝑛−2_{𝑗=𝑛−𝑘−1}𝑏(2[𝑗−𝑘−24𝑘+2]+1)^{𝑝𝑛−𝑗−1} (1.7) which are independent of the positive initial values 𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₀.

Our results extend those ones in [8], in which the case 𝑝=1 was considered, and also in some way improve those in [20], in which the case 𝑘=1 was considered.

On the other hand, inspired by the study in [19] we also investigate the quantitative bounds for the positive solutions to the following max-type recursive equation:𝑦_𝑛+1𝑦=max{𝐴,^𝑝𝑛𝑦^𝑝𝑘+1_𝑛−𝑘},𝑛∈ℕ₀, (1.8) where 𝐴,𝑝∈(0,+∞),𝑘∈ℕ, and some similar results are established.

We want to point out that the boundedness characters of (1.1) and (1.8) for the case 𝑘=1 and 𝑚∈ℕ, including our particular case, have been recently solved by Stević and presented at several conferences (see also [25]).

2. Auxiliary Results

In this section, we will present several preliminary lemmas needed to prove the main results in Section 3.

The following lemma can be easily proved.

Lemma 2.1. Equation (1.6) has a unique positive equilibrium point 𝑥>𝐴.

Now, let us define a first-order difference equation given by𝐴𝑢(𝑛+1)=(𝐴+𝑢(𝑛)^𝑟)^𝑝+1(𝐴+𝑢(𝑛)^𝑟)^𝑝𝑘+1,𝑛∈ℕ₀, (2.1) where 𝐴,𝑝>0 are identical to those of (1.6), 𝑟=∑^𝑘_𝑖=1𝑝^𝑖, and the initial value 𝑢(0)>0.

If 𝑝=1, then (2.1) reduces to the sequence {𝑥(𝑖)} defined in [8].

Lemma 2.2. Equation (2.1) has a unique positive equilibrium if 𝑝>1 and 𝐴≥[𝑟𝑝(𝑝^𝑘−1)]^1/𝑝 or 0<𝑝≤1.

Proof. Suppose that 𝑥>0 is an equilibrium point of (2.1), then we have 𝐴𝑥=(𝐴+𝑥^𝑟)^𝑝+1(𝐴+𝑥^𝑟)^𝑝𝑘+1. (2.2) Let 𝐹(𝑥)=𝑥(𝐴+𝑥^𝑟)^𝑝𝑘+1−𝐴(𝐴+𝑥^𝑟)^{𝑝(𝑝𝑘−1)}−1, then it suffices to show that 𝐹(𝑥) has only one positive fixed point. The derivative of 𝐹(𝑥) is 𝐹^′(𝑥)=(𝐴+𝑥^𝑟)^{𝑝𝑘+1−𝑝−1}[(𝐴+𝑥^𝑟)^𝑝+1+𝑟𝑥^𝑟𝑝^𝑘+1(𝐴+𝑥^𝑟)^𝑝−𝑟𝐴𝑝(𝑝^𝑘−1)𝑥^𝑟−1]=(𝐴+𝑥^𝑟)^{𝑝𝑘+1−𝑝−1}{(𝐴+𝑥^𝑟)^𝑝+1+𝑟𝑥^𝑟−1[𝑥𝑝^𝑘+1(𝐴+𝑥^𝑟)^𝑝−𝐴𝑝(𝑝^𝑘−1)]}. (2.3)(i)If 𝑝≤1, then obviously 𝐹^′(𝑥)>0 for 𝑥≥0.(ii)If 𝑝>1 and 𝑥≥1, then 𝐹^′(𝑥)>0 follows from (𝐴+1)^𝑝>𝐴.(iii)If 𝑝>1 and 0<𝑥<1, we have (𝐴+𝑥^𝑟)^𝑝+1+𝑟𝑥^𝑟−1[𝑥𝑝^𝑘+1(𝐴+𝑥^𝑟)^𝑝−𝐴𝑝(𝑝^𝑘−1)]>(𝐴+𝑥^𝑟)^𝑝+1−𝑟𝑝𝑥^𝑟−1𝐴(𝑝^𝑘−1)>𝐴(𝐴^𝑝−𝑟𝑝(𝑝^𝑘−1))≥0. (2.4) Hence 𝐹^′(𝑥)>0.
Through above analysis, if 𝑝>1 and 𝐴>[𝑟𝑝(𝑝^𝑘−1)]^1/𝑝 or 0<𝑝≤1, then 𝐹(𝑥) is monotonically increasing on (0,+∞). Hence the uniqueness of positive equilibrium of (2.1) follows from 𝐹(0)=−𝐴^{𝑝𝑘+1−𝑝+1}−1<0, and lim_𝑥→+∞𝐹(𝑥)=+∞.

