Quantitative Bounds for Positive Solutions of a Stevi ´c Difference Equation

Volume 2010, Article ID 235808,14pages doi:10.1155/2010/235808

Research Article

Quantitative Bounds for Positive Solutions of a Stevi ´c Difference Equation

Wanping Liu and Xiaofan Yang

College of Computer Science, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Wanping Liu,[email protected] Received 8 November 2009; Revised 5 March 2010; Accepted 7 April 2010 Academic Editor: Leonid Berezansky

Copyrightq2010 W. Liu and X. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies the behavior of positive solutions to the following particular case of a difference equation by Stevi´cxn1 Axn^p/x^p_n−k^k1, n ∈ N0, whereA,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 Stevi´c max-type difference equation.

1. Introduction

The study regarding the behavior of positive solutions to the difference equation

xnA x^p_n−k

x^q_n−m, n∈N0, 1.1

whereA, p, q ∈0,∞andk, m∈N, k /m,was put forward by Stevi´c at many conferences see, e.g.,1–3. For numerous papers in this area and some closely related results, see1–39 and the references cited therein.

In4,24, the authors proved some conditions for the global asymptotic stability of the positive equilibrium to the difference equation given by

y_n1A yn

yn−k, n∈N0, 1.2

withA >0, k∈N.

Motivated by these papers, the authors of8studied the quantitative bounds for the recursive equation1.2wherey_−k, . . . , y₋₁, y0, A >0, andk∈N\{1},and quantitative bounds of the formRi ≤ yi ≤ Si, i ≥ k1 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

in−3

1−

12 17

_i2/10

≤yn≤2 ⁿ⁻²

in−3

1

2 3

_2i−3/101

, n >6. 1.3

In20, Stevi´c investigated positive solutions of the following difference equation:

x_n1A x^pn

x_n−1^r , n∈N0, 1.4

whereA, p, r ∈0,∞, and gave a complete picture concerning the boundedness character of the positive solutions to1.4as well as of positive solutions of the following counterpart in the class of max-type difference equations:

yn1max

A, y^pn

y^r_n−1

, n∈N0, 1.5

whereA, p, rare positive real numbers.

Motivated by the above work and works in6,9,10,12,17,21,22, our aim in this paper is to discuss the quantitative bounds of the solutions to the following higher-order difference equation:

xn1A x^pn

x_n−k^pk1, n∈N0, 1.6

whereA, p ∈0,∞, k ∈ N, and the initial values are positive. Following the methods and ideas from8, we obtain theoretically computable explicit bounds of the form

A ⁿ⁻²

jn−k−1

a

2 jk−1 4k2

p^n−j−1

≤xn≤A ⁿ⁻²

jn−k−1

b

2 j−k−2 4k2

1

p^n−j−1

1.7

which are independent of the positive initial valuesx_−k, x_−k1, . . . , x0.

Our results extend those ones in8, in which the casep1 was considered, and also in some way improve those in20, in which the casek1 was considered.

On the other hand, inspired by the study in19we also investigate the quantitative bounds for the positive solutions to the following max-type recursive equation:

yn1max

⎧⎨

⎩A, y^pn

y^p_n−k^k1

⎫⎬

⎭, n∈N0, 1.8 whereA, p∈0,∞, k∈N,and some similar results are established.

We want to point out that the boundedness characters of1.1and1.8for the case k 1 and m ∈ N, including our particular case, have been recently solved by Stevi´c and presented at several conferencessee also25.

2. Auxiliary Results

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

The following lemma can be easily proved.

Lemma 2.1. Equation1.6has a unique positive equilibrium pointx > A.

Now, let us define a first-order difference equation given by

un1 A

Aun^r_p 1

Aun^r_pk1, n∈N0, 2.1 whereA, p >0 are identical to those of1.6,r_k

i1pⁱ, and the initial valueu0>0.

Ifp1, then2.1reduces to the sequence{xi}defined in8.

Lemma 2.2. Equation2.1has a unique positive equilibrium ifp > 1 andA ≥rpp^k−1^1/por 0< p≤1.

Proof. Suppose thatx >0 is an equilibrium point of2.1, then we have

x A

Ax^rp 1

Ax^rpk1. 2.2 LetFx xAx^rpk1−AAx^rppk−1−1, then it suffices to show thatFxhas only one positive fixed point. The derivative ofFxis

Fx Ax^rpk1−p−1

Ax^rp1rx^rp^k1Ax^rp−rAp p^k−1

x^r−1 Ax^rpk1−p−1

Ax^rp1rx^r−1

xp^k1Ax^rp−Ap

p^k−1 .

