Dynamics of a Higher-Order Nonlinear Difference Equation

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Volume 2010, Article ID 534947,15pages doi:10.1155/2010/534947

Research Article

Dynamics of a Higher-Order Nonlinear Difference Equation

Guo-Mei Tang,

¹

Lin-Xia Hu,

²

and Xiu-Mei Jia

³

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

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

3Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

Correspondence should be addressed to Guo-Mei Tang,[email protected] Received 20 April 2010; Accepted 18 July 2010

Academic Editor: Leonid Berezansky

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 αxn/ABxnx_n−k, n 0,1, . . ., where parameters are positive real numbers and initial conditionsx_−k, . . . , x0 are nonnegative real numbers,k ≥ 2. We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

1. Introduction and Preliminaries

In this paper, we will investigate the global behavior of solutions of the following nonlinear diﬀerence equation:

x_n1 αxn

ABx_nx_n−k, n0,1, . . ., 1.1 where parameters are positive real numbers and initial conditionsx_−k, . . . , x0are nonnegative real numbers,k≥2.

In 2003, the authors in1considered the diﬀerence equation

x_n1 αβxn

ABxnCx_n−1, n0,1, . . ., 1.2

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with nonnegative parameters α, β, A, B, C and nonnegative initial conditions x₋₁, x0. They obtained some global asymptotic stability results for the solutions of1.2. For1.2, we can also see2–4.

For the global behavior of solutions of some related equations, see 5–13. Other related results can be found in14–20. 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

y_n, y_n−1, . . . , y_n−k

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

yny, forn≥0, 1.6

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

Definition 1.2. Letybe an equilibrium point of1.4. Then the following are considered.

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 limn→ ∞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 ∈Iwith_i0

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

|yN−y| ≥r.

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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

∂u_i

y, . . . , y

y_n−i, n0,1,2, . . .. 1.8

Its characteristic equation is

λ^k1^k

io

∂f

∂u_i

y, . . . , y

λ^k−i. 1.9

Theorem 1.3see10. 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 1.4see10. 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_n1PynQy_n−k, n0,1, . . . . 1.11 Lemma 1.5see8. Consider the diﬀerence equation

y_n1f

yn, y_n−k

, n0,1, . . ., 1.12

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

f :a, b×a, b−→a, b 1.13

is a continuous function satisfying the following properties.

afx, yis nondecreasing inx∈a, bfor eachy∈a, b, andfx, yis nonincreasing in y∈a, bfor eachx∈a, b.

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bIfm, M∈a, b×a, bis a solution of the system

fm, M m, fM, m M, 1.14

thenmM.

Then1.12has a unique equilibriumx∈a, band every solution of 1.12converges tox.

Lemma 1.6see8. Consider the diﬀerence equation y_n1f

y_n, y_n−k

, n0,1, . . ., 1.15

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

f :a, b×a, b−→a, b 1.16

is a continuous function satisfying the following properties.

afx, yis nonincreasing in each of its arguments.

bIfm,M∈a, b×a, bis a solution of the system

mfM, M, Mfm, m, 1.17

thenmM.

Then1.15has a unique equilibriumy∈a, band every solution of 1.15converges toy.

2. Local Stability and Period-Two Solutions

The equilibria of1.1are the solutions of the equation

x αx

ABxx. 2.1

So1.1possesses the unique positive equilibrium

x 1−A

1−A²4αB1

2B1 . 2.2

The linearized equation associated with1.1about the positive equilibrium is z_n1 Bx−1

ABxxz_n x

ABxxz_n−k0. 2.3

The next result follows fromTheorem 1.4.

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Theorem 2.1. Assume that

eitherA≥1 orA <1, B−11−A²4B²α >0. 2.4

Then the positive equilibriumxof 1.1is locally asymptotically stable.

Theorem 2.2. Equation1.1has no nonnegative prime period-two solution.

Proof. Assume for the sake of contradiction that there exist distinct nonnegative real numbers φandψsuch that

. . . , φ, ψ, φ, ψ, . . . 2.5

is a prime period-two solution of1.1.

aAssume thatkis odd. Thenx_n1x_n−kandφ, ψsatisfy the following system:

φ αψ

ABψφ, ψ αφ

ABφψ. 2.6

Subtracting both sides of the above two equations, we obtain φ−ψ

φψ A1

0. 2.7

Ifφ /ψ, thenφψ−A1; this contradicts the hypothesis thatφ, ψ≥0.

bAssume thatkis even. Thenxnx_n−kandφ, ψsatisfy the following system:

φ αψ

ABψψ, ψ αφ

ABφφ. 2.8

Subtracting both sides of the above two equations, we obtain φ−ψ

A1 0. 2.9

Ifφ /ψ, thenA−1; this contradicts the hypothesis thatA≥0.