Lemma 2.3. If 𝑝>1 and 𝐴≥[𝑟𝑝(𝑝^𝑘−𝑝)]^1/𝑝 or 0<𝑝≤1, then the unique equilibrium point of (1.6) has the form 𝐴+𝜆^𝑟, where 𝜆>0 is the unique positive equilibrium of (2.1).

Proof. Defining 𝜌(𝑥)=𝑥(𝐴+𝑥^𝑟)^𝑝−(𝐴+𝑥^𝑟),𝑥>0, simply we have that 𝜌(𝑥) has a unique positive zero denoted by 𝜆, that is, 𝜆(𝐴+𝜆^𝑟)^𝑝=(𝐴+𝜆^𝑟).If 𝑝=1, then 𝜆=1, and thus 𝐴𝜆=(𝐴+𝜆^𝑟)^𝑝+1(𝐴+𝜆^𝑟)^𝑝𝑘+1,𝐴+𝜆^𝑟=𝐴+(𝐴+𝜆^𝑟)^𝑝(𝐴+𝜆^𝑟)^𝑝𝑘+1. (2.5)If 𝑝>0 and 𝑝≠1, then 𝐴+𝜆^𝑟=𝜆^1/(1−𝑝),𝜆(𝐴+𝜆^𝑟)^𝑝=𝐴+𝜆^𝑟=𝐴+(𝐴+𝜆^𝑟)^𝑝(𝐴+𝜆^𝑟)^𝑝𝑘+1. (2.6) Hence 𝐴𝜆=(𝐴+𝜆^𝑟)^𝑝+1(𝐴+𝜆^𝑟)^𝑝𝑘+1. (2.7)From above analysis, we conclude that 𝜆 and 𝐴+𝜆^𝑟 are the unique equilibriums of (2.1) and (1.6), respectively.

3. Quantitative Bounds of Solutions to (1.6)

In this section, through analyzing the boundedness of (1.6) we mainly present two explicit bounds for the positive solutions to (1.6).

Let the positive sequence {𝑥_𝑖}^∞_𝑖=−𝑘 be a solution to (1.6), then for 𝑛≥−𝑘 we define𝜃_𝑛=𝑥_𝑛+1𝑥^𝑝_𝑛. (3.1) It follows from (3.1) and (1.6) that𝜃_𝑛=𝐴𝑥^𝑝_𝑛+1𝑥^𝑝𝑘+1_𝑛−𝑘,𝑛∈ℕ₀. (3.2) Combining (3.1) and (1.6), we can simply obtain that𝑥_𝑛𝑥=𝐴+^𝑝𝑛−1𝑥^𝑝𝑘+1_{𝑛−𝑘−1}𝑥=𝐴+^𝑝𝑛−1𝑥^𝑝2_𝑛−2𝑥^𝑝2_𝑛−2𝑥^𝑝3_𝑛−3⋯𝑥^𝑝𝑘_𝑛−𝑘𝑥^𝑝𝑘+1_{𝑛−𝑘−1}=𝐴+𝜃^𝑝_𝑛−2𝜃^𝑝2_𝑛−3⋯𝜃^𝑝𝑘_{𝑛−𝑘−1},𝑛∈ℕ. (3.3) By (3.2) and (3.3), the identity𝜃_𝑛=𝐴(𝐴+𝜃^𝑝_𝑛−2𝜃^𝑝2_𝑛−3⋯𝜃^𝑝𝑘_{𝑛−𝑘−1})^𝑝+1(𝐴+𝜃^𝑝_{𝑛−𝑘−2}𝜃^𝑝2_{𝑛−𝑘−3}⋯𝜃^𝑝𝑘_{𝑛−2𝑘−1})^𝑝𝑘+1 (3.4) holds for all 𝑛≥𝑘+1.