2.3

iIfp≤1, then obviouslyFx>0 forx≥0.

iiIfp >1 andx≥1, thenFx>0 follows fromA1^p> A.

iiiIfp >1 and 0< x <1, we have Ax^rp1rx^r−1

xp^k1Ax^rp−Ap

p^k−1

>Ax^rp1−rpx^r−1A p^k−1

> A

A^p−rp

p^k−1

≥0.

2.4

HenceFx>0.

Through above analysis, ifp > 1 andA > rpp^k−1^1/p or 0 < p ≤ 1, thenFxis monotonically increasing on0,∞. Hence the uniqueness of positive equilibrium of2.1 follows fromF0 −A^pk1−p1−1<0, and lim_x→∞Fx ∞.

Lemma 2.3. Ifp >1 andA ≥ rpp^k−p^1/p or 0 < p≤ 1, then the unique equilibrium point of 1.6has the formAλ^r, whereλ >0 is the unique positive equilibrium of 2.1.

Proof. Definingρx xAx^rp−Ax^r, x > 0, simply we have thatρxhas a unique positive zero denoted byλ, that is,λAλ^rp Aλ^r.

Ifp1, thenλ1, and thus

λ A

Aλ^rp 1

Aλ^rpk1, Aλ^r A Aλ^rp

Aλ^rpk1. 2.5

Ifp >0 andp /1, then

Aλ^r λ^1/1−p, λAλ^rpAλ^r A Aλ^rp

Aλ^rpk1. 2.6

Hence

λ A

Aλ^rp 1

Aλ^rpk1. 2.7

From above analysis, we conclude thatλandAλ^rare the unique equilibriums of2.1and 1.6, respectively.

3. Quantitative Bounds of Solutions to 1.6

In this section, through analyzing the boundedness of1.6we mainly present two explicit bounds for the positive solutions to1.6.

Let the positive sequence{xi}^∞_i−kbe a solution to1.6, then forn≥ −kwe define

θn xn1

x^pn

. 3.1

It follows from3.1and1.6that

θn A x^pn

1

x^p_n−k^k1, n∈N0. 3.2

Combining3.1and1.6, we can simply obtain that

xnA x^p_n−1

x_n−k−1^pk1 Ax^p_n−1 x^p_n−2²

x^p_n−2²

x^p_n−3³ · · · x^p_n−k^k

x_n−k−1^pk1 Aθ_n−2^p θ^p_n−3² · · ·θ_n−k−1^pk , n∈N. 3.3

By3.2and3.3, the identity

θn A

Aθ_n−2^p θ^p_n−3² · · ·θ_n−k−1^pk _p 1

Aθ_n−k−2^p θ_n−k−3^p2 · · ·θ_n−2k−1^pk p^k1 3.4

holds for alln≥k1.

Note thatxi> Afori≥1, and hence it follows from3.2that

0< θi< A^1−p 1

A^pk1, i≥k1. 3.5

Let us define two sequences{Si}^∞_ik1and{Bi}^∞_ik1recursively in the following way:

Bi A

AS^p_i−2S^p_i−3² · · ·S^p_i−k−1^k p 1

AS^p_i−k−2S^p_i−k−3² · · ·S^p_i−2k−1^k p^k1,

Si A

AB_i−2^p B_i−3^p2 · · ·B^p_i−k−1^k p 1

AB_i−k−2^p B^p_i−k−3² · · ·B^p_i−2k−1^k p^k1

3.6

for alli≥3k3, and the initial values satisfy

Si0, BiA^1−p 1

A^pk1, k1≤i≤3k2. 3.7

Apparently Si ≤ θi ≤ Bi for i ≥ k 1, and the problem of bounding 1.6 reduces to consideration of the recursive dependent sequences{Si},{Bi}.

Lemma 3.1. The sequences{Si}and{Bi}are nondecreasing and nonincreasing, respectively.

