Global Stability of a Rational Difference Equation

Volume 2010, Article ID 432379,17pages doi:10.1155/2010/432379

Research Article

Global Stability of a Rational Difference Equation

Guo-Mei Tang,

Lin-Xia Hu,

and Gang Ma

1School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, China

2Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, China

Correspondence should be addressed to Guo-Mei Tang,[email protected] Received 23 October 2010; Accepted 29 November 2010

Academic Editor: Manuel De la Sen

Copyrightq2010 Guo-Mei Tang 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.

We consider the higher-order nonlinear difference equationx_n1 pqx_n−k/1xnrxn−k, n 0,1, . . .with the parameters, and the initial conditionsx−k, . . . , x0are nonnegative real numbers.

We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenovi´c and Ladas in their monographsee Kulenovi´c and Ladas, 2002.

1. Introduction and Preliminaries

Our aim in this paper is to investigate the global behavior of solutions of the following nonlinear difference equation:

x_n1 pqx_n−k

1xnrx_n−k, n0,1, . . . , 1.1

where the parameters p, q, r and the initial conditions x_−k, . . . , x0 are nonnegative real numbers,k∈ {1,2, . . .}.

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

Open Problem 1. Assume thatp, q, r∈0,∞andk∈ {2,3, . . .}. Investigate the global behavior of all positive solutions of1.1.

Consider the difference equation x_n1 αγx_n−1

ABxnCx_n−1, n0,1, . . . 1.2

with α, γ, A, B, C ∈ 0,∞, and the initial conditionsx₋₁, x0 are nonnegative real numbers.

Note that, the authors1,2investigated this equation and studied1.2.

In this paper, we will consider the above open problem. Actually, we will investigate the global asymptotic stability and the invariant interval for all positive solutions of1.1.

For the global behavior of solutions of some related equations, see3–9. Other related results can be found in10–19. For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel.

Definition 1.1. LetIbe some interval of real numbers and let

f:I^m1−→I 1.3

be a continuously differential function. Then for every set of initial conditionsy_−k, . . . , y₋₁, y₀∈I, the difference equation

y_n1f

yn, y_n−1, . . . , y_n−k

, n0,1, . . . , 1.4

has a unique solution{yn}^∞_n−k.

A pointyis called an equilibrium point of1.4if yf

y, y, . . . , y

, 1.5

that is,

y_ny forn≥0 1.6

is a solution of1.4, or equivalentlyyis a fixed point off.

Definition 1.2. Letybe an equilibrium point of1.4.

iThe equilibriumyis called locally stableor stableif for everyε >0, there exists δ >0 such that for ally_−k, . . . , y₋₁, y₀ ∈Iwith_i0

i−k|yi−y|< δ, we have|yn−y|< ε for alln≥k.

iiThe equilibriumyof1.4is called locally asymptotically stableasymptotic stable if it is locally stable, and if there existsγ >0 such that for ally_−k, . . . , y₋₁, y0∈Iwith _i0

i−k|yi−y|< γ, we have lim_{n→ ∞}y_ny.

iiiThe equilibriumyof1.4is called a global attractor if for everyy_−k, . . . , y₋₁, y₀∈I, we have limn→ ∞yny.

ivThe equilibriumyof1.4is globally asymptotically stable if it is locally stable and is a global attractor.

vThe equilibriumyof1.4is called unstable if it is not stable.

viThe equilibriumyof1.4is called a source, or a repeller, if there existsr >0 such that for ally_−k, . . . , y₋₁, y₀ ∈I with_i0

i−k|yi−y|< γ, there existsN ≥ 1 such that

|yN−y| ≥r.

An intervalJ⊆Iis called an invariant interval for1.4if

y_−k, . . . , y₀∈J⇒y_n∈J ∀n >0. 1.7

That is, every solution of1.4with initial conditions inJremains inJ.

