Dynamics of a Rational Difference Equation

doi:10.1155/2010/970720

Research Article

Dynamics of a Rational Difference Equation

Xiu-Mei Jia,

Lin-Xia Hu,

and Wan-Tong Li

1Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

3Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, China

Correspondence should be addressed to Wan-Tong Li,[email protected] Received 5 November 2009; Accepted 5 April 2010

Academic Editor: Martin Bohner

Copyrightq2010 Xiu-Mei Jia et al. 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.

The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equationxn1 αβxnγxn−k/1xn−k, n ∈ N0, where the parametersα, β, γ ∈ 0,∞, k ≥ 2 is an integer, and the initial conditions x−k, . . . , x0 ∈0,∞. It is shown that the unique positive equilibrium of the equation is globally asymptotically stable under the conditionβ ≤ 1. The result partially solves the open problem proposed by Kulenovi´c and Ladas in work2002.

1. Introduction

The aim of this paper is to study the dynamical behavior of the following rational difference equation:

xn1 αβx_nγx_n−k

1x_n−k , n∈N0, 1.1

where α, β, γ ∈ 0,∞, N0 {0,1,2,3, . . .}, k ≥ 2 is an integer, and the initial conditions x_−k, . . . , x0∈0,∞.The related case wherek1 was investigated in1.

In 2002, Kulenovi´c and Ladas1proposed the following open problem.

Open Problem 1. Assume thatα, β, γ ∈0,∞andk∈ {2,3, . . .}. Investigate the global behavior of all positive solutions of1.1.

If we allow the parameters to be satisfied,αβγ 0, then1.1contains, as special cases, six difference equations with positive parameters:

xn1 α

1x_n−k, n∈N0, 1.2

xn1 βxn

1x_n−k, n∈N0, 1.3

xn1 γxn−k

1x_n−k, n∈N0, 1.4

xn1 αβxn

1x_n−k, n∈N0, 1.5

xn1 βxnγxn−k

1xn−k , n∈N0, 1.6

x_n1 αγx_n−k

1xn−k , n∈N0. 1.7

Equations 1.2 and1.3can be reduced to 1.5, which was studied in 2. By the change of variablesx_n1/y_n, equation1.4can be reduced to the linear equation

yn 1 r

1

r

yn−k, n∈N0, 1.8

whose global behavior of solutions is easily derived. Equation1.6is investigated in3.

Equation 1.7 is essentially similar to the Riccati equation. In fact, if {x_n}^∞_n−k is a positive solution of 1.7, then {x_−kk1n}^∞_n0,{x_−k1k1n}^∞_n0, . . . ,{x_k1n}^∞_n0 are k 1 solutions of the following Riccati equations:

xn1 αγxn

1xn , n−k,−k1,−k2, . . . , x_n1 αγx_n

1xn , n−k1,−k2, . . . , ...

xn1 αγxn

1xn , n0,1,2, . . . ,

1.9

respectively.

Therefore, we only pay attention to investigating1.1with positive parameters and omit any further discussion of the above1.2–1.7.

For other related results on difference equations, one can refer to4–18.

Equation1.1has a unique equilibriumxwhich is positive and is given by

x βγ−1

βγ−1₂ 4α

2 . 1.10

In 19, the authors investigated the global asymptotic stability of the positive equilibriumxof1.1. We summarize their results in the following theorem.

Theorem 1.1. i Assume that β < 1. Then the positive equilibrium x of 1.1 is locally asymptotically stable.

ii Assume that β < 1 and γ ≥ α/1 − β. Then for every solution of 1.1 with initial conditions in invariant interval0,γ −α/β, the positive equilibrium point x is globally asymptotically stable.

iiiAssume thatβ <1 andα < γ < α/1−β. Then for every solution of 1.1with initial conditions in invariant intervalγ−α/β,αβγ²−αγ/βγ−α−β², the positive equilibrium pointxis globally asymptotically stable.

Reviewing the proof ofTheorem 1.1, one can easily find that the positive equilibrium xof1.1is locally asymptotically stable whenβ1. SoTheorem 1.1ican be improved as the following theorem.

Theorem 1.2. Assume thatβ≤1. Then the positive equilibriumxof 1.1is locally asymptotically stable.