Proof. It follows from3.7that fork1≤i≤3k1 we haveSi1 Si0 and

Bi1Bi 1

A^p−1 1

A^pk1. 3.8

Hence assume thatSi1 ≥SiandBi1 ≤Bifork1≤i < MM≥3k2. By induction, we have that

B_M1 A

AS^p_M−1S^p_M−2² · · ·S^p_M−k^k _p 1

AS^p_M−k−1S^p_M−k−2² · · ·S^p_M−2k^k p^k1

≤ A

AS^p_M−2S^p_M−3² · · ·S^p_M−k−1^k p 1

AS^p_M−k−2S^p_M−k−3² · · ·S^p_M−2k−1^k p^k1

BM.

3.9

Through similar calculations, we haveSM1 ≥SM, and by induction the lemma is proved.

Theorem 3.2. For2.1withu0 0, letS^∗_nu2nk−1/4k2andB^∗_nu2nk− 2/4k2 1forn≥k2. Then the inequality

S^∗_n≤θn≤B_n^∗ 3.10

holds for alln≥k2.

Proof. From3.7and the definitions ofS^∗_n, B_n^∗,we have thatSi S^∗_i andBi B_i^∗fork2 ≤ i≤3k2. Thus, assume thatSi≤S^∗_i andBi≤B_i^∗fork2≤i < MM≥3k3. Then

SM A

AB^p_M−2B_M−3^p2 · · ·B_M−k−1^pk p 1

AB^p_M−k−2B^p_M−k−3² · · ·B_M−2k−1^pk _p^k1

≥ A

AB^p···p_M−k−1^kp 1

AB^p···p_M−2k−1^k p^k1 ≥ A

AB^r_M−2k−1p 1

AB_M−2k−1^r p^k1

≥ A

A

B_M−2k−1^∗ rp 1 A

B_M−2k−1^∗ _rp^k1

A

A u2M−3k−3/4k2 1^r_p

1

A u2M−3k−3/4k2 1^rp^k1

u

2 Mk−1 4k2

S^∗_M.

3.11

Similar calculations lead toBM≤B_M^∗ , and inductively the theorem can be proved.

Theorem 3.3. If the solutionuito2.1withu0 0 converges to the unique equilibriumλunder the conditions inLemma 2.2 and there exist two sequences{ai} and {bi}which are lower and

upper bounds for2.1such thatai≤ui≤bi, i≥k2, and limi→∞ai limi→∞bi λ, then the solutions to1.6have explicit bounds of the following form:

A ⁿ⁻²

jn−k−1

a

2 jk−1 4k2

p^n−j−1

≤xn ≤A ⁿ⁻²

jn−k−1

b

2 j−k−2 4k2

1

p^n−j−1

3.12

for alln≥3k3.

Proof. The proof follows directly fromLemma 2.2andTheorem 3.2, and thus is omitted.

Note thatTheorem 3.3andLemma 2.3imply the following corollary.

Corollary 3.4. If the solutionuito2.1withu0 0 converges to the unique equilibriumλunder the conditions inLemma 2.2, then the unique equilibriumAλ^r of1.6is a global attractor.

ByTheorem 3.3, it suffices to determine the explicit bounds for2.1. In the following, a simple case would be taken. For example,

if the parametersA, p, kare fixed, then by2.1we get

u_n2 A

A

A/

Aun^r_p 1/

Aun^r_pk1_rp

1

A A/

Aun^rp1/

Aun^rp^k1_rp^k1, n∈N0.

3.13

Denoteui δi λfori≥0λthe unique equilibrium of2.1, then we have that, being forn≥0,

δn2 A

A

A/

A δn λ^rp1/

A δn λ^rp^k1_rp

1 A

A/

A δn λ^r_p 1/

A δn λ^r_p^k1rp^k1 −λγδn, 3.14

where the functionγis defined by

γx A

A

A/

A xλ^r_p 1/

A xλ^r_pk1_rp

1

A

A/

A xλ^rp1/

A xλ^rp^k1_rp^k1 −λ

3.15

forx >−λ.

Example 3.5A3, p1, k2. Thenγxreduces to the following form:

γx 4

3

4/3 x1²₂ −1, x >−1. 3.16

Obviously,γis monotonically increasing forx >−1 andγ0 0. By simplifyingγ, we have

γx x

x³4x²12x16

3x⁴12x³36x²48x64. 3.17

Having the functionϕdefined viaϕx γx/x, we get the derivative ofϕas follows:

ϕx −3x⁶24x⁵120x⁴384x³624x²640x

3x⁴12x³36x²48x64₂ <0 3.18

for allx >0.Thusϕx< ϕ0 1/4 forx >0 and γtⁿ

tⁿ² γtⁿ tⁿ

1 t² < 1

4t². 3.19

Thereforeγtⁿ < tⁿ² whenevert ≥ 1/2. In addition, for−1 ≤ x < 0, γx < 0 and both ξx x³4x²12x16 andηx 3x⁴12x³36x²48x64 are monotonically increasing.