The linearized equation associated with1.4about the equilibriumyis

y_n1^k

io

∂f

∂ui

y, . . . , y

y_n−i, n0,1, . . . . 1.8

Its characteristic equation is

λ^k1k

io

∂f

∂u_i

y, . . . , y

λ^k−i. 1.9

Theorem 1.3see20. Assume thatfis aC¹function and letybe an equilibrium of 1.4. Then the following statements are true:

iIf all the roots of 1.9lie in the open unit disk|λ|<1, then the equilibriumyof 1.4is asymptotically stable.

iiIf at least one root of 1.9has absolute value greater than one, then the equilibriumyof 1.4is unstable.

Theorem 1.4see20. Assume thatP, Q∈Randk∈ {1,2, . . .}. Then

|P||Q|<1 1.10

is a sufficient condition for the asymptotic stability of the difference equation

y_n1Py_nQy_n−k, n0,1, . . . . 1.11 Lemma 1.5see4. Letp ≥ 2 be a positive integer and assume that every positive solution of equation

x_n1 αβx_nγx_n−k

ABx_nCn_n−k, n0,1, . . . 1.12

is periodic with periodp. IfC >0, thenAB0.

Lemma 1.6see21. Consider the difference equation y_n1f

y_n, y_n−k

, n0,1, . . . , 1.13

wherek∈ {1,2, . . .}. LetI a, bbe some interval of real numbers and assume that

f :a, b×a, b−→a, b 1.14

is a continuous function satisfying the following properties:

afu, vis nonincreasing inuand nondecreasing inv, bifm, M∈a, b×a, bis a solution of the system

mfM, m, Mfm, M, 1.15

thenmM.

Then1.13has a unique equilibriumy∈a, b, and every solution of 1.13converges toy.

Lemma 1.7see21. Consider the difference equation y_n1f

yn, y_n−k

, n0,1, . . . , 1.16

wherek∈ {1,2, . . .}. LetI a, bbe some interval of real numbers, and assume that

f :a, b×a, b−→a, b 1.17

is a continuous function satisfying the following properties:

afu, vis nonincreasing in each of its arguments, bifm, M∈a, b×a, bis a solution of the system

mfM, M, Mfm, m, 1.18

thenmM.

Then1.16has a unique equilibriumy∈a, b, and every solution of 1.16converges toy.

2. The Special Case pqr 0

If the parameterspqr0, then1.1contains the following several equations. We now assume that all their parameters are positive

x_n1 qx_n−k

1x_nrx_n−k, n0,1, . . . , 2.1

x_n1 p

1x_nrx_n−k, n0,1, . . . , 2.2

x_n1 0, n0,1, . . . , 2.3

x_n1 p

1xn, n0,1, . . . , 2.4

x_n1 qx_n−k

1xn, n0,1, . . . , 2.5

x_n1 pqx_n−k

1x_n , n0,1, . . . . 2.6

Equation 2.2 was studied in 19, where it is shown that the unique positive equilibrium is a global attractor. Equation2.3is trivial. Equation2.4is the Riccati equation 1. Equation2.5can be reduces to2.6, which was discussed in22, and they showed that the unique positive equilibrium of2.6is globally asymptotically stable whenq <1. So, here we only consider2.1.

Clearly,x 0 is always an equilibrium of2.1and whenq > 1,2.1also possesses the unique positive equilibriumx q−1/1r.

The linearized equation associated with2.1about the zero equilibrium is

z_n1−qz_n−k0. 2.7

The linearized equation associated with2.1about the positive equilibrium is

z_n1 x

1xrxzn− q−rx

1xrxz_n−k0. 2.8

From this and byTheorem 1.4, we have the following result.

Theorem 2.1. aAssume thatq <1. Thenx0 of 2.1is locally asymptotically stable.

bAssume thatq >1 andr >1. Then the unique positive equilibriumx q−1/1rof 2.1is locally asymptotically stable.

Theorem 2.2. Assume that q > 1 andr > 1. Then the unique positive equilibriumx of 2.1 is globally asymptotically stable.

Proof. By Theorem 2.1, the positive equilibrium of 2.1is locally asymptotically stable. It suffices to show thatxis a global attractor.

Let

f x, y

qy

1xry forx, y∈0,∞, 2.9

thenfx, yis nonincreasing inxand nondecreasing iny. So 0≤f

x, y

≤ q

r forx, y∈0,∞. 2.10

From

qm

1Mrm m, qM

1mrM M, 2.11

we havemM.

