Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
VolumeΒ 2010, Article IDΒ 610467, 17 pages
http://dx.doi.org/10.1155/2010/610467
Research Article

Boundedness and Global Attractivity of a Higher-Order Nonlinear Difference Equation

1Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 5 November 2009; Accepted 4 February 2010

Academic Editor: GuangΒ Zhang

Copyright Β© 2010 Xiu-Mei Jia and Wan-Tong Li. 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.

Abstract

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: 𝑦𝑛+1=(π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜)/(π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜), π‘›βˆˆβ„•0, where the parameters 𝑝,π‘ž,π‘Ÿβˆˆ(0,∞),π‘˜βˆˆ{1,2,3,…} and the initial conditions π‘¦βˆ’π‘˜,…,𝑦0∈(0,∞). We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

1. Introduction and Preliminaries

Our aim in this paper is to study the dynamical behavior of the following rational difference equation

𝑦𝑛+1=π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,(1.1) where 𝑝,π‘ž,π‘Ÿβˆˆ(0,∞), β„•0∈{0,1,…}, π‘˜βˆˆ{1,2,3,…} and the initial conditions π‘¦βˆ’π‘˜,…,𝑦0∈(0,∞).

When π‘˜=1, (1.1) reduces to

𝑦𝑛+1=π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’1π‘žπ‘¦π‘›+π‘¦π‘›βˆ’1,π‘›βˆˆβ„•0.(1.2)

In [1] (see also [2]), the authors investigated the global convergence of solutions to (1.2) and they obtained the following result.

Theorem 1.1. Let 𝑝, π‘ž and π‘Ÿ be positive numbers. Then every solution of (1.2) converges to the unique equilibrium or to a prime-two solution.

The main purpose of this paper is to further consider the global attractivity of all positive solutions of (1.1). That is to say, we will prove that the unique positive equilibrium of (1.1) is a global attractor under certain conditions (see Theorem 4.10).

For the general theory of difference equations, one can refer to the monographes [3] and [2]. For other related results on nonlinear difference equations, see, for example, [1–18].

For the sake of convenience, we firstly present some definitions and known results which will be useful in the sequel.

Let 𝐼 be some interval of real numbers and let π‘“βˆΆπΌΓ—πΌβ†’πΌ be a continuously differentiable function. Then for initial conditions π‘₯βˆ’π‘˜,…,π‘₯0∈𝐼, the difference equation

π‘₯𝑛+1ξ€·π‘₯=𝑓𝑛,π‘₯π‘›βˆ’π‘˜ξ€Έ,π‘›βˆˆβ„•0(1.3) has a unique solution {π‘₯𝑛}βˆžπ‘›=βˆ’π‘˜.

A point π‘₯ is called an equilibrium of (1.3) if

ξ€·π‘₯=𝑓π‘₯,π‘₯ξ€Έ.(1.4)

That is, π‘₯𝑛=π‘₯ for 𝑛β‰₯0 is a solution of (1.3), or equivalently, π‘₯ is a fixed point of 𝑓.

An interval π½βŠ†πΌ is called an invariant interval of (1.3) if

π‘₯βˆ’π‘˜,…,π‘₯0∈𝐽⟹π‘₯π‘›βˆˆπ½βˆ€π‘›βˆˆβ„•0.(1.5)

That is, every solution of (1.3) with initial conditions in J remains in J.

Let

𝑃=πœ•π‘“ξ€·πœ•π‘’π‘₯,π‘₯ξ€Έ,𝑄=πœ•π‘“ξ€·πœ•π‘£π‘₯,π‘₯ξ€Έ(1.6)

denote the partial derivatives of 𝑓(𝑒,𝑣) evaluated at an equilibrium π‘₯ of (1.3). Then the linearized equation associated with (1.3) about the equilibrium π‘₯ is

𝑧𝑛+1=𝑃𝑧𝑛+π‘„π‘§π‘›βˆ’π‘˜,𝑛=0,1,…,(1.7) and its characteristic equation is

πœ†π‘˜+1βˆ’π‘ƒπœ†π‘˜βˆ’π‘„=0.(1.8)

Lemma 1.2 (see [3]). Assume that 𝑃,π‘„βˆˆπ‘… and π‘˜βˆˆ{1,2,…}. Then ||𝑃||+||𝑄||<1(1.9) is a sufficient condition for asymptotic stability of the difference equation (1.7). Suppose in addition that one of the following two cases holds: (i)π‘˜ odd and 𝑄>0,(ii)π‘˜ even and 𝑃𝑄>0.Then (1.9) is also a necessary condition for the asymptotic stability of the difference equation (1.7).

The following result will be useful in establishing the global attractivity character of the equilibrium of (1.1), and it is a reformulation of [2, 7].