By the change of variablesx_nγy_n,1.1reduces to the difference equation yn1 qry_ny_n−k

py_n−k , n∈N0, 1.11

wherep1/γ, qα/γ², andr β/γ.Its unique positive equilibrium is

y

r1−p

r1−p24q

2 . 1.12

Motivated by the above open problem, the purpose of this paper is to investigate the boundedness, local stability, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of1.11. We show that the unique positive equilibrium of1.11is a global attractor whenp≥rand our result solves the open problem whenβ≤1.

The organization of this paper is as follows. InSection 2, some basic definitions and lemmas regarding difference equations are given. The boundedness and invariant intervals of 1.11are discussed inSection 3. The semicycle analysis of1.11is presented inSection 4. The main results are formulated and proved inSection 5, where the global asymptotic stability of 1.11is investigated.

2. Some Lemmas

In this section, we recall some definitions and lemmas which will be useful in the sequel.

Let I be some interval of real numbers and let f : I ×I → I be a continuously differentiable function. Then for initial conditionsx_−k, . . . , x0∈I, the difference equation

xn1 fxn, xn−k, n∈N0, 2.1

has a unique solution{x_n}^∞_n−k.

An internalJ⊆Iis called an invariant interval of2.1if

x−k, . . . , x0 ∈J⇒xn∈J ∀n∈N. 2.2

That is, every solution of2.1with initial conditions inJremains inJ.

Definition 2.1. Letxbe an equilibrium point of2.1.

iThe equilibrium pointxof2.1is called locally stable if for everyε >0 there exists δ > 0 such that for allx−k, . . . , x0 ∈ I with ₀

i−k|xi−x| < δ, one has|xn −x| <

εfor alln≥ −k.

iiThe equilibrium pointxof2.1is called locally asymptotically stable if it is locally stable and if there existsγ >0 such that for allx−k, . . . , x0∈Iwith₀

i−k|xi−x|< γ one has lim_{n→ ∞}xnx.

iiiThe equilibrium pointxof2.1is called a global attractor if for everyx_−k, . . . , x0∈I one has lim_{n→ ∞}x_nx.

ivThe equilibrium pointxof2.1is called globally asymptotically stable if it is locally asymptotically stable and is a global attractor.

vThe equilibrium pointxof2.1is called unstable if it is not locally stable.

The following lemmas can be found in3,20, respectively; also see19,21–24.

Lemma 2.2see3. Leta, bbe an interval of real numbers and assume thatf:a, b×a, b → a, bis a continuous function satisfying the following properties.

ifx, yis nondecreasing in each of its arguments.

iiThe equation

fx, x x 2.3

has a unique positive solution in the intervala, b.

Then2.1has a unique positive equilibrium x∈a, band every solution of 2.1converges tox.

Lemma 2.3see20. Leta, bbe an interval of real numbers and assume thatf:a, b×a, b → a, bis a continuous function satisfying the following properties.

ifx, yis a nondecreasing function inxand a nonincreasing function iny.

iiIfm, M∈a, b×a, bis a solution of the following system:

mfm, M, MfM, m, 2.4

thenmM.

Then2.1has a unique equilibrium pointx∈a, band every solution of2.1converges tox.

The following result was proved in25.

Lemma 2.4see25. Assume thata >0 andA > b >0 hold. Then the positive equilibriumxof the difference equation

x_n1 abx_n

Axn−k, n∈N0, 2.5

is a global attractor of all positive solutions.

3. Boundedness and Invariant Intervals

The following result about boundedness of1.11can be found in19.

Theorem 3.1. Every solution of 1.11is bounded from above and from below by positive constants.

Let {yn}^∞_n−k be a nonnegative solution of 1.11. Then the following identities are easily established:

yn1−1 r y_n−

p−q /r

pyn−k , n∈N0, 3.1

yn1−p−q

r r

y_n− p−q

/r

1−

p−q

/r

py_n−k

pyn−k , n∈N0, 3.2

yn1− q p−r r

y_n− q/

p−r

1− q/

p−r y_n−k

py_n−k , n∈N0, 3.3

yn1−yn

p−r

q/

p−r

−yn

1−yn yn−k

py_n−k , n∈N0. 3.4

Ifpqr, then the unique equilibrium isy1 and3.1and3.4become

y_n1−1 r yn−1

py_n−k , n∈N0, 3.5

yn1−yn

1−y_n

qy_n−k

py_n−k , n∈N0, 3.6

respectively.