Note that 𝑥_𝑖>𝐴 for 𝑖≥1, and hence it follows from (3.2) that0<𝜃_𝑖<𝐴^1−𝑝+1𝐴^𝑝𝑘+1,𝑖≥𝑘+1. (3.5) Let us define two sequences {𝑆_𝑖}^∞_𝑖=𝑘+1 and {𝐵_𝑖}^∞_𝑖=𝑘+1 recursively in the following way:𝐵_𝑖=𝐴(𝐴+𝑆^𝑝_𝑖−2𝑆^𝑝2_𝑖−3⋯𝑆^𝑝𝑘_{𝑖−𝑘−1})^𝑝+1(𝐴+𝑆^𝑝_{𝑖−𝑘−2}𝑆^𝑝2_{𝑖−𝑘−3}⋯𝑆^𝑝𝑘_{𝑖−2𝑘−1})^𝑝𝑘+1,𝑆_𝑖=𝐴(𝐴+𝐵^𝑝_𝑖−2𝐵^𝑝2_𝑖−3⋯𝐵^𝑝𝑘_{𝑖−𝑘−1})^𝑝+1(𝐴+𝐵^𝑝_{𝑖−𝑘−2}𝐵^𝑝2_{𝑖−𝑘−3}⋯𝐵^𝑝𝑘_{𝑖−2𝑘−1})^𝑝𝑘+1 (3.6) for all 𝑖≥3𝑘+3, and the initial values satisfy𝑆_𝑖=0,𝐵_𝑖=𝐴^1−𝑝+1𝐴^𝑝𝑘+1,𝑘+1≤𝑖≤3𝑘+2. (3.7) Apparently 𝑆_𝑖≤𝜃_𝑖≤𝐵_𝑖 for 𝑖≥𝑘+1, and the problem of bounding (1.6) reduces to consideration of the recursive dependent sequences {𝑆_𝑖},{𝐵_𝑖}.

Lemma 3.1. The sequences {𝑆_𝑖} and {𝐵_𝑖} are nondecreasing and nonincreasing, respectively.

Proof. It follows from (3.7) that for 𝑘+1≤𝑖≤3𝑘+1 we have 𝑆_𝑖+1=𝑆_𝑖=0 and 𝐵_𝑖+1=𝐵_𝑖=1𝐴^𝑝−1+1𝐴^𝑝𝑘+1. (3.8) Hence assume that 𝑆_𝑖+1≥𝑆_𝑖 and 𝐵_𝑖+1≤𝐵_𝑖 for 𝑘+1≤𝑖<𝑀(𝑀≥3𝑘+2). By induction, we have that 𝐵_𝑀+1=𝐴(𝐴+𝑆^𝑝_𝑀−1𝑆^𝑝2_𝑀−2⋯𝑆^𝑝𝑘_𝑀−𝑘)^𝑝+1(𝐴+𝑆^𝑝_{𝑀−𝑘−1}𝑆^𝑝2_{𝑀−𝑘−2}⋯𝑆^𝑝𝑘_𝑀−2𝑘)^𝑝𝑘+1≤𝐴(𝐴+𝑆^𝑝_𝑀−2𝑆^𝑝2_𝑀−3⋯𝑆^𝑝𝑘_{𝑀−𝑘−1})^𝑝+1(𝐴+𝑆^𝑝_{𝑀−𝑘−2}𝑆^𝑝2_{𝑀−𝑘−3}⋯𝑆^𝑝𝑘_{𝑀−2𝑘−1})^𝑝𝑘+1=𝐵_𝑀. (3.9) Through similar calculations, we have 𝑆_𝑀+1≥𝑆_𝑀, and by induction the lemma is proved.