The proof is complete.

3. Boundedness and Invariant Interval

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

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Theorem 3.1. Every solution of 1.1is bounded from above and from below by positive constants.

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

x_n1≥ α

A B1M, forn≥ −k, 3.1

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 subsequence{xnm1}^∞_m0such that

mlim→ ∞nm∞, lim

m→ ∞xnm1∞ 3.2

and also

xnm1max{xn:n≤nm} form≥0. 3.3

From1.1we see that

x_n1< α A 1

Ax_n forn≥0, 3.4

and so

mlim→ ∞x_n_m₁ lim

m→ ∞x_n_m ∞. 3.5

Hence, for suﬃciently largem,

0≤x_n_m₁−x_n_m αxnm

ABxnmxnm−k −x_n_m α 1−A−Bxnm−xnm−kxnm

ABxnmxnm−k <0, 3.6 which is a contradiction.

Let

f x, y

αx

ABxy. 3.7

Then the following statements are true.

Lemma 3.2. aAssume thatA≥Bα.Thenfx, yis increasing inxfor eachyand decreasing in yfor eachx.

bAssume that A < Bα.Thenfx, yis decreasing in y for eachx, decreasing in xfor y∈0, Bα−A, and increasing inxfory∈Bα−A,∞.

Proof. The proofs ofaandbare simple and will be omitted.

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Theorem 3.3. Equation1.1possesses the following invariant intervals:

a 0,1/B,whenBα≤A;

b Bα−A,1/B,whenA < Bα < A1/B;

c 0, α/A,whenBαA1/B;

d 1/B, Bα−A,whenA1/B < Bα < Aα/A;

e 1/B, α/A,whenBα≥Aα/A.

Proof. aSetgx αx/ABx, sogxis nondecreasing forxandg1/B≤ 1/Bif Bα≤A, whenx_−k, . . . , x0∈0,1/B; then we have

x1 αx0

ABx₀x_−k ≤ αx0

ABx₀ ≤g 1 B

≤ 1

B. 3.8

The proof follows by induction.

bIn view ofLemma 3.2b, by using the monotonic character of the functionfx, y and the conditionA < Bα < A 1/B,whenx_−k, . . . , x₀∈Bα−A,1/B,we can get

x₁ αx₀

ABx0x_−k fxo, x_−k≥f Bα−A,1 B

> Bα−A, x₁ αx0

ABx₀x_−k fxo, x_−k≤f 1

B, Bα−A

1 B.

3.9

cSethx αx/ABxα/Aandgx αx/ABx, sohxis increasing andgxis decreasing forxifBαA1/B. In view ofLemma 3.2b, by using the monotonic character of the functionfx, y, whenx_−k, . . . , x₀ ∈0, α/A,we have

x₁ αx₀

ABx₀x_−k ≥ αx₀

ABx₀ α/A ≥h0>0, x₁ αx0

ABx₀x_−k ≤ αx0

ABx₀ ≤g0 α A.

3.10

dIn view ofLemma 3.2b, by using the monotonic character of the functionfx, y and the conditionA1/B < Bα < Aα/A, whenx_−k, . . . , x₀∈1/B, Bα−A,we obtain

x₁ αx₀

ABx0x_−k fx_o, x_−k≥fBα−A, Bα−A> 1 B, x₁ αx₀

ABx0x_−k fx_o, x_−k≤f 1 B, 1

B

Bα1

ABB1 < Bα−A.

3.11

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e In view of the conditionBα ≥ Aα/A, we can get Bα−A ≥ α/A; by using the monotonic character of the function fx, y and the condition Bα ≥ Aα/A, when x_−k, . . . , x₀∈1/B, α/A, we have

x₁ αx₀

ABx0x_−k fxo, x_−k≥fα A,α

A

Aαα A²Bαα≥ 1

B, x₁ αx₀

ABx₀x_−k fxo, x_−k≤f 1 B,1

B

Bα1 ABB1 < α

A.

3.12

4. Semicycles Analysis

We now give the definitions of positive and negative semicycles of a solution of1.4relative to an equilibrium pointx.

A positive semicycle of a solution{xn} of 1.4 consists of a string of terms {xl, x_l1, . . . , xm}, all greater than or equal to the equilibriumx, withl≥ −kandm≤ ∞and such that

eitherl−k orl >−k, x_l−1< x, either m∞orm <∞, x_m1< x.