Hence we have

γ−tⁿ

−tⁿ² <

ξ0 η−1

1 t² 16

43 1

t², 3.20

andγ−tⁿ>−tⁿ²whenevert²≥16/43.

Now setπn 1/2ⁿ, forn ≥ 0, andπ⁻n 16/43^n/2. Note that δ0 −1

−π⁻0 and δ1 1/3 < π1. Thus suppose that −1 ≤ −π⁻2i ≤ δ2i ≤ 0 and 0 ≤ δ2i1≤π2i1, for 0≤i≤NN≥0. Then by induction we have

δ2N2 γδ2N≥γ−π⁻2N γ

− 16

43 N

≥ −π⁻2N2,

δ2N3 γδ2N1≤γπ2N1 γ 1

2

_2N1

≥π2N3.

3.21

Therefore since the fact thatδi≤0 forieven andδi≥0 foriodd, we obtain that, fori≥0,

− 16

43 _i/2

≤δi≤ 1

2 _i

, 1− 16

43 _i/2

≤ui≤1 1

2 _i

. 3.22

EmployingTheorem 3.3, we get the bounds

3 ⁿ⁻²

in−3

1−

16 43

_i1/10

≤xn≤3 ⁿ⁻²

in−3

1

1 2

_2i−4/101

, n≥9. 3.23

4. Quantitative Bounds for Solutions to 1.8

In this section, the upper and lower bounds of solutions to1.8are given, and first we present a lemma concerning the equilibrium points of1.8.

Lemma 4.1. IfA >1, then1.8has a unique positive equilibriumy A; and if 0 < A≤1, then 1.8has a unique positive equilibriumy1.

The proof is simple and thus omitted.

Suppose that{yi}^p_i−kis a positive solution to1.8, and by the transformation

βn y_n1 yn^p

, n≥ −k, 4.1

we have that

βnmax

⎧⎨

⎩A y^pn

, 1 y_n−k^pk1

⎫⎬

⎭, n∈N0. 4.2

It follows from4.1and1.8that

ynmax

⎧⎨

⎩A, y^p_n−1 y_n−k−1^pk1

⎫⎬

⎭max

A, β^p_n−2β^p_n−3² · · ·β^p_n−k−1^k

, n∈N. 4.3

Employing4.2and4.3, we obtain that, forn≥ −k,

βnmax

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, β^p_n−2β^p_n−3² · · ·β^p_n−k−1^k p, 1 max

A, β_n−k−2^p β^p_n−k−3² · · ·β_n−2k−1^pk p^k1

⎫⎪

⎪⎬

⎪⎪

⎭. 4.4

For two nonnegative sequences{Li}and{Hi}, letLi≤βi≤Hifork1≤i < T T ≥3k2.

Then according to4.4and the sequences

LTmax

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, H_T−2^p H_T−3^p2 · · ·H_T−k−1^pk _p, 1 max

A, H_T−k−2^p H_T−k−3^p2 · · ·H_T−2k−1^pk p^k1

⎫⎪

⎪⎬

⎪⎪

⎭,

HT max

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, L^p_T−2H_T−3^p2 · · ·L^p_T−k−1^k _p, 1 max

A, L^p_T−k−2L^p_T−k−3² · · ·L^p_T−2k−1^k p^k1

⎫⎪

⎪⎬

⎪⎪

⎭, 4.5

we have thatLT ≤βT≤HT.

Note thatyi≥Afori≥1, and thus from4.2we get

0< βi≤max 1

A^p−1, 1 A^pk1

⎧⎪

⎪⎨

⎪⎪

⎩ 1

A^pk1, A <1, 1

A^p−1, A≥1

4.6

fork1≤i≤3k2.

Lemma 4.2. LetLi0 andHimax{1/A^p−1,1/A^pk1}fork1≤i≤3k2, then the sequences {Li}and{Hi}are nondecreasing and nonincreasing, respectively.