Theorem 3.2. For (2.1) with 𝑢(0)=0, let 𝑆^∗_𝑛=𝑢(2[(𝑛+𝑘−1)/(4𝑘+2)]) and 𝐵^∗_𝑛=𝑢(2[(𝑛+𝑘−2)/(4𝑘+2)]+1) for 𝑛≥k+2. Then the inequality 𝑆^∗_𝑛≤𝜃_𝑛≤𝐵^∗_𝑛 (3.10) holds for all 𝑛≥𝑘+2.

Proof. From (3.7) and the definitions of 𝑆^∗_𝑛,𝐵^∗_𝑛, we have that 𝑆_𝑖=𝑆^∗_𝑖and𝐵_𝑖=𝐵^∗_𝑖 for 𝑘+2≤𝑖≤3𝑘+2. Thus, assume that 𝑆_𝑖≤𝑆^∗_𝑖 and 𝐵_𝑖≤𝐵^∗_𝑖 for 𝑘+2≤𝑖<𝑀(𝑀≥3𝑘+3). Then 𝑆_𝑀=𝐴(𝐴+𝐵^𝑝_𝑀−2𝐵^𝑝2_𝑀−3⋯𝐵^𝑝𝑘_{𝑀−𝑘−1})^𝑝+1(𝐴+𝐵^𝑝_{𝑀−𝑘−2}𝐵^𝑝2_{𝑀−𝑘−3}⋯𝐵^𝑝𝑘_{𝑀−2𝑘−1})^𝑝𝑘+1≥𝐴(𝐴+𝐵^{𝑝+⋯+𝑝𝑘}_{𝑀−𝑘−1})^𝑝+1(𝐴+𝐵^{𝑝+⋯+𝑝𝑘}_{𝑀−2𝑘−1})^𝑝𝑘+1≥𝐴(𝐴+𝐵^𝑟_{𝑀−2𝑘−1})^𝑝+1(𝐴+𝐵^𝑟_{𝑀−2𝑘−1})^𝑝𝑘+1≥𝐴(𝐴+(𝐵^∗_{𝑀−2𝑘−1})^𝑟)^𝑝+1(𝐴+(𝐵^∗_{𝑀−2𝑘−1})^𝑟)^𝑝𝑘+1=𝐴(𝐴+(𝑢(2[(𝑀−3𝑘−3)/(4𝑘+2)]+1))^𝑟)^𝑝+1(𝐴+(𝑢(2[(𝑀−3𝑘−3)/(4𝑘+2)]+1))^𝑟)^𝑝𝑘+1=𝑢(2[𝑀+𝑘−14𝑘+2])=𝑆^∗_𝑀. (3.11) Similar calculations lead to 𝐵_𝑀≤𝐵^∗_𝑀, and inductively the theorem can be proved.

Theorem 3.3. If the solution 𝑢(𝑖) to (2.1) with 𝑢(0)=0 converges to the unique equilibrium 𝜆 under the conditions in Lemma 2.2 and there exist two sequences {𝑎(𝑖)} and {𝑏(𝑖)} which are lower and upper bounds for (2.1) such that 𝑎(𝑖)≤𝑢(𝑖)≤𝑏(𝑖),𝑖≥𝑘+2, and lim_𝑖→+∞𝑎(𝑖)=lim_𝑖→+∞𝑏(𝑖)=𝜆, then the solutions to (1.6) have explicit bounds of the following form: 𝐴+^𝑛−2_{𝑗=𝑛−𝑘−1}𝑎(2[𝑗+𝑘−14𝑘+2])^{𝑝𝑛−𝑗−1}≤𝑥_𝑛≤𝐴+^𝑛−2_{𝑗=𝑛−𝑘−1}𝑏(2[𝑗−𝑘−24k+2]+1)^{𝑝𝑛−𝑗−1} (3.12) for all 𝑛≥3𝑘+3.

Proof. The proof follows directly from Lemma 2.2 and Theorem 3.2, and thus is omitted.

Note that Theorem 3.3 and Lemma 2.3 imply the following corollary.

Corollary 3.4. If the solution 𝑢(𝑖) to (2.1) with 𝑢(0)=0 converges to the unique equilibrium 𝜆 under the conditions in Lemma 2.2, then the unique equilibrium 𝐴+𝜆^𝑟 of (1.6) is a global attractor.