Hence byLemma 1.6the proof is complete.

In the following sections we assume that all parameters in1.1are positive.

3. Local Stability and Period-Two Solutions

The equilibria of1.1are the solutions of the equation x pqx

1xrx. 3.1

So1.1possesses the unique positive equilibrium

x q−1 q−1₂

4pr1

2r1 . 3.2

The linearized equation associated with1.1about the positive equilibrium is

z_n1 x

1xrxzn− q−rx

1xrxz_n−k0. 3.3

ByTheorem 1.4, it is sufficient to show that in x

1xrx

q−rx 1xrx

<1. 3.4 Thus

q−rx<1rx. 3.5

Ifq−rx <0, then we haverx−q <1rx, and it clearly holds.

Ifq−rx≥0, then we haveq−rx <1rx, and

q−1<2rx. 3.6

Ifq≤1, the inequality3.6obviously holds. Supposeq >1, then we can get q−1< r

q−1₂

4pr1, q−1₂

< r² q−1₂

4pr²r1,

3.7

from which it follows that

r−1 q−1₂

4pr²>0. 3.8 So we have the following result.

Theorem 3.1. Assume that

eitherq≤1 orq >1, r−1

q−124pr²>0. 3.9 Then the positive equilibriumxof 1.1is locally asymptotically stable.

Theorem 3.2. aAssume thatkis odd. Then1.1has a nonnegative prime period-two solution if and only if

q >1, r−1 q−1₂

4pr²<0. 3.10 Further when3.10holds, the period-two solution is “unique” and the value ofφ₁ andφ₂ are the positive roots of the quadratic equation

t²−q−1 r t p

1−r 0. 3.11

bAssume thatkis even. Equation1.1has no nonnegative prime period-two solution.

Proof. aAssume thatkis odd, thenx_n1x_n−k. Let

. . . , φ₁, φ₂, φ₁, φ₂, . . . 3.12

be a nonnegative prime period-two solution of1.1. Thenφ₁, φ₂satisfy the following system:

φ₁ pqφ₁

1φ₂rφ₁, φ₂ pqφ₂

1φ₁rφ₂. 3.13

Substituting the above two equations, we obtain φ₁−φ₂

φ₁φ₂−q−1

r 0. 3.14

Thus

φ₁φ₂ q−1

r . 3.15

Adding them and using the above equations, we can get φ₁φ₂ p

1−r. 3.16

Clearly, in this case the discriminant of3.11is positive, that is

Δ r−1 q−1₂

4pr²

r²r−1 >0, 3.17

and soφ₁, φ₂are the positive roots of3.11.

bAssume thatk is even, thenx_n x_n−k. If there exist distinctive nonnegative real numberφ₁andφ₂, such that

. . . , φ₁, φ₂, φ₁, φ₂, . . . 3.18

is a prime period-two solution of1.1andφ₁, φ₂satisfy the following system:

φ₁ pqφ₂

1φ₂rφ₂, φ₂ pqφ₁

1φ₁rφ₁, 3.19

which is equivalent to

φ₁φ₁φ₂rφ₁φ₂pqφ₂, φ₂φ₁φ₂rφ₁φ₂pqφ₁. 3.20 Subtracting these two equation, we can get

φ1−φ2

q1

0. 3.21 Thenφ₁φ₂. This contradicts the hypothesis thatφ₁/φ₂.

The proof is complete.

Further, applyingLemma 1.5, we have the following result about the period solutions.

Theorem 3.3. There is no integerp ≥ 2 such that every positive solution of 1.1is periodic with periodp.

4. Boundedness and Invariant Interval

In this section, we will investigate the boundedness and invariant interval of1.1.

Theorem 4.1. Every solution of 1.1is bounded from above and from below by positive constants.