4.1

A negative semicycle of a solution {xn} of 1.4 consists of a string of terms {xl, x_l1, . . . , x_m}, all less than the equilibriumx, withl≥ −kandm≤ ∞and such that

eitherl−k orl >−k, xl−1≥x, eitherm∞ orm <∞, x_m1≥x.

4.2

Theorem 4.1 see 12. Assume that f ∈ C0,∞ ×0,∞,0,∞ is such that fx, y is increasing inxfor each fixedyand is decreasing inyfor each fixedx. Letxbe a positive equilibrium of 1.12. Then the following are considered.

aIfk1, then every solution of 1.12has semicycles of length at least two.

bIfk≥2, then every solution of 1.12has semicycles that are either of length at leastk1 or of length at mostk−1.

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Let{xn}be a positive solution of1.1. Then one has the following identities:

x_n1− 1 B 1

B

Bα−A−x_n−k

ABxnx_n−k , n∈N0, 4.3

x_n1− α A −1

A

Bα−Axnαx_n−k

ABxnx_n−k , n∈N0, 4.4

x_n1−Bα−A B1/B−Bα−AxnBα1/B−x_n−k Axn−k−Bα−A

ABx_nx_n−k , n∈N0,

4.5 x_n1−x xx−x_n−k Bx−1/Bx−x_n

ABxnx_n−k , n∈N0, 4.6

x_2k1n1−x_2k1n

Bx_2k1n1−1

1/B−x_2k1n

Ax_2k1nBx_2k1nk

ABx_2k1n1−1

ABx_n1x_2k1n

αx_2k1nk

1ABx_2k1nk

αB/1AB−x_2k1n A

α−Ax_2k1n−x²_2k1n

ABx_2k1n1−1

ABx_2k1nkx_2k1n

αx_2k1nk , n∈N0.

4.7

If 1/BBα−A, thenx1/Band4.3,4.7change into

x_n1− 1 B 1

B

1/B−x_n−k

ABx_nx_n−k, n∈N0, 4.8

x_2k1n1−x_2k1n

1/B−x_2k1n

ABx_2k1n1−1Bx_2k1nx_2k1n1−1

ABx_2k1n1−1

ABx_2k1nkx_2k1n

αx_2k1nk

1/B−x_2k1n

B²x_2k1nkx_2k1n1−1B²αx_2k1nkAx_2k1nABα

ABx_2k1n1−1

ABx_2k1nkx_2k1n

αx_2k1nk ,

n∈N0. 4.9

The following lemmas are straightforward consequences of identities4.3−4.9.

Lemma 4.2. Assume that

Bα≤A, 4.10

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and let{xn}be a solution of 1.1. Then the following statements are true.

ixn≤1/Bfor 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 <1/B.

A < Bα < A 1

B, 4.11

iIf for someN≥0,x_N< Bα−A, thenx_Nk1>1/B.

iiIf for someN≥0,x_NBα−A, thenx_Nk11/B.

iiiIf for someN≥0,x_N> Bα−A, thenx_Nk1<1/B.

ivIf for someN≥0,Bα−A < x_N<1/B, thenBα−A < x_Nk1<1/B.

vIf for someN≥0,x_N−k≤xandx_N≥x, thenx_N1≥x.

viIf for someN≥0,x_N−k> xandx_N< x, thenx_N1< x.

viiIf for someN≥0,x_2k1N< Bα−A, thenx_2k1N1> x_2k1N. viiiIf for someN≥0,x_2k1N>1/B, thenx_2k1N1< x_2k1N.

ixBα−A < x <1/B.

BαA 1

B, 4.12

iIf for someN≥0,x_N>1/B, thenx_Nk1 <1/B.

iiIf for someN≥0,x_N1/B, thenx_Nk1 1/B.

iiiIf for someN≥0,x_N<1/B, thenx_Nk1 >1/B.

ivIf for someN≥0,x_2k1N>1/B, thenx_2k1N1< x_2k1N. vIf for someN≥0,x_2k1N<1/B, thenx_2k1N1> x_2k1N. vix1/B.