Ifpr, then the unique equilibrium isy 1

14q/2 and3.4becomes yn1−yn q

1−y_n y_n−k

py_n−k , n∈N0. 3.7

Set

f x, y

qrxy

py . 3.8

Then we have the following monotone character for the functionfx, y.

Lemma 3.2. Letfx, ybe defined by3.8. Then the following statements hold true.

iAssume thatp > q. Then fx, yis strictly increasing in each of its arguments forx <

p−q/rand it is strictly increasing inxand decreasing inyforx≥p−q/r. iiAssume thatp≤q. Thenfx, yis strictly increasing inxand decreasing inyforx≥0.

Proof. By calculating, the partial derivatives of the functionfx, yare f_x

x, y r

py, f_y x, y

p−q−rx

py₂ , 3.9

from whichiandiieasily follow.

3.1. The Casep > r

Lemma 3.3. Assume thatp > qr and {y_n}^∞_n−k is a nonnegative solution of 1.11. Then the following statements are true.

iIf for someN≥0, yN<p−q/r, thenyn<1 forn > N.

iiIf for someN≥0, y_N p−q/r, theny_N1 1.

iiiIf for someN≥0, yN>p−q/r, thenyN1 >1.

ivIf for someN≥0, y_N≥q/p−r, theny_n> q/p−rforn > N.

vIf for someN≥0, yN≤q/p−r, thenyN1 > yN. viIf for someN≥0, y_N≥1, theny_N1< y_N.

viiEquation1.11possesses an invariant intervalq/p−r,p−q/randy ∈ q/p− r,p−q/r; moreover, the intervalq/p−r,1is also an invariant interval of 1.11 andy∈q/p−r,1.

Proof. i–viClearly, in this caseq/p−r < 1 < p−q/r holds. The statements directly follow by using the identities3.1,3.3and3.4.

viiByLemma 3.2ithe functionfx, yis strictly increasing in each of its arguments forx <p−q/r. Given thaty−k, . . . , y−1, y0∈q/p−r,p−q/r, then we get

y1 f y0, y_−k

≥f

q

p−r, q p−r

q

qr/

p−r

q/

p−r p

q/

p−r q

p1 p²−prq > q

p−r, y1f

y0, y−k

≤f p−q

r ,p−q r

q

p−q

p−q /r

p

p−q

/r 1< p−q r ,

3.10 which implies thaty1∈q/p−r,1⊂q/p−r,p−q/r. By the induction,y_n∈q/p− r,1⊂q/p−r,p−q/rfor everyn∈N, as claimed.

On the other hand,p > qrimplies that

y

r1−p

r1−p₂ 4q

2 <

r1−p

r1−p₂ 4

p−r

2 1. 3.11

Furthermore,yis the unique positive root of the quadratic equation

y²

p−r−1

y−q0. 3.12

Since

q

p−r ₂

p−r−1 q

p−r −q q

qr−p

p−r2 <0, 3.13

then we have thaty > q/p−r. That is,y∈q/p−r,1⊂q/p−r,p−q/r, finishing the proof.

Whenpqr, identities3.5and3.6imply that the following results hold.

Lemma 3.4. Assume thatp qr and {y_n}^∞_n−k is a nonnegative solution of 1.11. Then the following statements are true.

iIf for someN≥0, y_N>1, theny_n>1 forn > N.

iiIf for someN≥0, yN1, thenyn1 forn > N.

iiiIf for someN≥0, yN<1, thenyn<1 forn > N.

ivIf for someN≥0, y_N>1, theny_N1< y_N. vIf for someN≥0, y_N<1, theny_N1> y_N.

Lemma 3.5. Assume thatr < p < qr holds and{y_n}^∞_n−kis a nonnegative solution of equation 1.11. Then the following statements are true

iIf for someN≥0, y_N≤q/p−r, theny_n< q/p−rforn > N.

iiIf for someN≥0, y_N≥q/p−r, theny_N1 < y_N. iiiIf for someN≥0, yN≤1, thenyN1> yN.

ivEquation 1.11possesses an invariant interval 1, q/p−rand y ∈ 1, q/p−r.

Further, ifp≤q, thenyn ≥1 for alln∈N, and ifp > q, then the following statements are also true.

vIf for someN≥0, yN>p−q/r, thenyn>1 forn > N.

viIf for someN≥0, y_N p−q/r, theny_N1 1.

viiIf for someN≥0, y_N<p−q/r, theny_N1 <1.