Proof. Let{xn}^∞_n−kbe a solution of1.1. Clearly, if the solution is bounded from above by a constantM, then

x_n1≥ p

1 r1M ∀n≥ −k, 4.1

and so it is also bounded from below. Now for the sake of contradiction assume that the solution is not bounded from above. Then there exists a subsequences{xnm1}^∞_m0such that

mlim→ ∞n_m∞, lim

m→ ∞x_nm1∞, 4.2

and also

xnm1max{xn:n≤nm} form≥0. 4.3 From1.1we see that

x_n1< pqx_n−k forn≥0, 4.4

and so

m→ ∞limx_nm1 lim

m→ ∞x_nm−k∞. 4.5

Hence, for sufficiently largem,

0≤xnm1−xnm−k pqx_nm−k

1x_nmrx_nm−k −xnm−k p q−1

−x_nm−rx_nm−k x_nm−k 1x_nmrx_nm−k <0,

4.6

which is a contradiction. The proof is complete.

Let

fu, v pqv

1urv. 4.7

Then the following statements are true.

Lemma 4.2. aAssume thatpr≤q. Thenfu, vis decreasing inufor eachvand increasing inv for eachu.

bAssume thatpr > q. Thenfu, vis decreasing inufor eachvand decreasing invfor u∈0,pr−q/q, and increasing invforu∈pr−q/q,∞.

Proof. The proofs ofaandbare simple and will be omitted.

Lemma 4.3. Equation1.1possesses the following invariant intervals:

a 0, q/r, whenpr≤q;

b pr−q/q, q/r, whenq < pr < qq²/r;

c 0, p, whenprqq²/r;

d q/r,pr−q/q, whenqq²/r < pr < pqq;

e q/r, p, whenpr≥pqq.

Proof. aSetgx pqx/1rx, sogxis increasing forxandgq/r ≤q/r. Using 1.1we see that whenx_−k, . . . , x₋₁, x0 ∈0, q/r, then

x₁ pqx_−k

1x₀rx_−k ≤ pqx_−k 1rx_−k ≤g

q r ≤ q

r. 4.8

The proof follows by induction.

b Using the monotonic character of the function fu, v which is described by Lemma 4.2b and the condition that q < pr < q q²/r, when x_−k, . . . , x₋₁, x0 ∈ pr − q/q, q/r, we can get

pr−q q ≤f

q r,pr−q

q ≤x₁ pqx_−k

1x₀rx_−k fxo, x_−k≤f

pr−q q ,q

r q

r. 4.9

The proof follows by induction.

cSethx pqx/1prx,gx pqx/1rx, thenhxis increasing for xandgxis decreasing forxwhenpr qq²/r, we see that whenx_−k, . . . , x₋₁, x0∈0, p, then

x₁ pqx_−k

1x₀rx_−k ≥ pqx_−k

1prx_−k ≥h0 p 1p >0, x1 pqx_−k

1x₀rx_−k ≤ pqx_−k

1rx_−k ≤g0 p.

4.10

The proof follows by induction.

d Using the monotonic character of the function fu, v which is described by Lemma 4.2band the conditionqq²/r < pr < pqq, whenx_−k, . . . , x₋₁, x₀∈q/r,pr−q/q, we obtain

q r f

pr−q q ,pr−q

q ≤x1 pqx_−k

1x₀rx_−k fxo, x_−k≤f q

r,q

r prq²

r r1q < pr−q q . 4.11 The proof follows by induction.

eIn view of the conditionpr ≥ pqq, we can getpr−q/q > p > q/r. By using the monotonic character of the functionfu, vwhich is described byLemma 4.2band the conditionpr≥pqq, whenx_−k, . . . , x₋₁, x₀ ∈q/r, p, we have

q

r ≤ pqp 1ppr f

p, p

≤x1 pqx_−k

1x₀rx_−k fxo, x_−k≤f q

r,q

r prq² rqqr < p.

4.12 The proof follows by induction.

The proof is complete.

5. Semicycles Analysis

Let{xn}^∞_n−kbe a positive solution of1.1. Then we have the following equations:

x_n1−q r q

r

pr−q /q−xn

1x_nrx_n−k , forn≥0, 5.1

x_n1−p−pxn pr−q

x_n−k

1x_nrx_n−k , forn≥0, 5.2

x_n1−pr−q q

1/q

qq²/r−pr

pr−q /q

q/r−x_n

r/q

qq²/r−pr x_n−k

1xnrx_n−k , forn≥0,

5.3 x_n1−x r

x−q/r

x−x_n−k xx−x_n

1x_nrx_n−k , forn≥0, 5.4

x_n2−x_n rx_n1−k

q/r−xn

rxn−kxn1 qr

x_n−k pr/

qr

−xn

p−xn−x²_n 1x_nrx_n−k1rx_n1−k pqx_n−k ,

forn≥0.