Lemma 1.3. Suppose that a continuous function π‘“βˆΆ[π‘Ž,𝑏]Γ—[π‘Ž,𝑏]β†’[π‘Ž,𝑏] satisfies one of (i)–(iii):

(i)𝑓(π‘₯,𝑦) is nonincreasing in π‘₯,𝑦, and []Γ—[]βˆ€(π‘š,𝑀)βˆˆπ‘Ž,π‘π‘Ž,𝑏,(𝑓(π‘š,π‘š)=𝑀,𝑓(𝑀,𝑀)=π‘š)βŸΉπ‘š=𝑀,(1.10)(ii)𝑓(π‘₯,𝑦) is nondecreasing in π‘₯ and nonincreasing in 𝑦, and []Γ—[]βˆ€(π‘š,𝑀)βˆˆπ‘Ž,π‘π‘Ž,𝑏,(𝑓(π‘š,𝑀)=π‘š,𝑓(𝑀,π‘š)=𝑀)βŸΉπ‘š=𝑀,(1.11)(iii)𝑓(π‘₯,𝑦) is nonincreasing in π‘₯ and nondecreasing in 𝑦, and []Γ—[]βˆ€(π‘š,𝑀)βˆˆπ‘Ž,π‘π‘Ž,𝑏,(𝑓(𝑀,π‘š)=π‘š,𝑓(π‘š,𝑀)=𝑀)βŸΉπ‘š=𝑀.(1.12)

(Note that for π‘˜ odd this is equivalent to (1.3) having no prime period-two solution)

Then (1.3) has a unique equilibrium in [π‘Ž,𝑏] and every solution with initial values in [π‘Ž,𝑏] converges to the equilibrium.

This work is organized as follows. In Section 2, the local stability and periodic character are discussed. In Section 3, the boundedness, invariant intervals of (1.1) are presented. Our main results are formulated and proved in Section 4, where the global attractivity of (1.1) is investigated.

2. Local Stability and Period-Two Solutions

The unique positive equilibrium of (1.1) is

βˆšπ‘¦=(1+𝑝)+(1+𝑝)2+4π‘Ÿ(1+π‘ž)2.(1+π‘ž)(2.1)

The linearized equation associated with (1.1) about 𝑦 is

𝑧𝑛+1βˆ’(π‘βˆ’π‘ž)π‘¦βˆ’π‘žπ‘Ÿξ€Ί(π‘ž+1)π‘Ÿ+(𝑝+1)𝑦𝑧𝑛+(π‘βˆ’π‘ž)𝑦+π‘Ÿξ€Ί(π‘ž+1)π‘Ÿ+(𝑝+1)π‘¦ξ€»π‘§π‘›βˆ’π‘˜=0,(2.2)

and its characteristic equation is

πœ†π‘˜+1βˆ’(π‘βˆ’π‘ž)π‘¦βˆ’π‘žπ‘Ÿξ€Ί(π‘ž+1)π‘Ÿ+(𝑝+1)π‘¦ξ€»πœ†π‘˜+(π‘βˆ’π‘ž)𝑦+π‘Ÿξ€Ί(π‘ž+1)π‘Ÿ+(𝑝+1)𝑦=0.(2.3)

From this and Lemma 1.2, we have the following result.

Theorem 2.1. Assume that 𝑝,π‘ž,π‘Ÿβˆˆ(0,∞) and initial conditions π‘¦βˆ’π‘˜,…,𝑦0∈(0,∞). Then the following stataments are true. (i)If (π‘βˆ’π‘ž)π‘¦βˆ’π‘žπ‘Ÿβ‰₯0,(π‘βˆ’3π‘žβˆ’π‘π‘žβˆ’1)𝑦<2π‘žπ‘Ÿ,(2.4) then the unique positive equilibrium 𝑦 of (1.1) is locally asymptotically stable;(ii) If (π‘βˆ’π‘ž)π‘¦βˆ’π‘žπ‘Ÿ<0<(π‘βˆ’π‘ž)𝑦+π‘Ÿ,(2.5) then the unique positive equilibrium 𝑦 of (1.1) is locally asymptotically stable. In particular, if π‘˜ is even, then the equilibrium 𝑦 is locally asymptotically stable if and only if (2.5) holds;(iii)If (π‘βˆ’π‘ž)𝑦+π‘Ÿβ‰€0,(2.6) then the unique positive equilibrium 𝑦 of (1.1) is locally asymptotically stable.

In the following, we will consider the period-two solutions of (1.1).

Let

…,πœ™,πœ“,πœ™,πœ“,…(2.7)

be a period-two solution of (1.1), where πœ™ and πœ“ are two arbitrary positive real numbers.

If π‘˜ is even, then 𝑦𝑛=π‘¦π‘›βˆ’π‘˜, and πœ™ and πœ“ satisfy the following system:

πœ™=π‘Ÿ+π‘πœ“+πœ“π‘žπœ“+πœ“,πœ“=π‘Ÿ+π‘πœ™+πœ™,π‘žπœ™+πœ™(2.8)

then (πœ™βˆ’πœ“)(𝑝+1)=0, we have πœ™=πœ“, which is a contradiction.

If π‘˜ is odd, then 𝑦𝑛+1=π‘¦π‘›βˆ’π‘˜, and πœ™ and πœ“ satisfy the following system:

πœ™=π‘Ÿ+π‘πœ“+πœ™π‘žπœ“+πœ™,πœ“=π‘Ÿ+π‘πœ™+πœ“,π‘žπœ™+πœ“(2.9)

then πœ™+πœ“=1βˆ’π‘, πœ™πœ“=𝑝(1βˆ’π‘)/(π‘žβˆ’1). By calculating, (1.1) has prime period-two solution if and only if

𝑝<1,π‘ž>1,4π‘Ÿ<(1βˆ’π‘)(π‘žβˆ’1βˆ’π‘π‘žβˆ’3𝑝).(2.10)

From the above discussion, we have the following result.

