doi:10.1155/2010/970720
Research Article
Dynamics of a Rational Difference Equation
Xiu-Mei Jia,
1, 2Lin-Xia Hu,
3and Wan-Tong Li
21Department 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
αβxnγxn−k
1xn−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.
If we allow the parameters to be satisfied,αβγ 0, then1.1contains, as special cases, six difference equations with positive parameters:
xn1 1αx change of variablesxn1/yn, equation1.4can be reduced to the linear equation
yn 1r
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 {xn}∞n−k is
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−124α
2 .
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 <1andα < 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 variablesxnγyn,1.1reduces to the difference equation
yn1
qrynyn−k
pyn−k , n∈N0, 1.11
wherep1/γ, qα/γ2, andr β/γ.Its unique positive equilibrium is
y
r1−pr1−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
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 0i−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∈Iwith
0
i−k|xi−x|< γ
one has limn→ ∞xnx.
iiiThe equilibrium pointxof2.1is called a global attractor if for everyx−k, . . . , x0∈I
one has limn→ ∞xnx.
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.
Lemma 2.4see25. Assume thata >0andA > b >0hold. Then the positive equilibriumxof the difference equation
xn1 Aaxbxn
n−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
ryn−p−q/r
pyn−k , n∈N0, 3.1
yn1−p−r q
ryn−p−q/r1−p−q/rpyn−k
pyn−k , n∈N0, 3.2
yn1−
q p−r
ryn−q/p−r1−q/p−ryn−k
pyn−k , n∈N0, 3.3
yn1−yn
p−rq/p−r−yn1−ynyn−k
pyn−k , n∈N0. 3.4
Ifpqr, then the unique equilibrium isy1 and3.1and3.4become
yn1−1
ryn−1
pyn−k , n∈N0, 3.5
yn1−yn
1−ynqyn−k
pyn−k , n∈N0, 3.6
respectively.
Ifpr, then the unique equilibrium isy 114q/2 and3.4becomes
yn1−yn
q1−ynyn−k
pyn−k , n∈N0. 3.7
Set
fx, y qrxy
py . 3.8
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
Furthermore,yis the unique positive root of the quadratic equation
y2p−r−1y−q0. 3.12
Since
q
p−r 2
p−r−1 q
p−r −q
qqr−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 {yn}∞n−k is a nonnegative solution of 1.11. Then the following statements are true.
iIf for someN≥0, yN>1, thenyn>1forn > N.
iiIf for someN≥0, yN1, thenyn1forn > N.
iiiIf for someN≥0, yN<1, thenyn<1forn > N.
ivIf for someN≥0, yN>1, thenyN1< yN.
vIf for someN≥0, yN<1, thenyN1> yN.
Lemma 3.5. Assume thatr < p < qr holds and{yn}∞n−kis a nonnegative solution of equation
1.11. Then the following statements are true
iIf for someN≥0, yN≤q/p−r, thenyn< q/p−rforn > N.
iiIf for someN≥0, yN≥q/p−r, thenyN1 < yN.
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 ≥1for alln∈N, and ifp > q, then the following statements are
also true.
vIf for someN≥0, yN>p−q/r, thenyn>1forn > N.
viIf for someN≥0, yN p−q/r, thenyN1 1.
Proof. In this casep−q/r <1< q/p−rholds.
y >1. Further, ifp > q, then the following statements are also true.
iIf for someN≥0, yN>p−q/r, thenyn>1forn > 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
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 {xl, xl1, . . . , xm}, 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{xl, xl1, . . . , xm}, all less than the equilibrium pointx,
withl≥ −kandm≤ ∞such that
either l−k or l >−k, xl−1< x,
either m∞ or m <∞, xm1< 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 leastk1or 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 {yn}∞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≥0oryi−1<0, i −k, . . . ,0.
iiiIfr < p < qr, then, except possibly for the first semicycle, every oscillatory solution of
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 leastk1or 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 that p ≥ 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{yn}∞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{yn}is strictly decreasing in the intervalp−q/r,∞. Hence, limn→ ∞ynexists and limn→ ∞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 thatyN0 ∈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 thatyn< yn1< q/p−rforn∈N,
which means that limn→ ∞yn exists and limn→ ∞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
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{yn}∞n−kis a nontrivial nonnegative solution of 1.11. Then the sequence{yn}∞n0is monotonic andlimn→ ∞yn1.
Proof. In this case, the unique positive equilibrium isy1. ByLemma 3.4it easy to see that if
y0>1 then{yn}∞n0is decreasing and bounded below by 1, ify0<1 then{yn}∞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{yn}∞n−kis a nonnegative solution of 1.11. If
y0 ∈1, q/p−r, thenyn ∈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 thatyN∈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 thatyN >1, then the proof follows from the first assertion. Now assume for the sake of
contradiction that all terms of{yn}never enter the interval1, q/p−r, then{yn}would lie in the interval0,1forn∈N. UsingLemma 3.5iii, we obtain thatyn< yn1<1 forn≥1,
from which it follows that limn→ ∞yn exists and limn→ ∞yn ≤ 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 thatyN1 ∈ 0,1, then the proof
also follows from Case1. Now assume for the sake of contradiction thatyn> q/p−rfor all
n∈ N, then byLemma 3.5ii, we have thatyn > yn1 > q/p−rforn ∈N, from which it
follows that limn→ ∞ynexists and limn→ ∞yn≥q/p−r. This contradictsLemma 3.5iv.
The proof is complete.
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 qprmMM, M qprMmm, 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 qprynyyn−k
n−k ≥
qyn−k
pyn−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 thatyn <p−q/r for alln∈ N. Using identity3.7, we get thatyn < yn1 <p−q/r for
n≥1, from which it follows that the sequence{yn}is strictly increasing and there is a finite limn→ ∞yn≤p−q/r; this contradicts the fact thaty 114q/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 thatyn ≥1 forn≥ N.
Therefore the change of variables
ynzn1 5.4
transforms1.11to the difference equation
zn1
qpzn
p1zn−k, n≥N. 5.5
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.
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