Proof. Assume thatLi≤Li1andHi1≤Hifork1≤i < T T ≥3k2. Then we have that

HT1max

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, L^p_T−1L^p_T−2² · · ·L^p_T−k^k p, 1 max

A, L^p_T−k−1L^p_T−k−2² · · ·L^p_T−2k^k p^k1

⎫⎪

⎪⎬

⎪⎪

⎭

≤max

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, L^p_T−2L^p_T−3² · · ·L^p_T−k−1^k _p, 1 max

A, L^p_T−k−2L^p_T−k−3² · · ·L^p_T−2k−1^k p^k1

⎫⎪

⎪⎬

⎪⎪

⎭ HT.

4.7

Analogous argument gives thatLT ≤L_T1, and the lemma can be proved inductively.

Now we will take in the other first-order recursive equation

vn1 max

⎧⎨

⎩ A max

A,vn^rp, 1 max

A,vn^r_pk1

⎫⎬

⎭, n∈N0, 4.8

whereA, pequal those of1.8,r pp²· · ·p^k, and the initial valuev0 0.

ThroughLemma 4.2, simpler bounds for{βi}are given below.

Theorem 4.3. The inequality

L^∗_n≤βn≤H_n^∗ 4.9

holds forn≥k2, whereL^∗_nv2nk−1/4k2andH_n^∗v2nk−2/4k2 1.

Proof. LetLi0 andHi max{1/A^p−1,1/A^pk1}fork2≤i≤3k2, and by the definitions of{L^∗_n}and{H_n^∗}we have thatL^∗_i LiandH_i^∗Hifork2≤i≤3k2.

Assume thatL^∗_i ≤LiandH_i^∗≥Hifork2≤i< T T ≥3k3, then

HT max

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, L^p_T−2L^p_T−3² · · ·L^p_T−k−1^k p, 1 max

A, L^p_T−k−2L^p_T−k−3² · · ·L^p_T−2k−1^k _p^k1

⎫⎪

⎪⎬

⎪⎪

⎭

≤max

⎧⎪

⎪⎨

⎪⎪

⎩

A max

A, L^p···p_T−k−1^kp, 1 max

A, L^p···p_T−2k−1^kp^k1

⎫⎪

⎪⎬

⎪⎪

⎭

≤max

⎧⎨

⎩ A max

A, L^r_T−2k−1p, 1 max

A, L^r_T−2k−1p^k1

⎫⎬

⎭

≤max

⎧⎨

⎩

A max

A,

L^∗_T−2k−1rp, 1 max

A,

L^∗_T−2k−1rp^k1

⎫⎬

⎭ max

A

max

A,v2T−k−2/4k2^r_p, 1

max

A,v2T−k−2/4k2^r_p^k1

⎫⎬

⎭ v

2 T−k−2 4k2

1

H_T^∗.

4.10

Similar computations lead to the inequalityL^∗_T ≤LT, and the theorem follows by induction.

Theorem 4.3and4.8imply the following result.

Theorem 4.4. Suppose that there exist two positive sequences{cj}and{dj}which are bounds for a positive solution to4.8withv0 0 such that

c j

≤v j

≤d j

, j≥k2. 4.11

Then the solutions to1.8have explicit upper and lower bounds of the following form:

max

A, n−2 in−k−1

c

2 ik−1 4k2

pⁿ⁻ⁱ⁻¹

≤yn≤max

A, n−2 in−k−1

d

2 i−k−2 4k2

1

pⁿ⁻ⁱ⁻¹

4.12 for alln≥3k3.

5. Conclusion

In this paper, we investigate a particular case of a higher-order difference equation by Stevi´c which is a natural extension of that one in8, 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 parametersA, 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:

un1 A

Aun^r_p 1

Aun^r_p^k1, n∈N0, vn1 max

⎧⎨

⎩ A max

A,vn^r_p, 1 max

A,vn^rp^k1

⎫⎬

⎭, n∈N0,

5.1

whereA, p∈0,∞, k∈N, andr Σ^k_i1pⁱ.

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.

References

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9 K. S. Berenhaut and S. Stevi´c, “A note on positive non-oscillatory solutions of the difference equation xn1αx^p_n−k/x^pn,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 495–499, 2006.

10 K. S. Berenhaut and S. Stevi´c, “The behaviour of the positive solutions of the difference equation xn1Ax_n−2^p /x^p_n−1,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909–918, 2006.