5.5 Ifpr qq²/r, then the unique positive equilibrium isx q/r, and5.1and5.5 change into

x_n1−q r q

r

q/r−x_n

1xnrx_n−k, forn≥0, 5.6

x_n2−xn

q/r−xn

r²x_n1−kx_n−krxnx_n1−krx_n1−k pr²/q

x_n−kxnq/r1

1xnrx_n−k1rx_n1−k pqx_n−k , forn≥0.

5.7

The following lemma is straightforward consequences of identities5.1–5.7.

Lemma 5.1. Assume that pr ≤ q and let {xn}^∞_n−k be a solution of 1.1. Then the following statements are true:

ixn≤q/rfor alln≥1;

iiif for someN≥0,x_N−k≤xandxN ≥x, thenx_N1≤x;

iiiif for someN≥0,x_N−k> xandxN < x, thenx_N1> x;

iv0≤x≤q/r.

Lemma 5.2. Assume that q < pr < qq²/r and let {xn}^∞_n−k be a solution of 1.1. Then the following statements are true:

iif for someN≥0,x_N< q/r, thenx_N1>pr−q/q;

iiif for someN≥0,x_N<pr−q/q, thenx_N1 > q/r;

iiiif for someN≥0,x_N pr−q/q, thenx_N1 q/r;

ivif for someN≥0,x_N>pr−q/q, thenx_N1 < q/r;

vif for someN≥0,pr−q/q≤x_N≤q/r, thenpr−q/q≤x_n≤q/rforn≥N;

viif for someN≥0,x_N−k≤x, andx_N≥x, thenx_N1≤x;

viiif for someN≥0,x_N−k> x, andx_N< x, thenx_N1> x;

viiiif for someN≥0,x_N<pr−q/q, thenx_N2 > x_N; ixif for someN≥0,xN> q/r, thenx_N2< xN;

x pr−q/q≤x≤q/r.

Lemma 5.3. Assume thatprqq²/r and let{xn}^∞_n−kbe a solution of 1.1. Then the following statements are true:

iif for someN≥0,xN> q/r, thenx_N1< q/r;

iiif for someN≥0,xNq/r, thenx_N1q/r;

iiiif for someN≥0,xN< q/r, thenx_N1> q/r;

ivif for someN≥0,xN≥x, thenx_N1≤x;

vif for someN≥0,xN< x, thenx_N1> x;

viif for someN≥0,xN> q/r, thenx_N2< xN; viiif for someN≥0,xN< q/r, thenx_N2> xN; viiixq/r.

Lemma 5.4. Assume thatqq²/r < pr < pqqand let{xn}^∞_n−kbe a solution of 1.1. Then the following statements are true:

iif for someN≥0,x_N> q/r, thenx_N1<pr−q/q;

iiif for someN≥0,x_N<pr−q/q, thenx_N1 > q/r;

iiiif for someN≥0,xN pr−q/q, thenx_N1 q/r;

ivif for someN≥0,x_N>pr−q/q, thenx_N1 < q/r;

vif for someN≥0,q/r≤xN ≤pr−q/q, thenq/r ≤xn ≤pr−q/qforn≥N;

viif for someN≥0,x_N−k≤xandx_N ≤x, thenx_N1≥x;

viiif for someN≥0,x_N−k> xandxN > x, thenx_N1< x;

viiiif for someN≥0,x_N<pr−q/q, thenx_N1 > q/r;

ixif for someN≥0,xN< q/r, thenx_N2> xN; xif for someN≥0,x_N>pr−q/q, thenx_N2 < x_N; xiq/r≤x≤pr−q/q.