By Theorem 3.3, it suffices to determine the explicit bounds for (2.1). In the following, a simple case would be taken. For example,

if the parameters 𝐴,𝑝,𝑘 are fixed, then by (2.1) we get𝑢_𝑛+2=𝐴(𝐴+(𝐴/(𝐴+𝑢(𝑛)^𝑟)^𝑝+1/(𝐴+𝑢(𝑛)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝+1(𝐴+(𝐴/(𝐴+𝑢(𝑛)^r)^𝑝+1/(𝐴+𝑢(𝑛)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝𝑘+1,𝑛∈ℕ₀. (3.13)

Denote 𝑢(𝑖)=𝛿(𝑖)+𝜆 for 𝑖≥0 (𝜆 the unique equilibrium of (2.1)), then we have that, being for 𝑛≥0,𝐴𝛿(𝑛+2)=(𝐴+(𝐴/(𝐴+(𝛿(𝑛)+𝜆)^𝑟)^𝑝+1/(𝐴+(𝛿(𝑛)+𝜆)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝+1(𝐴+(𝐴/(𝐴+(𝛿(𝑛)+𝜆)^𝑟)^𝑝+1/(𝐴+(𝛿(𝑛)+𝜆)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝𝑘+1−𝜆=𝛾(𝛿(𝑛)), (3.14) where the function 𝛾 is defined by𝐴𝛾(𝑥)=(𝐴+(𝐴/(𝐴+(𝑥+𝜆)^𝑟)^𝑝+1/(𝐴+(𝑥+𝜆)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝+1(𝐴+(𝐴/(𝐴+(𝑥+𝜆)^𝑟)^𝑝+1/(𝐴+(𝑥+𝜆)^𝑟)^𝑝𝑘+1)^𝑟)^𝑝𝑘+1−𝜆 (3.15) for 𝑥>−𝜆.

Example 3.5 ((𝐴=3,𝑝=1,𝑘=2)). Then 𝛾(𝑥) reduces to the following form: 4𝛾(𝑥)=3+(4/(3+(𝑥+1)²))²−1,𝑥>−1. (3.16) Obviously, 𝛾 is monotonically increasing for 𝑥>−1 and 𝛾(0)=0. By simplifying 𝛾, we have 𝛾(𝑥)=𝑥(𝑥³+4𝑥²+12𝑥+16)3𝑥⁴+12𝑥³+36𝑥².+48𝑥+64 (3.17) Having the function 𝜑 defined via 𝜑(𝑥)=𝛾(𝑥)/𝑥, we get the derivative of 𝜑 as follows: 𝜑^′(𝑥)=−3𝑥⁶+24𝑥⁵+120𝑥⁴+384𝑥³+624𝑥²+640𝑥(3𝑥⁴+12𝑥³+36𝑥²+48𝑥+64)²<0 (3.18) for all 𝑥>0. Thus 𝜑(𝑥)<𝜑(0)=1/4 for 𝑥>0 and 𝛾(𝑡^𝑛)𝑡^𝑛+2=𝛾(𝑡^𝑛)𝑡^𝑛1𝑡²<14𝑡². (3.19) Therefore 𝛾(𝑡^𝑛)<𝑡^𝑛+2 whenever 𝑡≥1/2. In addition, for −1≤𝑥<0,𝛾(𝑥)<0 and both 𝜉(𝑥)=𝑥³+4𝑥²+12𝑥+16 and 𝜂(𝑥)=3𝑥⁴+12𝑥³+36𝑥²+48𝑥+64 are monotonically increasing. Hence we have |𝛾(−𝑡^𝑛)−𝑡^𝑛+2|<|𝜉(0)|1𝜂(−1)𝑡²=16143𝑡², (3.20) and 𝛾(−𝑡^𝑛)>−𝑡^𝑛+2 whenever 𝑡²≥16/43.
Now set 𝜋⁺(𝑛)=(1/2)^𝑛, for 𝑛≥0, and 𝜋⁻(𝑛)=(16/43)^𝑛/2. Note that 𝛿(0)=−1=−𝜋⁻(0) and 𝛿(1)=1/3<𝜋⁺(1). Thus suppose that −1≤−𝜋⁻(2𝑖)≤𝛿(2𝑖)≤0 and 0≤𝛿(2𝑖+1)≤𝜋⁺(2𝑖+1), for 0≤𝑖≤𝑁(𝑁≥0). Then by induction we have 𝛿(2𝑁+2)=𝛾(𝛿(2𝑁))≥𝛾(−𝜋⁻(2𝑁))=𝛾(−(16)43^𝑁)≥−𝜋⁻(2𝑁+2),𝛿(2𝑁+3)=𝛾(𝛿(2𝑁+1))≤𝛾(𝜋⁺1(2𝑁+1))=𝛾((2)^2𝑁+1)≥𝜋⁺(2𝑁+3). (3.21) Therefore since the fact that 𝛿(𝑖)≤0 for 𝑖 even and 𝛿(𝑖)≥0 for 𝑖 odd, we obtain that, for 𝑖≥0, −(16)43^𝑖/21≤𝛿(𝑖)≤(2)^𝑖,1−(16)43^𝑖/21≤𝑢(𝑖)≤1+(2)^𝑖. (3.22) Employing Theorem 3.3, we get the bounds 3+^𝑛−2_{𝑖=𝑛−3}(1−(16)43^{[(𝑖+1)/10]})≤𝑥_𝑛≤3+^𝑛−2_{𝑖=𝑛−3}1(1+(2)^{2[(𝑖−4)/10]+1}),𝑛≥9. (3.23)