11 R. Devault, V. L. Kocic, and D. Stutson, “Global behavior of solutions of the nonlinear difference equationxn1pnxn−1/xn,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 707–

719, 2005.

12 H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equationxn1 αx_n−k^p /x^pn,” Journal of Applied Mathematics & Computing, vol. 12, no. 1-2, pp. 31–

37, 2003.

13 L. Gutnik and S. Stevi´c, “On the behaviour of the solutions of a second-order difference equation,”

Discrete Dynamics in Nature and Society, vol. 2007, Article ID 27562, 14 pages, 2007.

14 A. E. Hamza and A. Morsy, “On the recursive sequencexn1 Axn−1/x^k_n,” Applied Mathematics Letters, vol. 22, no. 1, pp. 91–95, 2009.

15 B. Iriˇcanin and S. Stevi´c, “On a class of third-order nonlinear difference equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479–483, 2009.

16 M. R. S. Kulenovi´c, G. Ladas, and C. B. Overdeep, “On the dynamics ofxn1 pnxn−1/xn with a period-two coefficient,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 905–914, 2004.

17 S. Stevi´c, “Boundedness character of a class of difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 839–848, 2009.

18 S. Stevi´c, “On the difference equationxn1αxn−1/xn,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159–1171, 2008.

19 S. Stevi´c, “On the recursive sequencexn1max{c, x^pn/x_n−1^p },” Applied Mathematics Letters, vol. 21, no.

8, pp. 791–796, 2008.

20 S. Stevi´c, “On the recursive sequencexn1Ax^pn/x^r_n−1,” Discrete Dynamics in Nature and Society, vol.

2007, Article ID 40963, 9 pages, 2007.

21 S. Stevi´c, “On the recursive sequencexn1Ax^p_n−1/x_n^p,” Discrete Dynamics in Nature and Society, vol.

2007, Article ID 34517, 9 pages, 2007.

22 S. Stevi´c, “On the recursive sequencexn1αx^p_n−1/x^pn,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229–234, 2005.

23 S. Stevi´c, “On the recursive sequencexn1 αxn−1/xn. II,” Dynamics of Continuous, Discrete &

Impulsive Systems. Series A, vol. 10, no. 6, pp. 911–916, 2003.

24 S. Stevi´c, “On the recursive sequencexn1 A/ ^k_i0xn−i1/ ^2k1_jk2xn−j,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249–259, 2003.

25 S. Stevi´c, “On a nonlinear generalized max-type difference equation,” preprint, 2009.

26 S. Stevi´c, “On the recursive sequencexn1 _k

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j1βjxn−qj,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 39404, p. 7, 2007.

27 S. Stevi´c, “Boundedness character of a fourth order nonlinear difference equation,” Chaos, Solitons &

Fractals, vol. 40, no. 5, pp. 2364–2369, 2009.

28 S. Stevi´c, “Boundedness character of two classes of third-order difference equations,” Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1193–1209, 2009.

29 S. Stevi´c, “On a class of higher-order difference equations,” Chaos, Solitons & Fractals, vol. 42, no. 1, pp. 138–145, 2009.

30 S. Stevi´c, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1841–1849, 2010.

31 F. Sun, X. Yang, and C. Zhang, “On the recursive sequencexnAx^p_n−k/x^r_n−1,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 608976, 8 pages, 2009.

32 T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation xn1 max{1/xn, An/xn−1},” Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009.

33 H. Xi, T. Sun, W. Yu, and J. Zhao, “On boundedness of solutions of the difference equationxn1 pxnqxn−1/1xnforq >1p >1,” Advances in Difference Equations, vol. 2009, Article ID 463169, 11 pages, 2009.

34 T. Sun, H. Xi, and M. Xie, “Global stability for a delay difference equation,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 367–372, 2009.

35 T. Sun, H. Xi, and W. Quan, “Existence of monotone solutions of a difference equation,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 917560, 8 pages, 2008.

36 T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equationxn1xn−1/p xn,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 20652, 7 pages, 2006.

37 T. Sun and H. Xi, “On the global behavior of the nonlinear difference equation xn1 fpn, xn−m, xn−tk11,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 90625, 12 pages, 2006.

38 X. Yang, Y. Y. Tang, and J. Cao, “Global asymptotic stability of a family of difference equations,”

Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2643–2649, 2008.

39 X. Yang, Y. Yang, and J. Luo, “On the difference equationxn pxn−s/qxn−txn−s,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 918–926, 2007.