Proof. In this casep−q/r <1< q/p−rholds.

i–iiiUsing the identities3.3and3.4, one can see that the results follow.

ivByLemma 3.2the functionfx, yis strictly increasing inxand decreasing iny in1, q/p−r. Then we can get

y1f

y0, y−k

≤f

q

p−r,1

≤f

q

p−r,p−q r

pqr p−r

p−q p−r

prp−q < q p−r, y1f

y0, y_−k

≥f

1, q p−r

≥f p−q

r , q p−r

1> p−q r ,

3.14

which implies thaty1∈1, q/p−r. By the induction,yn∈1, q/p−rfor everyn∈N.

On the other hand,p < qrimplies that

y

r1−p

r1−p24q

2 >

r1−p

r1−p24 p−r

2 1. 3.15

Then as same as the argument inLemma 3.3viiit can be proved thaty∈1, q/p−r.

Further, ifp≤ q, then the identity3.1implies thatyn ≥1 for alln∈N, and ifp > q, the resultsv–viican also follow from3.1.

The proof is complete.

3.2. The Casep≤r

Lemma 3.6. Assume thatp ≤ r. Then the interval 1,∞ is an invariant interval of 1.11and y >1. Further, ifp > q, then the following statements are also true.

iIf for someN≥0, y_N>p−q/r, theny_n>1 forn > N.

iiIf for someN≥0, yN p−q/r, thenyN1 1.

iiiIf for someN≥0, yN<p−q/r, thenyN1 <1.

Proof. Notice that in this case the inequalityp−q/r <1 holds; then the results can easily be obtained by using the identity3.1. Further,p≤rimplies that

y

r1−p

r1−p₂ 4q

2 ≥ 1

14q

2 >1, 3.16

finishing the proof.

4. Semicycle Analysis

Here, we present some results regarding the semicycle analysis of the solutions of1.11. Now recall two definitions from25.

Definition 4.1. Let{xn}^∞_n−kbe a solution of2.1. A positive semicycle of the solution{xn}^∞_n−k of 2.1 consists of a “string” of terms {x_l, x_l1, . . . , x_m}, all greater than or equal to the equilibrium pointx, withl≥ −kandm≤ ∞such that

either l−k or l >−k, xl−1< x,

either m∞ or m <∞, xm1< x. 4.1

Definition 4.2. Let{xn}^∞_n−kbe a solution of2.1. A negative semicycle of the solution{xn}^∞_n−k of2.1consists of a “string” of terms{x_l, x_l1, . . . , x_m}, all less than the equilibrium pointx, withl≥ −kandm≤ ∞such that

either l−k or l >−k, xl−1< x,

either m∞ or m <∞, x_m1< x. 4.2

The next two lemmas can be found in26and19, respectively.

Lemma 4.3see26. Assume thatf ∈C0,∞×0,∞,0,∞and thatfx, yis increasing in both arguments. Letxbe a positive equilibrium of2.1. Then except possibly for the first semicycle, every oscillatory solution of 2.1has semicycles of length at mostk.

Lemma 4.4see19. Assume thatf ∈C0,∞×0,∞,0,∞and thatfx, yis increasing inxfor each fixedyand is decreasing inyfor each fixedx. Letxbe a positive equilibrium of2.1. If k≥2, then every oscillatory solution of 2.1has semicycles that are either of length at leastk1 or of length at mostk−1.

Using the monotonic character of the function fx, y from Lemma 3.2, in each of the intervals in Lemmas3.3–3.6, together with Lemmas4.3and4.4, it is easy to obtain the following results concerning semicycle analysis.

Theorem 4.5. Assume that p > r and {y_n}^∞_n−k is a nonnegative solution of 1.11. Then the following statements are true.

iIfp > qr, then, except possibly for the first semicycle, every oscillatory solution of 1.11 which lies in the invariant intervalq/p−r,p−q/rhas semicycles of length at most k.

iiIfpqr, then1.11does not have oscillatory solution withyi−1≥0 oryi−1<0, i

−k, . . . ,0.

iiiIfr < p < qr, then, except possibly for the first semicycle, every oscillatory solution of 1.11which lies in the invariant interval1, q/p−rhas semicycles that are either of length at leastk1 or of length at mostk−1.