4. Quantitative Bounds for Solutions to (1.8)

In this section, the upper and lower bounds of solutions to (1.8) are given, and first we present a lemma concerning the equilibrium points of (1.8).

Lemma 4.1. If 𝐴>1, then (1.8) has a unique positive equilibrium 𝑦=𝐴; and if 0<𝐴≤1, then (1.8) has a unique positive equilibrium 𝑦=1.

The proof is simple and thus omitted.

Suppose that {𝑦_𝑖}^𝑝_𝑖=−𝑘 is a positive solution to (1.8), and by the transformation𝛽_𝑛=𝑦_𝑛+1𝑦^𝑝_𝑛,𝑛≥−𝑘, (4.1) we have that𝛽_𝑛𝐴=max{𝑦^𝑝_𝑛,1𝑦^𝑝𝑘+1_𝑛−𝑘},𝑛∈ℕ₀. (4.2) It follows from (4.1) and (1.8) that𝑦_𝑛𝑦=max{𝐴,^𝑝𝑛−1𝑦^𝑝𝑘+1_{𝑛−𝑘−1}}=max{𝐴,𝛽^𝑝_𝑛−2𝛽^𝑝2_𝑛−3⋯𝛽^𝑝𝑘_{𝑛−𝑘−1}},𝑛∈ℕ. (4.3) Employing (4.2) and (4.3), we obtain that, for 𝑛≥−𝑘,𝛽_𝑛𝐴=max{max{𝐴,𝛽^𝑝_𝑛−2𝛽^𝑝2_𝑛−3⋯𝛽^𝑝𝑘_{𝑛−𝑘−1}}^𝑝,1max{𝐴,𝛽^𝑝_{𝑛−𝑘−2}𝛽^𝑝2_{𝑛−𝑘−3}⋯𝛽^𝑝𝑘_{𝑛−2𝑘−1}}^𝑝𝑘+1}. (4.4) For two nonnegative sequences {𝐿_𝑖} and {𝐻_𝑖}, let 𝐿_𝑖≤𝛽_𝑖≤𝐻_𝑖 for 𝑘+1≤𝑖<𝑇(𝑇≥3𝑘+2).

Then according to (4.4) and the sequences𝐿_𝑇𝐴=max{max{𝐴,𝐻^𝑝_𝑇−2𝐻^𝑝2_𝑇−3⋯𝐻^𝑝𝑘_{𝑇−𝑘−1}}^𝑝,1max{𝐴,𝐻^𝑝_{𝑇−𝑘−2}𝐻^𝑝2_{𝑇−𝑘−3}⋯𝐻^𝑝𝑘_{𝑇−2𝑘−1}}^𝑝𝑘+1𝐻},_𝑇𝐴=max{max{𝐴,𝐿^𝑝_𝑇−2𝐻^𝑝2_𝑇−3⋯𝐿^𝑝𝑘_{𝑇−𝑘−1}}^𝑝,1max{𝐴,𝐿^𝑝_{𝑇−𝑘−2}𝐿^𝑝2_{𝑇−𝑘−3}⋯𝐿^𝑝𝑘_{𝑇−2𝑘−1}}^𝑝𝑘+1}, (4.5) we have that 𝐿_𝑇≤𝛽_𝑇≤𝐻_𝑇.