Theorem 2.2. Equation (1.1) has a positive prime period-two solution …,πœ™,πœ“,πœ™,πœ“,…(2.11) if and only if π‘˜isodd,𝑝<1,π‘ž>1,4π‘Ÿ<(1βˆ’π‘)(π‘žβˆ’π‘π‘žβˆ’3π‘βˆ’1).(2.12) Furthermore, if (2.12) holds, then the prime period-two solution of (1.1) is β€œunique” and the values of πœ™ and πœ“ are the positive roots of the quadratic equation 𝑑2βˆ’(1βˆ’π‘)𝑑+π‘Ÿ+𝑝(1βˆ’π‘)π‘žβˆ’1=0.(2.13)

3. Boundedness and Invariant Intervals

In this section, we discuss the boundedness, invariant intervals of (1.1).

3.1. Boundedness

Theorem 3.1. All positive solutions of (1.1) are bounded.

Proof. Equation (1.1) can be written as 𝑦𝑛+1=π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜β‰₯𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜β‰₯ξ€·min{(𝑝/π‘ž),1}π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜ξ€Έπ‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜ξ‚»π‘=minπ‘žξ‚Ό,1(3.1) for all 𝑛β‰₯0. We denote 𝑝𝐾=minπ‘žξ‚Ό,1.(3.2) Then 𝑦𝑛+1=π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜β‰€π‘Ÿ+𝑝𝑦𝑛+π‘¦π‘›βˆ’π‘˜(π‘ž/2)𝐾+(𝐾/2)+(π‘ž/2)𝑦𝑛+(1/2)π‘¦π‘›βˆ’π‘˜β‰€ξ€·max{π‘Ÿ,𝑝,1}1+𝑦𝑛+π‘¦π‘›βˆ’π‘˜ξ€Έξ€·min{(π‘ž/2)𝐾+(𝐾/2),(π‘ž/2),(1/2)}1+𝑦𝑛+π‘¦π‘›βˆ’π‘˜ξ€Έ=max{π‘Ÿ,𝑝,1}min{(π‘ž/2)𝐾+(𝐾/2),(π‘ž/2),(1/2)}(3.3) for all 𝑛>π‘˜. The proof is complete.

Let {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ be a positive solution of (1.1). Then the following identities are easily established:

𝑦𝑛+1ξ€·βˆ’1=(π‘žβˆ’π‘)(π‘Ÿ/(π‘žβˆ’π‘))βˆ’π‘¦π‘›ξ€Έπ‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,𝑦(3.4)𝑛+1βˆ’π‘π‘ž=ξ€·((π‘βˆ’π‘ž)/π‘ž)(π‘žπ‘Ÿ/(π‘βˆ’π‘ž))βˆ’π‘¦π‘›βˆ’π‘˜ξ€Έπ‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,𝑦(3.5)𝑛+1βˆ’π‘žπ‘Ÿ=π‘π‘βˆ’π‘žξ€·ξ€·2βˆ’π‘π‘žβˆ’π‘ž2π‘Ÿξ€Έπ‘¦/(π‘βˆ’π‘ž)𝑛+(1/π‘ž)π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜+𝑦((π‘βˆ’π‘žβˆ’π‘žπ‘Ÿ)/(π‘βˆ’π‘ž))π‘›βˆ’π‘˜ξ€Έβˆ’(𝑝/π‘ž)π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,𝑦(3.6)𝑛+1βˆ’π‘+π‘Ÿπ‘ž=π‘Ÿξ€·1βˆ’π‘¦π‘›ξ€Έ+((π‘žβˆ’π‘βˆ’π‘Ÿ)/π‘ž)π‘¦π‘›βˆ’π‘˜π‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,𝑦(3.7)π‘›βˆ’π‘¦π‘›+2(π‘˜+1)=ξ€·π‘¦π‘›π‘žβˆ’(𝑝/π‘ž)ξ€Έξ€·2𝑦𝑛+π‘˜π‘¦π‘›+2π‘˜+1+π‘žπ‘¦π‘›π‘¦π‘›+2π‘˜+1ξ€Έπ‘žπ‘¦π‘›+2π‘˜+1ξ€·π‘žπ‘¦π‘›+π‘˜+𝑦𝑛+ξ€·π‘Ÿ+𝑝𝑦𝑛+π‘˜+𝑦𝑛+𝑝𝑦𝑛+π‘˜ξ€·π‘¦π‘›ξ€Έ+ξ€·π‘¦βˆ’((𝑝+π‘žπ‘Ÿ)/𝑝)2π‘›βˆ’π‘¦π‘›ξ€Έβˆ’π‘Ÿπ‘žπ‘¦π‘›+2π‘˜+1ξ€·π‘žπ‘¦π‘›+π‘˜+𝑦𝑛+ξ€·π‘Ÿ+𝑝𝑦𝑛+π‘˜+𝑦𝑛,π‘›βˆˆβ„•0.(3.8)

When π‘ž=𝑝+π‘Ÿ, the unique positive equilibrium of (1.1) is 𝑦=1, (3.4) becomes

𝑦𝑛+1π‘Ÿξ€·βˆ’1=1βˆ’π‘¦π‘›ξ€Έπ‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0.(3.9)