A 1

B < Bα < A α

A, 4.13

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iIf for someN≥0,x_N< Bα−A, thenx_Nk1>1/B.

iiIf for someN≥0,x_NBα−A, thenx_Nk11/B.

iiiIf for someN≥0,x_N> Bα−A, thenx_Nk1<1/B.

ivIf for someN≥0, 1/B < x_N< Bα−A, then 1/B < x_Nk1< Bα−A.

vIf for someN≥0,x_N−k≤xandx_N≤x, thenx_N1≥x.

viIf for someN≥0,x_N−k> xandx_N> x, thenx_N1< x.

viiIf for someN≥0,x_2k1N<1/B, thenx_2k1N1> x_2k1N. viiiIf for someN≥0,x_2k1N> Bα−A, thenx_2k1N1< x_2k1N.

ix1/B < x < Bα−A.

Bα≥A α

A, 4.14

ixn< α/Afor alln≥1.

iiIf for someN≥0,x_N< α/A, thenx_Nk1>1/B.

iiiIf for someN≥0, 1/B < x_N< α/A, then 1/B < x_Nk1< α/A.

ivIf for someN≥0,x_N−k≤xandx_N≤x, thenx_N1≥x.

vIf for someN≥0,x_N−k> xandx_N> x, thenx_N1< x.

viIf for someN≥0,x_2k1N<1/B, thenx_2k1N1> x_2k1N. viiIf for someN≥0,x_2k1N> α/A, thenx_2k1N1< x_2k1N. viii1/B < x < α/A.

The following result is a consequence ofTheorem 4.1and Lemmas4.2−4.6.

Theorem 4.7. Let{xn}^∞_n−kbe a nontrivial solution of 1.1. Then the following statements are true.

aAssume thatBα≤A.Then, except possibly for the first semicycle, every oscillatory solution of 1.1has semicycles that are either of length at leastk1, or of length at mostk−1.

bAssume that A < Bα < A1/B. Then, except possibly for the first semicycle, every oscillatory solution of 1.1 which lies in the invariant interval Bα − A,1/B has semicycles that are either of length at leastk1, or of length at mostk−1.

cAssume that Bα A1/B. Then, except possibly for the first semicycle,{xn}^∞_n−k is oscillatory and the sum of the lengths of two consecutive semicycles is equal to 2k1.

dAssume thatA1/B < Bα < Aα/A.Then, except possibly for the first semicycle, every oscillatory solution of 1.1which lies in the invariant interval1/B, Bα−Ahas semicycles at mostk1.

eAssume thatBα≥Aα/A.Then, except possibly for the first semicycle, every oscillatory solution of 1.1which lies in the invariant interval1/B, α/Ahas semicycles at most k1.

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5. Global Stability Proof

In this section, we will investigate the global stability of all positive solutions of1.1.

Theorem 5.1. Let{xn}^∞_n−kbe a positive solution of1.1. Then the following statements are true.

aAssume thatBα≤A.Then every solution of 1.1eventually enters the interval0,1/B.

bAssume thatA < Bα < A1/B.Then every solution of 1.1eventually enters the interval Bα−A,1/B.

cAssume thatA1/B < Bα < Aα/A.Then every solution of 1.1eventually enters the interval1/B, Bα−A.

dAssume thatBα≥ Aα/A.Then every solution of 1.1eventually enters the interval 1/B, α/A.

Proof. aIn view ofLemma 4.2, we know thatx_n≤1/Bfor alln≥1 andx∈0,1/B; that is, all solutions of1.1eventually enter the interval0,1/B.

b If x_−k, . . . , x0 ∈ Bα− A,1/B, by Theorem 3.3b, then we have xn ∈ Bα − A,1/B,for alln ≥ 0. If the initial conditions are not in the interval Bα− A,1/B, then we consider the 2k1th subsequences {x_2k1nj}^2k1_j0 of the solution {xn}. We will give the proof for the subsequence{x_2k1n}. The proof for all the other subsequences is similar and will be omitted. Without loss of generality, we assume that there existsN suﬃciently large such thatx_2k1N < Bα−Aif x2k1N > 1/B, then the proof is similar and will be omitted; then in view of Lemmas4.3iiandiv, we know thatx_k12N1 >1/B > Bα−A

and x_2k1N1 < 1/B. If x_2k1N1 ≥ Bα− A, then, by induction, we know that the

former assertion implies that the result is true. Ifx_2k1N1 < Bα−A, byLemma 4.3viii, then we can get x_2k1N1 > x_2k1N. It follows by induction that the subsequence {x2k1Nm}^∞_m0is increasing, and becausex_2k1Nm< Bα−A, so limm→ ∞x_2k1Nmexists and limm→ ∞x_2k1Nm≤Bα−A. However, taking limits by4.7, we get a contradiction.

cThe proof is similar tob, so will be omitted.