Note that 𝑦_𝑖≥𝐴 for 𝑖≥1, and thus from (4.2) we get0<𝛽_𝑖1≤max{𝐴^𝑝−1,1𝐴^𝑝𝑘+11}=𝐴1^𝑝𝑘+1,𝐴<1,𝐴^𝑝−1,𝐴≥1 (4.6) for 𝑘+1≤𝑖≤3𝑘+2.

Lemma 4.2. Let 𝐿_𝑖=0and𝐻_𝑖=max{1/𝐴^𝑝−1,1/A^𝑝𝑘+1} for 𝑘+1≤𝑖≤3𝑘+2, then the sequences {𝐿_𝑖} and {𝐻_𝑖} are nondecreasing and nonincreasing, respectively.

Proof. Assume that 𝐿_𝑖≤𝐿_𝑖+1 and 𝐻_𝑖+1≤𝐻_𝑖 for 𝑘+1≤𝑖<𝑇(𝑇≥3𝑘+2). Then we have that 𝐻_𝑇+1𝐴=max{max{𝐴,𝐿^𝑝_𝑇−1𝐿^𝑝2_𝑇−2⋯𝐿^𝑝𝑘_𝑇−𝑘}^𝑝,1max{𝐴,𝐿^𝑝_{𝑇−𝑘−1}𝐿^𝑝2_{𝑇−𝑘−2}⋯𝐿^𝑝𝑘_𝑇−2𝑘}^𝑝𝑘+1}𝐴≤max{max{𝐴,𝐿^𝑝_𝑇−2𝐿^𝑝2_𝑇−3⋯𝐿^𝑝𝑘_{𝑇−𝑘−1}}^𝑝,1max{𝐴,𝐿^𝑝_{𝑇−𝑘−2}𝐿^𝑝2_{𝑇−𝑘−3}⋯𝐿^𝑝𝑘_{𝑇−2𝑘−1}}^𝑝𝑘+1}=𝐻_𝑇. (4.7) Analogous argument gives that 𝐿_𝑇≤𝐿_𝑇+1, and the lemma can be proved inductively.

Now we will take in the other first-order recursive equation𝐴𝑣(𝑛+1)=max{max{𝐴,(𝑣(𝑛))^𝑟}^𝑝,1max{𝐴,(𝑣(𝑛))^𝑟}^𝑝𝑘+1},𝑛∈ℕ₀, (4.8) where 𝐴,𝑝 equal those of (1.8), 𝑟=𝑝+𝑝²+⋯+𝑝^𝑘, and the initial value 𝑣(0)=0.

Through Lemma 4.2, simpler bounds for {𝛽_𝑖} are given below.

Theorem 4.3. The inequality 𝐿^∗_𝑛≤𝛽_𝑛≤𝐻^∗_𝑛 (4.9) holds for 𝑛≥𝑘+2, where 𝐿^∗_𝑛=𝑣(2[(𝑛+𝑘−1)/(4𝑘+2)]) and 𝐻^∗_𝑛=𝑣(2[(𝑛+𝑘−2)/(4𝑘+2)]+1).