When βˆšπ‘=π‘ž(1+1+4π‘Ÿ)/2, the unique positive equilibrium is 𝑦=𝑝/π‘ž, (3.5) becomes

𝑦𝑛+1βˆ’π‘π‘ž=ξ€·(π‘βˆ’π‘ž)/π‘ž(𝑝/π‘ž)βˆ’π‘¦π‘›βˆ’π‘˜ξ€Έπ‘žπ‘¦π‘›+π‘¦π‘›βˆ’π‘˜,π‘›βˆˆβ„•0,(3.10) and (3.8) becomes

π‘¦π‘›βˆ’π‘¦π‘›+2(π‘˜+1)=ξ€·π‘¦π‘›π‘žβˆ’(𝑝/π‘ž)ξ€Έξ€·2𝑦𝑛+π‘˜π‘¦π‘›+2π‘˜+1+π‘žπ‘¦π‘›π‘¦π‘›+2π‘˜+1+𝑝𝑦𝑛+π‘˜+𝑦𝑛+(π‘βˆ’π‘ž)/π‘žπ‘žπ‘¦π‘›+2π‘˜+1ξ€·π‘žπ‘¦π‘›+π‘˜+𝑦𝑛+ξ€·π‘Ÿ+𝑝𝑦𝑛+π‘˜+𝑦𝑛,π‘›βˆˆβ„•0.(3.11)

Set

𝑓(π‘₯,𝑦)=π‘Ÿ+𝑝π‘₯+𝑦.π‘žπ‘₯+𝑦(3.12)

Lemma 3.2. Assume that 𝑓(π‘₯,𝑦) is defined in (3.12). Then the following statements are true: (i)assume 𝑝<π‘ž. Then 𝑓(π‘₯,𝑦) is strictly decreasing in π‘₯ and increasing in 𝑦 for π‘₯β‰₯π‘Ÿ/(π‘žβˆ’π‘); and it is strictly decreasing in each of its arguments for π‘₯<π‘Ÿ/(π‘žβˆ’π‘);(ii)assume 𝑝>π‘ž. Then 𝑓(π‘₯,𝑦) is increasing in π‘₯ and strictly decreasing in 𝑦 for 𝑦β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž); and it is strictly decreasing in each of its arguments for 𝑦<π‘žπ‘Ÿ/(π‘βˆ’π‘ž).

Proof. By calculating the partial derivatives of the function 𝑓(π‘₯,𝑦), we have π‘“ξ…žπ‘₯(π‘₯,𝑦)=(π‘βˆ’π‘ž)π‘¦βˆ’π‘žπ‘Ÿ(π‘žπ‘₯+𝑦)2,π‘“ξ…žπ‘¦(π‘₯,𝑦)=(π‘žβˆ’π‘)π‘₯βˆ’π‘Ÿ(π‘žπ‘₯+𝑦)2,(3.13) from which these statements easily follow.

3.2. Invariant Interval

In this subsection, we discuss the invariant interval of (1.1).

3.2.1. The Case 𝑝<π‘ž

Lemma 3.3. Assume that 𝑝<π‘ž, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)𝑦𝑛>𝑝/π‘ž for all 𝑛β‰₯1;(ii)If for some 𝑁β‰₯0, 𝑦𝑁>π‘Ÿ/(π‘žβˆ’π‘), then 𝑦𝑁+1<1;(iii)If for some 𝑁β‰₯0, 𝑦𝑁=π‘Ÿ/(π‘žβˆ’π‘), then 𝑦𝑁+1=1;(iv)If for some 𝑁β‰₯0, 𝑦𝑁<π‘Ÿ/(π‘žβˆ’π‘), then 𝑦𝑁+1>1;(v)If 𝑝<π‘ž<𝑝+π‘Ÿ, then (1.1) possesses an invariant interval [1,π‘Ÿ/(π‘žβˆ’π‘)] and π‘¦βˆˆ(1,π‘Ÿ/(π‘žβˆ’π‘));(vi)If 𝑝+π‘Ÿ<π‘ž<𝑝+π‘žπ‘Ÿ/𝑝, then (1.1) possesses an invariant interval [π‘Ÿ/(π‘žβˆ’π‘),1] and π‘¦βˆˆ(π‘Ÿ/(π‘žβˆ’π‘),1);(vii)If π‘žβ‰₯𝑝+π‘žπ‘Ÿ/𝑝, then (1.1) possesses an invariant interval [𝑝/π‘ž,1] and π‘¦βˆˆ(𝑝/π‘ž,1).