dIn view ofLemma 4.6, we know thatxn < α/Afor alln≥ 1; that is, all solutions of1.1eventually enter the interval 0, α/A. Furthermore, byTheorem 3.3,1/B, α/Ais an invariant interval of 1.1. Now, assume for the sake of contradiction that all solutions never enter the interval1/B, α/A, then the subsequence{x_2k1Nm}^∞_m0enters the interval 0,1/B. Because x_2k1N ≤ 1/Band Bα ≥ Aα/A, then, by Lemma 4.6, we know that x_2k1N1 > x_2k1N; it follows by induction that the subsequence {x_2k1Nm}^∞_m0 is increasing in the interval0,1/B. So limm→ ∞x_2k1Nm exists and lim_m_{→ ∞}x_2k1Nm ≤ 1/B, which is a contradiction because1.1has no equilibrium point in the interval0,1/B.

Theorem 5.2. Assume that2.4holds. Then the positive equilibriumxof 1.1is a global attractor of all positive solutions of1.1.

We consider the following five cases.

Case 1. Assume thatBα≤A. By Theorems3.3aand5.1a, we know that1.1possesses an invariant interval0,1/Band every solution of1.1eventually enters the interval0,1/B.

Further, it is easy to see thatfx, yincreases inxand decreases inyin0,1/B.

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Letm, M∈0,1/Bbe a solution of the system αm

ABmM m, αM

ABMmM, 5.1

which is equivalent to

αmAmBm²Mm, αMAMBM²Mm. 5.2

Hence

m−M1−A−BMm 0. 5.3

Now ifmM / 1−A/B, thenmM. For instance, this is the case ifA≥1 is satisfied.

IfmM 1−A/B, thenmandMsatisfy the system mM 1−A

B , mM α

1−B 5.4

and the equation

BB−1m² B−1A−1m−Bα0, 5.5

whose discriminant is

Δ B−1

B−1A−1²4B²α

. 5.6

Clearly, in this case,B < 1, and in view of condition2.4, we have Δ < 0, from which it follows thatmM. In view ofLemma 1.5,1.1has a unique equilibriumx∈0,1/Band every solution of1.1converges tox.

Case 2. Assume thatA < Bα < A1/B. By Theorems3.3band5.1b, we know that1.1 possesses an invariant intervalBα−A,1/Band every solution of1.1eventually enters the intervalBα−A,1/B. Further, it is easy to see thatfx, yincreases inxand decreases in yinBα−A,1/B. Then using the same argument in Case1,1.1has a unique equilibrium x∈Bα−A,1/Band every solution of1.1converges tox.

Case 3. Assume thatBαA1/B. Considering the 2k1th subsequences{x_2k1nj}^∞_n0j∈ {0,1, . . . ,2k 1}, n ≥ 0, then by Lemma 4.4, we know that each one of the 2k 1th subsequences is above 1/B, below 1/B, or identically equal to 1/B. Furthermore, by the identity4.9, we can get that all 2k1th subsequences converge monotonically to limits, and for alln∈N,

x_2k1n1x_2k1n iﬀx_2k1n 1

B. 5.7

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So all the 2k1th subsequences{x2k1n1j}^∞_n0j ∈ {0,1, . . . ,2k1}converge to 1/B.

That is,x1/Bis a global attractor of1.1.

Case 4. Assume thatA1/B < Bα < Aα/A.By Theorems3.3dand5.1c, we know that 1.1possesses an invariant interval 1/B, Bα−A and every solution of1.1 eventually enters the interval 1/B, Bα− A. Furthermore, it is easy to see that the function fx, y decreases in each of its arguments in the interval1/B, Bα−A. Letm, M∈1/B, Bα−Abe a solution of the system

αm

ABmm M, αM

ABMM m, 5.8

that is, the solution of the system

αmAM B1mM, αMAm B1mM. 5.9

Thenm−MA1 0, which implies thatmM. EmployingLemma 1.6, we see that1.1 has a unique equilibriumx∈1/B, Bα−Aand every solution of1.1converges tox.

Case 5. Assume that Bα ≥ Aα/A. By Theorems 3.3e and 5.1d, we know that 1.1 possesses an invariant interval1/B, α/Aand every solution of1.1eventually enters the interval1/B, α/A. Further, it is clear to see that the functionfx, ydecreases in each of its arguments in the interval1/B, α/A. Then, using the same argument as in Case4,1.1has a unique equilibriumx∈1/B, α/Aand every solution of1.1converges tox.

In view of Theorems2.1and5.2, we have the following result.

Theorem 5.3. Assume that2.4holds. Then the unique positive equilibriumxof 1.1is globally asymptotically stable.

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