Proof. Let 𝐿_𝑖=0and𝐻_𝑖=max{1/𝐴^𝑝−1,1/𝐴^𝑝𝑘+1} for 𝑘+2≤𝑖≤3𝑘+2, and by the definitions of {𝐿^∗_𝑛} and {𝐻^∗_𝑛} we have that 𝐿^∗_𝑖=𝐿_𝑖 and 𝐻^∗_𝑖=𝐻_𝑖 for 𝑘+2≤𝑖≤3𝑘+2.
Assume that 𝐿^∗_𝑖≤𝐿_𝑖 and 𝐻^∗_𝑖≥𝐻_𝑖 for 𝑘+2≤i<𝑇(𝑇≥3𝑘+3), then 𝐻_𝑇𝐴=max{max{𝐴,𝐿^𝑝_𝑇−2𝐿^𝑝2_𝑇−3⋯𝐿^𝑝𝑘_{𝑇−𝑘−1}}^𝑝,1max{𝐴,𝐿^𝑝_{𝑇−𝑘−2}𝐿^𝑝2_{𝑇−𝑘−3}⋯𝐿^𝑝𝑘_{𝑇−2𝑘−1}}^𝑝𝑘+1}𝐴≤max{max{𝐴,𝐿^{𝑝+⋯+𝑝𝑘}_{𝑇−𝑘−1}}^𝑝,1max{𝐴,𝐿^{𝑝+⋯+𝑝𝑘}_{𝑇−2𝑘−1}}^𝑝𝑘+1}𝐴≤max{max{𝐴,𝐿^𝑟_{𝑇−2𝑘−1}}^𝑝,1max{𝐴,𝐿^𝑟_{𝑇−2𝑘−1}}^𝑝𝑘+1}𝐴≤max{max{𝐴,(𝐿^∗_{𝑇−2𝑘−1})^𝑟}^𝑝,1max{𝐴,(𝐿^∗_{𝑇−2𝑘−1})^𝑟}^𝑝𝑘+1}𝐴=max{max{𝐴,(𝑣(2[(𝑇−𝑘−2)/(4𝑘+2)]))^𝑟}^𝑝,1max{𝐴,(𝑣(2[(𝑇−𝑘−2)/(4𝑘+2)]))^𝑟}^𝑝𝑘+1}=𝑣(2[𝑇−𝑘−24𝑘+2]+1)=𝐻^∗_𝑇. (4.10) Similar computations lead to the inequality 𝐿^∗_𝑇≤𝐿_𝑇, and the theorem follows by induction.

Theorem 4.3 and (4.8) imply the following result.

Theorem 4.4. Suppose that there exist two positive sequences {𝑐(𝑗)} and {𝑑(𝑗)} which are bounds for a positive solution to (4.8) with 𝑣(0)=0 such that 𝑐(𝑗)≤𝑣(𝑗)≤𝑑(𝑗),𝑗≥𝑘+2. (4.11) Then the solutions to (1.8) have explicit upper and lower bounds of the following form: max{𝐴,^𝑛−2_{𝑖=𝑛−𝑘−1}𝑐(2[𝑖+𝑘−14𝑘+2])^{𝑝𝑛−𝑖−1}}≤𝑦_𝑛≤max{𝐴,^𝑛−2_{𝑖=𝑛−𝑘−1}𝑑(2[𝑖−𝑘−24𝑘+2]+1)^{𝑝𝑛−𝑖−1}} (4.12) for all 𝑛≥3𝑘+3.

5. Conclusion

In this paper, we investigate a particular case of a higher-order difference equation by Stević which is a natural extension of that one in [8], and mainly present improved results which give computable approaches for quantitative bounds of solutions to (1.6). However, the methods are only effective for small parameters, because complex polynomials will arise in the process of computing for large parameters 𝐴,𝑝,𝑘.

On the basis of Corollary 3.4 and Theorem 4.4, we suggest to study the behaviors, particularly the convergence and stability, of positive solutions to the following two recursive equations:𝐴𝑢(𝑛+1)=(𝐴+𝑢(𝑛)^𝑟)^𝑝+1(𝐴+𝑢(𝑛)^𝑟)^𝑝𝑘+1,𝑛∈ℕ₀,𝐴𝑣(𝑛+1)=max{max{𝐴,(𝑣(𝑛))^𝑟}^𝑝,1max{𝐴,(𝑣(𝑛))^𝑟}^𝑝𝑘+1},𝑛∈ℕ₀, (5.1) where 𝐴,𝑝∈(0,+∞),𝑘∈ℕ, and 𝑟=Σ^𝑘_𝑖=1𝑝^𝑖.

Acknowledgment

The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation of the consequences in the paper.