Proof. The proofs of (i)–(iv) are straightforward consequences of the identities (3.5) and (3.4). So we only prove (v)–(vii). By the condition (i) of Lemma 3.2, the function 𝑓(π‘₯,𝑦) is strictly decreasing in π‘₯ and increasing in 𝑦 for π‘₯β‰₯π‘Ÿ/(π‘žβˆ’π‘); and it is strictly decreasing in both arguments for π‘₯<π‘Ÿ/(π‘žβˆ’π‘).
(v) Using the decreasing character of 𝑓, we obtain
ξ‚΅π‘Ÿ1=𝑓,π‘Ÿπ‘žβˆ’π‘ξ‚Άπ‘žβˆ’π‘<𝑓(π‘₯,𝑦)<𝑓(1,1)=π‘Ÿ+𝑝+1<π‘Ÿπ‘ž+1π‘žβˆ’π‘.(3.14) The inequalities π‘Ÿ1<,π‘žβˆ’π‘π‘Ÿ+𝑝+1<π‘Ÿπ‘ž+1π‘žβˆ’π‘(3.15) are equivalent to the inequality π‘ž<𝑝+π‘Ÿ.
On the other hand, 𝑦 is the unique positive root of quadratic equation
(π‘ž+1)𝑦2βˆ’(𝑝+1)π‘¦βˆ’π‘Ÿ=0.(3.16) Since ξ‚΅π‘Ÿ(π‘ž+1)ξ‚Άπ‘žβˆ’π‘2π‘Ÿβˆ’(𝑝+1)π‘žβˆ’π‘βˆ’π‘Ÿ=π‘Ÿ(π‘ž+1)(𝑝+π‘Ÿβˆ’π‘ž)(π‘žβˆ’π‘)2(>0,π‘ž+1)βˆ’(𝑝+1)βˆ’π‘Ÿ=π‘žβˆ’π‘βˆ’π‘Ÿ<0,(3.17) then we have that π‘¦βˆˆ(1,π‘Ÿ/(π‘žβˆ’π‘)).
(vi) By using the monotonic character of 𝑓, we obtain
(π‘žβˆ’π‘)(𝑝+π‘Ÿ)+π‘Ÿπ‘ž2ξ‚΅π‘Ÿβˆ’π‘π‘ž+π‘Ÿ=𝑓1,ξ‚Άξ‚΅π‘Ÿπ‘žβˆ’π‘β‰€π‘“(π‘₯,𝑦)β‰€π‘“ξ‚Άπ‘žβˆ’π‘,1=1.(3.18) The inequalities (π‘žβˆ’π‘)(𝑝+π‘Ÿ)+π‘Ÿπ‘ž2>π‘Ÿβˆ’π‘π‘ž+π‘Ÿ,π‘Ÿπ‘žβˆ’π‘π‘žβˆ’π‘<1(3.19) follow from the inequality π‘ž>𝑝+π‘Ÿ.
On the other hand, similar to (v) it can be proved that π‘¦βˆˆ(π‘Ÿ/(π‘žβˆ’π‘),1).
(vii) In this case note that π‘Ÿ/(π‘žβˆ’π‘)≀𝑝/π‘ž<1 holds, and using the monotonic character of 𝑓, we obtain
π‘π‘ž<π‘žπ‘Ÿ+π‘π‘ž+π‘π‘ž2𝑝+𝑝=𝑓1,π‘žξ‚Άξ‚΅π‘β‰€π‘“(π‘₯,𝑦)β‰€π‘“π‘žξ‚Ά=,1π‘žπ‘Ÿ+𝑝2+π‘žπ‘ž(𝑝+1)≀1.(3.20) Furthermore, similar to (v) it follows π‘¦βˆˆ(𝑝/π‘ž,1). The proof is complete.

When π‘ž=𝑝+π‘Ÿ, (3.9) implies that the following result holds.

Lemma 3.4. Assume that π‘ž=𝑝+π‘Ÿ, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)If for some 𝑁β‰₯0, 𝑦𝑁>1, then 𝑦𝑁+1<1;(ii)If for some 𝑁β‰₯0, 𝑦𝑁=1, then 𝑦𝑁+1=1;(iii)If for some 𝑁β‰₯0, 𝑦𝑁<1, then 𝑦𝑁+1>1.

3.2.2. The Case 𝑝>π‘ž

Lemma 3.5. Assume that 𝑝>π‘ž, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)𝑦𝑛>1 for all 𝑛β‰₯1;(ii)If for some 𝑁β‰₯0, 𝑦𝑁<π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑦𝑁+π‘˜+1>𝑝/π‘ž;(iii)If for some 𝑁β‰₯0, 𝑦𝑁=π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑦𝑁+π‘˜+1=𝑝/π‘ž;(iv)If for some 𝑁β‰₯0, 𝑦𝑁>π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑦𝑁+π‘˜+1<𝑝/π‘ž;(v)If βˆšπ‘ž<𝑝<π‘ž(1+1+4π‘Ÿ)/2, then (1.1) possesses an invariant interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)] and π‘¦βˆˆ(𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž));(vi)If βˆšπ‘ž(1+1+4π‘Ÿ)/2<𝑝<π‘ž+π‘žπ‘Ÿ, then (1.1) possesses an invariant interval [π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž] and π‘¦βˆˆ(π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž);(vii)If 𝑝β‰₯π‘ž+π‘žπ‘Ÿ, then (1.1) possesses an invariant interval [1,𝑝/π‘ž] and π‘¦βˆˆ(1,𝑝/π‘ž).