Theorem 4.6. Assume thatp ≤ r and {yn}^∞_n−kis a nonnegative solution of 1.11. Then, except possibly for the first semicycle, every oscillatory solution of 1.11which lies in the invariant interval 1,∞has semicycles that are either of length at leastk1 or of length at mostk−1.

5. Global Asymptotic Stability for the Case p ≥ r

In this section, we discuss the global attractivity of the positive equilibrium of1.11. We show thatyis a global attractor of all nonnegative solutions of1.11whenp≥r. Further, the unique positive equilibriumxof1.1is globally asymptotically stable whenβ≤1.

Theorem 5.1. Assume thatp ≥ r. Then the unique positive equilibrium y of 1.11is a global attractor.

The proof is finished by considering the following four cases; see Theorems5.3,5.5, 5.7, and5.9.

Theorem 5.2. Assume thatp > qr holds, and{y_n}^∞_n−k is a nonnegative solution of 1.11. If y0 ∈ q/p−r,p−q/r, thenyn ∈ q/p−r,p−q/r forn ∈ N. Furthermore, every nonnegative solution of 1.11lies eventually in the intervalq/p−r,p−q/r.

Proof. Firstly, note that in this caseq/p−r<1<p−q/rholds.

Ify0 ∈q/p−r,p−q/r, then byLemma 3.3iandiv, we have thatq/p−r<

yn<1<p−q/rforn≥1, the first assertion follows.

To complete the proof it remains to show that wheny0/∈q/p−r,p−q/rthere existsN >0 such thatyN∈q/p−r,p−q/r. There are two cases to be considered.

Case 1y0∈p−q/r,∞. Lemma 3.3iiandiiiimplies thaty1<1. Ify1≤p−q/r, then the proof follows from the first assertion. Now assume for the sake of contradiction that all terms of{yn}never enter the intervalq/p−r,p−q/r; then{yn}would lie in the interval p−q/r,∞forn∈N. UsingLemma 3.3vi, we obtainyn> yn1>p−q/rforn≥1, from which it follows that the sequence{y_n}is strictly decreasing in the intervalp−q/r,∞.

Hence, lim_{n→ ∞}ynexists and lim_{n→ ∞}yn≥p−q/r, which is a contradiction, because, in view ofLemma 3.3vii,1.11has no equilibrium points in the intervalp−q/r,∞.

Case 2y0∈0, q/p−r. If there existsN0 >0 such thaty_N0 ∈q/p−r,p−q/r, then the proof follows from the first assertion. If there existsN1>0 such thatyN1∈p−q/r,∞, then the proof also follows from Case1. Now assume for the sake of contradiction thatyn <

q/p−rfor alln∈N, then byLemma 3.3v, we have thaty_n< y_n1< q/p−rforn∈N, which means that lim_{n→ ∞}yn exists and lim_{n→ ∞}yn ≤ q/p−r; this contradictsLemma 3.3 vii.

The proof is complete.

Theorem 5.3. Assume thatp > qr holds. Then the unique positive equilibriumyof 1.11is a global attractor of all nonnegative solutions of 1.11.

Proof. Theorem 5.2 and Lemma 3.3 vii imply that every nonnegative solution of 1.11 eventually enters the interval q/p −r,p− q/r. Furthermore, the function fx, y is

increasing in each of its arguments inq/p−r,p−q/rand the equation

qryy

py y 5.1

has a unique positive solution on the interval q/p − r,p − q/r. The proof now immediately follows by applyingLemma 2.2.

Theorem 5.4. Assume thatpqr, and{y_n}^∞_n−kis a nontrivial nonnegative solution of 1.11.

Then the sequence{yn}^∞_n0is monotonic and lim_{n→ ∞}yn1.

Proof. In this case, the unique positive equilibrium isy1. ByLemma 3.4it easy to see that if y0>1 then{y_n}^∞_n0is decreasing and bounded below by 1, ify0<1 then{y_n}^∞_n0is increasing and bounded above by 1, and ify0 1 thenyn 1 forn∈N. Hence, in all case the sequence {yn}^∞_n0converges to 1, as desired.

ByTheorem 5.4, it is easy to see that the following result is true.

Theorem 5.5. Assume thatpqr. Then the unique positive equilibriumyis a global attractor of all nonnegative solutions of1.11.

Theorem 5.6. Assume thatr < p < qrholds and{y_n}^∞_n−kis a nonnegative solution of 1.11. If y0 ∈1, q/p−r, theny_n ∈1, q/p−rforn∈N. Furthermore, every nonnegative solution of 1.11lies eventually in the interval1, q/p−r.