Lemma 5.5. Assume thatpr ≥ pqqand let{xn}^∞_n−k be a solution of 1.1. Then the following statements are true:

ixn≤pfor alln≥1;

iiif for someN≥0,x_N≤p, thenx_N1≥q/r;

iiiif for someN≥0,q/r≤xN ≤p, thenq/r≤xn≤pforn≥N;

ivif for someN≥0,x_N−k≤xandx_N ≤x, thenx_N1≥x;

vif for someN≥0,x_N−k> xandxN > x, thenx_N1< x;

viif for someN≥0,x_N< q/r, thenx_N2> x_N; viiif for someN≥0,xN> p, thenx_N2< xN; viiiq/r≤x≤p.

The following results are consequences of Lemmas5.1–5.5.

Theorem 5.6. Let{xn}^∞_n−kbe a nontrivial solution of 1.1andxis the unique positive equilibrium point of 1.1. Then the following statements are true.

aAssume thatpr≤q. Then except possibly for the first semicycle, every oscillatory solution of 1.1has semicycles of length at mostk.

bAssume that q < pr < q q²/r. Then, except possibly for the first semicycle, every oscillatory solution of 1.1 which lies in the invariant interval pr −q/q, q/r has semicycles of length at mostk.

cAssume thatprqq²/r. Then after the first semicycle, every oscillatory solution of1.1 about the equilibrium pointxwith semicycle of length one.

dAssume thatqq²/r < pr < pqq. Then, except possibly for the first semicycle, every oscillatory solution of 1.1 which lies in the invariant interval q/r,pr −q/q has semicycles that is either of length at leastk−1, or of length at mostk1.

eAssume thatpr ≥ pqq. Then, except possibly for the first semicycle, every oscillatory solution of 1.1which lies in the invariant intervalq/r, phas semicycles that is either of length at leastk−1, or of length at mostk1.

6. Global Stability

In this section, we will investigate global stability of the positive equilibrium pointxof1.1.

Theorem 6.1. Assume that3.9holds and let{xn}^∞_n−k be a positive solution of 1.1. Then every solution of 1.1eventually enters the invariant interval

a 0, q/rifpr≤q;

b pr−q/q, q/rifq < pr < qq²/r; c q/r,pr−q/qifqq²/r < pr < pqq;

d q/r, pifpr≥pqq.

Proof. aThe proof is a direct consequence ofLemma 5.1.

bFromLemma 5.2vwe know that if there exist an integerNsuch thatx_N ∈pr− q/q, q/r, thenx_n∈pr−q/q, q/rforn≥Nand the result follows. Now assume for the sake of contradiction that all terms of{xn}never enter the invariant intervalpr−q/q, q/r forn≥0. Notice thatLemma 5.2iiimplies thatx_n1> q/rforx_n<pr−q/q. Further using Lemma 5.2viiiand ix, we obtain that the subsequence{x2n}^∞_n0 and {x_2n1}^∞_n0 are both monotonous. If one of them is decreasing, then it is bounded above bypr−q/q, and if one of them is decreasing, then it is bounded below byq/r. Thus lim_{n→ ∞}x_2n and limn→ ∞x_2n1 exist. Set

nlim→ ∞x2nL, lim

n→ ∞x_2n1M, 6.1

thenL≤pr−q/qandM≥q/r, or vice versa. From which it follows that

. . . , L, M, L, M, . . . 6.2

is a period-two solution of1.1, which is a contradiction, since when2.3holds,1.1has no period-two solution.

cThe proof is similar tob, so will be omitted.

dIn view ofLemma 5.5iandiii, we know thatx_n ≤ pfor alln ≥1 andq/r, p is an invariant interval of 1.1. If there exist an integerN such that x_N ∈ q/r, p, then xn∈q/r, pforn≥N, from which it follows that the result is true. Now assume for the sake of contradiction that terms of{xn}never enter the invariant intervalq/r, p, then they all lie in the interval0, q/r. Noticing thatx₁≤q/randpr≥pqqhold, we get

x₂−x₁ p−x₁rx_1−k

q/r−x₁

−x²₁

1x₁rx_1−k ≥ p−q/r− q/r₂

1x₁rx_1−k >0, 6.3 from which it follows by induction that the sequence{xn}is increasing in the interval0, q/r.