Proof. The proofs of (i)–(iv) are direct consequences of the identities (3.4) and (3.5). So we only give the proofs (v)–(vii). By Lemma 3.2 (ii), the function 𝑓(π‘₯,𝑦) is increasing in π‘₯ and strictly decreasing in 𝑦 for 𝑦β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž); and it is strictly decreasing in each of its arguments for 𝑦<π‘žπ‘Ÿ/(π‘βˆ’π‘ž).
(v) Using the decreasing character of 𝑓, we obtain
π‘π‘žξ‚΅=π‘“π‘žπ‘Ÿ,π‘βˆ’π‘žπ‘žπ‘Ÿξ‚Άξ‚΅π‘π‘βˆ’π‘žβ‰€π‘“(π‘₯,𝑦)β‰€π‘“π‘ž,π‘π‘žξ‚Ά=π‘žπ‘Ÿ+𝑝(𝑝+1)≀𝑝(π‘ž+1)π‘žπ‘Ÿπ‘βˆ’π‘ž.(3.21) The inequalities π‘žπ‘Ÿ+𝑝(𝑝+1)≀𝑝(π‘ž+1)π‘žπ‘Ÿ,π‘π‘βˆ’π‘žπ‘ž<π‘žπ‘Ÿπ‘βˆ’π‘ž(3.22) are equivalent to the inequality βˆšπ‘<π‘ž(1+1+4π‘Ÿ)/2. That is, [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)] is an invariant interval of (1.1).
On the other hand, similar to Lemma 3.3 (v), it can be proved that π‘¦βˆˆ(𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)).
(vi) By using the monotonic character of 𝑓, we obtain
π‘žπ‘Ÿβ‰€π‘βˆ’π‘ž(π‘žπ‘Ÿ+𝑝)(π‘βˆ’π‘ž)+π‘π‘ž2π‘Ÿπ‘ž3ξ‚΅π‘Ÿ+𝑝(π‘βˆ’π‘ž)=π‘“π‘žπ‘Ÿ,π‘π‘βˆ’π‘žπ‘žξ‚Άξ‚΅π‘β‰€π‘“(π‘₯,𝑦)β‰€π‘“π‘ž,π‘žπ‘Ÿξ‚Ά=π‘π‘βˆ’π‘žπ‘ž.(3.23) The inequalities (π‘žπ‘Ÿ+𝑝)(π‘βˆ’π‘ž)+π‘π‘ž2π‘Ÿπ‘ž3β‰₯π‘Ÿ+𝑝(π‘βˆ’π‘ž)π‘žπ‘Ÿ,π‘βˆ’π‘žπ‘žπ‘Ÿβ‰€π‘π‘βˆ’π‘žπ‘ž(3.24) are equivalent to the inequality βˆšπ‘>π‘ž(1+1+4π‘Ÿ)/2.
On the other hand, similar to Lemma 3.3 (v) it can be proved that π‘¦βˆˆ(π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž).
(vii) In this case note that π‘žπ‘Ÿ/(π‘βˆ’π‘ž)≀1<𝑝/π‘ž holds. By the monotonic character of 𝑓, we have
1<π‘žπ‘Ÿ+π‘π‘ž+π‘π‘ž2𝑝+𝑝=𝑓1,π‘žξ‚Άξ‚΅π‘β‰€π‘“(π‘₯,𝑦)β‰€π‘“π‘žξ‚Ά=,1π‘žπ‘Ÿ+𝑝2+π‘žβ‰€π‘π‘ž(𝑝+1)π‘ž.(3.25) The inequalities π‘žπ‘Ÿ+π‘π‘ž+π‘π‘ž2+𝑝>1,π‘žπ‘Ÿ+𝑝2+π‘žβ‰€π‘π‘ž(𝑝+1)π‘ž(3.26) are equivalent to the inequality 𝑝β‰₯π‘ž+π‘žπ‘Ÿ.
Furthermore, similar to Lemma 3.3 (v), it follows π‘¦βˆˆ(1,𝑝/π‘ž). The proof is complete.

Lemma 3.6. Assume that βˆšπ‘=π‘ž(1+1+4π‘Ÿ)/2, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)If for some 𝑁β‰₯0, 𝑦𝑁<𝑝/π‘ž, then 𝑦𝑁+π‘˜+1>𝑝/π‘ž;(ii)If for some 𝑁β‰₯0, 𝑦𝑁=𝑝/π‘ž, then 𝑦𝑁+π‘˜+1=𝑝/π‘ž;(iii)If for some 𝑁β‰₯0, 𝑦𝑁>𝑝/π‘ž, then 𝑦𝑁+π‘˜+1<𝑝/π‘ž;(iv)If for some 𝑁β‰₯0, 𝑦𝑁>𝑝/π‘ž, then 𝑦𝑁>𝑦𝑁+2(π‘˜+1)>𝑝/π‘ž;(v)If for some 𝑁β‰₯0, 𝑦𝑁<𝑝/π‘ž, then 𝑦𝑁<𝑦𝑁+2(π‘˜+1)<𝑝/π‘ž.

Proof. In this case, we have that π‘žπ‘Ÿ/(π‘βˆ’π‘ž)=𝑝/π‘ž. These results follow from the identities (3.10) and (3.11) and the details are omitted.

4. Global Attractivity

In this section, we discuss the global attractivity of the positive equilibrium of (1.1). We show that the unique positive equilibrium 𝑦 of (1.1) is a global attractor when 𝑝=π‘ž or 𝑝<1 and 𝑝<π‘žβ‰€π‘π‘ž+1+3𝑝 or π‘ž<𝑝≀1.

4.1. The Case 𝑝=π‘ž

In this subsection, we discuss the behavior of positive solutions of (1.1) when 𝑝=π‘ž.

Theorem 4.1. Assume that 𝑝=π‘ž holds, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the unique positive equilibrium 𝑦 of (1.1) is a global attractor.