Proof. Firstly, note that in this casep−q/r <1< q/p−rholds.

Ify0 ∈1, q/p−r, then byLemma 3.5ivandi, we have that 1≤yn≤q/p−r forn≥1; the first assertion follows.

To complete the proof it remains to show that wheny0/∈1, q/p−rthere existsN >0 such thaty_N∈1, q/p−r. There are two cases to be considered.

Case 1y0 ∈ 0,1. Lemma 3.5iimplies thatyn < q/p−rforn ≥ 1. If there existsN such thaty_N >1, then the proof follows from the first assertion. Now assume for the sake of contradiction that all terms of{y_n}never enter the interval1, q/p−r, then{y_n}would lie in the interval0,1forn∈N. UsingLemma 3.5iii, we obtain thatyn< yn1<1 forn≥1, from which it follows that lim_{n→ ∞}y_n exists and lim_{n→ ∞}y_n ≤ 1, which is a contradiction, because in view ofLemma 3.5iv,1.11has no equilibrium points in the interval0,1.

Case 2y0∈q/p−r,∞. If there existsN0>0 such thatyN0 ∈1, q/p−r, then the proof follows from the first assertion. If there existsN1 > 0 such thaty_N1 ∈ 0,1, then the proof also follows from Case1. Now assume for the sake of contradiction thaty_n> q/p−rfor all n∈ N, then byLemma 3.5ii, we have thatyn > yn1 > q/p−rforn ∈N, from which it follows that lim_{n→ ∞}y_nexists and lim_{n→ ∞}y_n≥q/p−r. This contradictsLemma 3.5iv.

The proof is complete.

Theorem 5.7. Assume thatr < p < qrholds. Then the unique positive equilibriumyof 1.11is a global attractor of all nonnegative solutions of1.11.

Proof. Theorem 5.6implies that every nonnegative solution of1.11lies eventually in the invariant interval1, q/p−r.

Further, the functionfx, yis nondecreasing inxand nonincreasing inyin1, q/p− r. Letm, M∈1, q/p−rbe a solution of the system

m qrmM

pM , M qrMm

pm , 5.2

thenm−Mp−r1 0, from which we get thatmM, and the proof now follows by applyingLemma 2.3.

Theorem 5.8. Assume thatprholds and that{yn}^∞_n−kis a nonnegative solution of 1.11. Then every nonnegative solution of 1.11lies eventually in the invariant interval1,∞.

Proof. Whenp≤q,

yn1 qrynyn−k

py_n−k ≥ qyn−k

py_n−k ≥1, n∈N0. 5.3

Hence the assertion is true for the casep≤q. There remains to consider the casep > q. In this casep−q/r <1 holds.

Ify0≥p−q/r, then byLemma 3.6iandii, we have thatyn ≥1 forn∈N, and the assertion follows.

Given thaty0<p−q/r, then byLemma 3.6iii, we have thaty1<1. Ify1≥p−q/r, then the assertion is true from the above proof. Now assume for the sake of contradiction thaty_n <p−q/r for alln∈ N. Using identity3.7, we get thaty_n < y_n1 <p−q/r for n≥1, from which it follows that the sequence{y_n}is strictly increasing and there is a finite lim_{n→ ∞}yn≤p−q/r; this contradicts the fact thaty 1

14q/2>1.

The proof is complete.

Theorem 5.9. Assume thatprholds. Then the unique positive equilibriumyof1.11is a global attractor of all nonnegative solutions of 1.11.

Proof. Theorem 5.8implies that there exist a positive integerNsuch thaty_n ≥1 forn≥ N.

Therefore the change of variables

y_nz_n1 5.4

transforms1.11to the difference equation

z_n1 qpz_n

p1z_n−k, n≥N. 5.5

Clearly,q > 0 and p1 > p > 0 hold, and the remainder proof now is a straightforward consequence ofLemma 2.4.

In view ofTheorem 5.1, we know that the unique positive equilibriumxof1.1is a global attractor whenβ≤ 1. From this andTheorem 1.2, we have the following main result, which partially solves Open Problem1.

Theorem 5.10. Assume that β ≤ 1. Then the unique positive equilibrium x of 1.1 is globally asymptotically stable.

References

1 M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problem and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.