Hence, lim_{n→ ∞}x_nexists and lim_{n→ ∞}x_n ≤q/r, which is a contradiction because1.1has no equilibrium point in the interval0, q/r.

The proof is complete.

Theorem 6.2. Assume that3.9holds. Then the positive equilibriumxis a global attractor of 1.1.

Proof. The proof is finished by considering the following five cases.

Case 1whenpr≤q. ByLemma 4.3aandTheorem 6.1a, we know that1.1possesses an invariant interval0, q/rand every solution of1.1eventually enters the interval0, q/r.

Further, it is easy to see thatfu, vdecreases inuand increases invin0, q/r.

Finally observe that when3.9holds, the only solution of the system pqm

1Mrm m, pqM

1mrM M, 6.4

is m M. Further, Lemma 5.1implies that 1.1 has a unique equilibrium x ∈ 0, q/r.

Thus, in view ofLemma 1.6, every solution of1.1converges tox. So the unique positive equilibriumxis a global attractor of1.1.

Case 2 when q < pr < q q²/r. By Lemma 4.3b and Theorem 6.1b, we know that 1.1possesses an invariant intervalpr−q/q, q/rand every solution of1.1eventually enters the intervalpr−q/q, q/r. Further, it is easy to see thatfu, vdecreases inuand increases invinpr−q/q, q/r. Then using the same argument in Case1,1.1has a unique equilibriumx ∈pr−q/q, q/rand every solution of1.1converges tox. So the unique positive equilibriumxis a global attractor of1.1.

Case 3 when pr q q²/r. In view of part c of Theorem 5.6, we know that, after the first semicycle, the nontrivial solution oscillates aboutxwith semicycles of length one.

Considering the subsequences{x2n}^∞_n0and{x_2n1}^∞_n0, we have x2n> q

r, x_2n1< q

r forn≥0, 6.5

or

x2n< q

r, x_2n1> q

r forn≥0. 6.6

Let us consider Case1. Case2can be handled in a similar way. In view ofTheorem 4.1and Lemma 5.3, we know that{xn}^∞_n−kis bounded andx_n1q/rifxnq/rand

x₁< x₃< x₅ <· · ·< q

r <· · ·< x₄< x₂< x₀. 6.7

So limn→ ∞x2nand limn→ ∞x_2n1exist.

Let

L lim

n→ ∞x_2n, M lim

n→ ∞x_2n1, 6.8

thenL ≥ q/r, M ≤ q/r. IfL /M, thenL, M is a period-two solution of1.1. Furthermore, the condition3.9holds. This contradictsTheorem 3.2. ThusL Mand lim_{n→ ∞}xn q/r.

Soxq/ris a global attractor of1.1.

Case 4 whenqq²/r < pr < pq q. By Theorem 6.1c, we know that every solution of 1.1 eventually enters the interval q/r,pr−q/q. Furthermore, it is easy to see that the functionfu, vdecreases in each of its arguments in the intervalq/r,pr−q/q. Let m, M∈q/r,pr−q/qis a solution of the system

pqm

1mrm M, pqM

1MrM m, 6.9

that is, the solution of the system

pqmMmMrmM, pqMmmMrmM. 6.10

Thenm−Mq1 0, which implies thatmM. EmployingLemma 1.7, we see that1.1 has a unique equilibriumx ∈ q/r,pr−q/qand every solution of 1.1converges tox.

Thus the unique positive equilibriumxis a global attractor of1.1.

Case 5. Whenpr≥pqq. ByTheorem 6.1d, we know that every solution of1.1eventually enters the intervalq/r, p. Further, it is clear to see that the functionfu, vdecreases in each of its arguments in the intervalq/r, p. Then, using the same argument as in Case4,1.1has a unique equilibriumx∈q/r, pand every solution of1.1converges tox. Thus the unique positive equilibriumxis a global attractor of1.1.

The proof is complete.

In view of Theorems3.1and 6.2, we have the following result, which solves Open Problem1when conditions3.9holds.

Theorem 6.3. Assumed that 3.9 holds. Then the positive equilibrium of 1.1 is globally asymptotically stable.

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