Proof. By the change of variables π‘¦π‘›π‘Ÿ=1+𝑒𝑝+1𝑛,(4.1) Equation (1.1) reduces to the difference equation 𝑒𝑛+1=1ξ€·1+π‘π‘Ÿ/(𝑝+1)2𝑒𝑛+ξ€·π‘Ÿ/(𝑝+1)2ξ€Έπ‘’π‘›βˆ’π‘˜,π‘›βˆˆβ„•0.(4.2) The unique positive equilibrium 𝑒 of (4.2) is βˆšπ‘’=βˆ’(𝑝+1)+(𝑝+1)2+4π‘Ÿ(𝑝+1).2π‘Ÿ(4.3) Applying Lemma 1.3 in interval [0,1], then every positive solution of (1.1) converges to 𝑒. That is, 𝑒 is a global attractor. So, 𝑦 is a global attractor.

4.2. The Case 𝑝<π‘ž

In this subsection, we present global attractivity of (1.1) when 𝑝<π‘ž.

The following result is straightforward consequence of the identity (3.7).

Lemma 4.2. Assume that 𝑝<π‘ž holds, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)Suppose that π‘ž<𝑝+π‘Ÿ. If for some 𝑁β‰₯0, 𝑦𝑁>1, then 𝑦𝑁+1<(𝑝+π‘Ÿ)/π‘ž;(ii)Suppose that π‘ž>𝑝+π‘Ÿ. If for some 𝑁β‰₯0, 𝑦𝑁<1, then 𝑦𝑁+1>(𝑝+π‘Ÿ)/π‘ž.

Theorem 4.3. Assume that 𝑝<π‘ž, 𝑝<1 and π‘žβ‰€π‘π‘ž+1+3𝑝 hold. Let {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ be a positive solution of (1.1). Then the following statements hold true: (i)Suppose π‘ž<𝑝+π‘Ÿ. If 𝑦0∈[1,π‘Ÿ/(π‘žβˆ’π‘)], then π‘¦π‘›βˆˆ[1,π‘Ÿ/(π‘žβˆ’π‘)] for 𝑛β‰₯1. Furthermore, every positive solution of (1.1) lies eventually in the interval [1,π‘Ÿ/(π‘žβˆ’π‘)].(ii)Suppose π‘ž>𝑝+π‘Ÿ. If 𝑦0∈[π‘Ÿ/(π‘žβˆ’π‘),1], then π‘¦π‘›βˆˆ[π‘Ÿ/(π‘žβˆ’π‘),1] for 𝑛β‰₯1. Furthermore, every positive solution of (1.1) lies eventually in the interval [π‘Ÿ/(π‘žβˆ’π‘),1].