2 M. Dehghan and R. Mazrooei-Sebdani, “Dynamics of a higher-order rational difference equation,”

Applied Mathematics and Computation, vol. 178, no. 2, pp. 345–354, 2006.

3 W.-T. Li and H.-R. Sun, “Dynamics of a rational difference equation,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 577–591, 2005.

4 R. P. Agarwal, W.-T. Li, and P. Y. H. Pang, “Asymptotic behavior of a class of nonlinear delay difference equations,” Journal of Difference Equations and Applications, vol. 8, no. 8, pp. 719–728, 2002.

5 A. M. Amleh, E. Camouzis, and G. Ladas, “On the boundedness character of rational equations. II,”

Journal of Difference Equations and Applications, vol. 12, no. 6, pp. 637–650, 2006.

6 D. Chen, X. Li, and Y. Wang, “Dynamics for nonlinear difference equationxn1 αxn−k/βγx_n−1^p ,”

Advances in Difference Equations, vol. 2009, Article ID 235691, 13 pages, 2009.

7 G. H. Gibbons, M. R. S. Kulenovi´c, and G. Ladas, “On the dynamics ofxn1 αβxnγxn−1/A Bxn,” in New Trends in Difference Equations, pp. 141–158, Taylor & Francis, London, UK, 2002.

8 L.-X. Hu, W.-T. Li, and S. Stevi´c, “Global asymptotic stability of a second order rational difference equation,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 779–797, 2008.

9 L.-X. Hu, W.-T. Li, and H.-W. Xu, “Global asymptotical stability of a second order rational difference equation,” Computers & Mathematics with Applications, vol. 54, no. 9-10, pp. 1260–1266, 2007.

10 Y. S. Huang and P. M. Knopf, “Boundedness of positive solutions of second-order rational difference equations,” Journal of Difference Equations and Applications, vol. 10, no. 11, pp. 935–940, 2004.

11 V. L. Koci´c, G. Ladas, and I. W. Rodrigues, “On rational recursive sequences,” Journal of Mathematical Analysis and Applications, vol. 173, no. 1, pp. 127–157, 1993.

12 M. R. S. Kulenovi´c, G. Ladas, L. F. Martins, and I. W. Rodrigues, “The dynamics of xn1 α βxn/ABxnCxn−1: facts and conjectures,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 1087–1099, 2003.

13 W. Li, Y. Zhang, and Y. Su, “Global attractivity in a class of higher-order nonlinear difference equation,” Acta Mathematica Scientia, vol. 25, no. 1, pp. 59–66, 2005.

14 S. Stevi´c, “On the difference equationxn1 αxn−1/xn,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159–1171, 2008.

15 S. Stevi´c, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006.

16 Y.-H. Su, W.-T. Li, and S. Stevi´c, “Dynamics of a higher order nonlinear rational difference equation,”

Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 133–150, 2005.

17 Y. H. Su and W. T. Li, “Global asymptotic stability for a higher order nonlinear difference equations,”

Journal of Difference Equations and Applications, vol. 11, pp. 947–958, 2005.

18 X.-X. Yan, W.-T. Li, and Z. Zhao, “Global asymptotic stability for a higher order nonlinear rational difference equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1819–1831, 2006.

19 M. Dehghan and N. Rastegar, “On the global behavior of a high-order rational difference equation,”

Computer Physics Communications, vol. 180, no. 6, pp. 873–878, 2009.

20 R. DeVault, W. Kosmala, G. Ladas, and S. W. Schultz, “Global behavior ofyn1 pyn−k/qxn yn−k,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4743–4751, 2001.

21 E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, vol. 5 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008.

22 M. Jaberi Douraki, M. Dehghan, and J. Mashreghi, “Dynamics of the difference equationxn1 xn pxn−k/xnq,” Computers & Mathematics with Applications, vol. 56, no. 1, pp. 186–198, 2008.

23 R. Mazrooei-Sebdani and M. Dehghan, “The study of a class of rational difference equations,” Applied Mathematics and Computation, vol. 179, no. 1, pp. 98–107, 2006.

24 M. Saleh and S. Abu-Baha, “Dynamics of a higher order rational difference equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 84–102, 2006.

25 V. L. Koci´c and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.

26 R. Mazrooei-Sebdani and M. Dehghan, “Dynamics of a non-linear difference equation,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 250–261, 2006.