Quantitative Bounds for Positive Solutions of a Stević Difference Equation

This paper studies the behavior of positive solutions to the following particular case of a difference equation by Stević xn 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.


Introduction
The study regarding the behavior of positive solutions to the difference equation 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 by Motivated by these papers, the authors of 8 studied the quantitative bounds for the recursive equation 1.2 where y −k , . . ., y −1 , y 0 , 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: , n > 6. 1.3 In 20 , Stević investigated positive solutions of the following difference equation: 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: 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: 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 form 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: 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 .

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 by where A, p > 0 are identical to those of 1.6 , r k i 1 p 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
Proof.Suppose that x > 0 is an equilibrium point of 2.1 , then we have 2.2 Let F x x A x r p k 1 − A A x r p p k −1 − 1, then it suffices to show that F x has only one positive fixed point.The derivative of F x is 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
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 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

2.7
From above analysis, we conclude that λ and A λ r are the unique equilibriums of 2.1 and 1.6 , respectively.

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 define

3.1
It follows from 3.1 and 1.6 that Combining 3.1 and 1.6 , we can simply obtain that By 3.2 and 3.3 , the identity holds for all n ≥ k 1.
Note that x i > A for i ≥ 1, and hence it follows from 3.2 that Let us define two sequences {S i } ∞ i k 1 and {B i } ∞ i k 1 recursively in the following way: for all i ≥ 3k 3, and the initial values satisfy Proof.It follows from 3.7 that for k 1 ≤ i ≤ 3k 1 we have S i 1 S i 0 and

3.8
Hence assume that S i 1 ≥ S i and By induction, we have that 3.9 Through similar calculations, we have S M 1 ≥ S M , and by induction the lemma is proved.
Then the inequality holds for all n ≥ k 2.
Proof.From 3.7 and the definitions of S * n , B * n , we have that S i S * i and

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:
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 get

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,

3.14
where the function γ is defined by Example 3.5 A 3, p 1, k 2 .Then γ x reduces to the following form:

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,

3.22
Employing Theorem 3.3, we get the bounds , 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 .The proof is simple and thus omitted.Suppose that {y i } p i −k is a positive solution to 1.8 , and by the transformation we have that It follows from 4.1 and 1.8 that Employing 4.2 and 4.3 , we obtain that, for n ≥ −k, For two nonnegative sequences {L i } and Then according to 4.4 and the sequences we have that L T ≤ β T ≤ H T .Note that y i ≥ A for i ≥ 1, and thus from 4.2 we get Lemma 4.2.Let L i 0 and H i max{1/A p−1 , 1/A p k 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 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 equation where A, p equal those of 1.8 , r p p

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: 4.12 for all n ≥ 3k 3.

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:

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.

u n 1 AA u n r p 1 A u n r p k 1 ,
p ∈ 0, ∞ , k ∈ N, and r Σ k i 1 p i .
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. 2 • • • p k , and the initial value v 0 0. Through Lemma 4.2, simpler bounds for {β i } are given below.Let L i 0 and H i max{1/A p−1 , 1/A p k 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