Proof. We only give the proof of (i), the proof of (ii) is similar and will be omitted. First, note that in this case 𝑝/π‘ž<1<(𝑝+π‘Ÿ)/π‘ž<π‘Ÿ/(π‘žβˆ’π‘) holds.
If 𝑦0∈[1,π‘Ÿ/(π‘žβˆ’π‘)], then by Lemma 3.3 (iv), we have that 𝑦1>1, and by Lemma 4.2 (i), we obtain that 𝑦1<(𝑝+π‘Ÿ)/π‘ž<π‘Ÿ/(π‘žβˆ’π‘), which implies that 𝑦1∈[1,π‘Ÿ/(π‘žβˆ’π‘)], by induction, we have π‘¦π‘›βˆˆ[1,π‘Ÿ/(π‘žβˆ’π‘)], for 𝑛β‰₯1.
Now, to complete the proof it remains to show that when 𝑦0βˆ‰[1,π‘Ÿ/(π‘žβˆ’π‘)], there exists 𝑁>0 such that π‘¦π‘βˆˆ[1,π‘Ÿ/(π‘žβˆ’π‘)].
If 𝑦0βˆ‰[1,π‘Ÿ/(π‘žβˆ’π‘)], then we have the following two cases to be considered:
(a)𝑦0>π‘Ÿ/(π‘žβˆ’π‘);(b)𝑦0<1.
Case (a). From Lemma 3.3 (ii), we see that 𝑦1<1. Thus, in the sequel, we only consider case (b).
If 𝑦0<1, then by Lemma 3.3 (iv), we have 𝑦1>1, and from Lemma 4.2 (i), we have 𝑦2<(𝑝+π‘Ÿ)/π‘ž<π‘Ÿ/(π‘žβˆ’π‘). So 𝑦3>1 and 𝑦4<π‘Ÿ/(π‘žβˆ’π‘). By induction, there exists exactly one term greater than 1, which is followed by exactly one term less than π‘Ÿ/(π‘žβˆ’π‘), which is followed by exactly one term greater than 1, and so on. If for some 𝑁>0, 1β‰€π‘¦π‘β‰€π‘Ÿ/(π‘žβˆ’π‘), then the former assertion implies that the result is true.
So assume for the sake of contradiction, that for all 𝑛β‰₯1, 𝑦𝑛 never enter the interval [1,π‘Ÿ/(π‘žβˆ’π‘)], then the sequence {𝑦𝑛}βˆžπ‘›=1 will oscillate relative to the interval [1,π‘Ÿ/(π‘žβˆ’π‘)] with semicycles of length one. Consider the subsequence {𝑦2𝑛}βˆžπ‘›=1 and {𝑦2𝑛+1}βˆžπ‘›=1 of solution {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜, we have
𝑦2𝑛<1,𝑦2𝑛+1>π‘Ÿπ‘žβˆ’π‘for𝑛β‰₯1.(4.4) Let 𝐿=limπ‘›β†’βˆžsup𝑦2𝑛,𝑙=limπ‘›β†’βˆžinf𝑦2𝑛,𝑀=limπ‘›β†’βˆžsup𝑦2𝑛+1,π‘š=limπ‘›β†’βˆžinf𝑦2𝑛+1,(4.5) which in view of Theorem 3.1 exist as finite numbers, such that πΏβ‰€π‘Ÿ+π‘π‘š+π‘™π‘žπ‘š+𝑙,𝑙β‰₯π‘Ÿ+𝑝𝑀+𝐿,π‘žπ‘€+𝐿(4.6)π‘€β‰€π‘Ÿ+𝑝𝑙+π‘šπ‘žπ‘™+π‘š,π‘šβ‰₯π‘Ÿ+𝑝𝐿+π‘€π‘žπΏ+𝑀.(4.7) From (4.6), we have π‘ž(πΏπ‘šβˆ’π‘™π‘€)≀𝑝(π‘šβˆ’π‘€)+(π‘™βˆ’πΏ), which implies that πΏπ‘šβˆ’π‘™π‘€β‰€0. Also, from (4.7), we have π‘ž(π‘™π‘€βˆ’πΏπ‘š)≀𝑝(π‘™βˆ’πΏ)+(π‘šβˆ’π‘€), which implies that π‘™π‘€βˆ’πΏπ‘šβ‰€0. Thus π‘™π‘€βˆ’πΏπ‘š=0 and 𝐿=𝑙, 𝑀=π‘š hold, from which it follows that limπ‘›β†’βˆžπ‘¦2𝑛 and limπ‘›β†’βˆžπ‘¦2𝑛+1 exist.
Set
limπ‘›β†’βˆžπ‘¦2𝑛=𝐿,limπ‘›β†’βˆžπ‘¦2𝑛+1=𝑀,(4.8) then 𝐿≀1, 𝑀β‰₯π‘Ÿ/(π‘žβˆ’π‘) and 𝐿, 𝑀 satisfies the system 𝐿=π‘Ÿ+𝑝𝑀+πΏπ‘žπ‘€+𝐿,𝑀=π‘Ÿ+𝑝𝐿+𝑀,π‘žπΏ+𝑀(4.9) which implies that 𝐿, 𝑀 is a period-two solution of (1.1). Furthermore, in view of Theorem 2.2, (1.1) has no period-two solution when 𝑝<1 and π‘žβ‰€π‘π‘ž+1+3𝑝 hold. This is a contradiction, as desired. The proof is complete.

Theorem 4.4. Assume that 𝑝<π‘ž, 𝑝<1 and π‘žβ‰€π‘π‘ž+1+3𝑝 hold. Then the unique positive equilibrium 𝑦 of (1.1) is a global attractor.

Proof. To complete the proof, there are four cases to be considered.
Case (i). π‘ž<𝑝+π‘Ÿ.
By Theorem 4.3 (i), we know that all solutions of (1.1) lies eventually in the invariant interval [1,π‘Ÿ/(π‘žβˆ’π‘)]. Furthermore, the function 𝑓(π‘₯,𝑦) is non-increasing in each of its arguments in the interval [1,π‘Ÿ/(π‘žβˆ’π‘)]. Thus, applying Lemma 1.3, every solution of (1.1) converges to 𝑦, that is, 𝑦 is a global attractor.
Case (ii). π‘ž=𝑝+π‘Ÿ.
In this case, the only positive equilibrium is 𝑦=1. In view of Lemma 3.4, we see that, after the first semicycle, the nontrivial solution oscillates about 𝑦 with semicycles of length one. Consider the subsequences {𝑦2𝑛}βˆžπ‘›=1 and {𝑦2𝑛+1}βˆžπ‘›=1 of any nontrivial solution {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ of (1.1). We have
𝑦2𝑛<1,𝑦2𝑛+1>1,for𝑛β‰₯1,(4.10) or vice versa. Here, we may assume, without loss of generality, that 𝑦2𝑛<1 and 𝑦2𝑛+1>1, for 𝑛β‰₯1.
Let
𝐿=limπ‘›β†’βˆžsup𝑦2𝑛,𝑙=limπ‘›β†’βˆžinf𝑦2𝑛,𝑀=limπ‘›β†’βˆžsup𝑦2𝑛+1,π‘š=limπ‘›β†’βˆžinf𝑦2𝑛+1,(4.11) which, in view of Theorem 3.1, exist. Then as the same argument in Theorem 4.3, we can see that limπ‘›β†’βˆžπ‘¦2𝑛 and limπ‘›β†’βˆžπ‘¦2𝑛+1 exist.
Set
limπ‘›β†’βˆžπ‘¦2𝑛=𝐿,limπ‘›β†’βˆžπ‘¦2𝑛+1=𝑀,(4.12) then 𝐿≀1, 𝑀β‰₯1. If 𝐿≠𝑀, then, also as the same argument in Theorem 4.3, we can see that 𝐿, 𝑀 is a period-two solution of (1.1), which contradicts Theorem 2.2. Thus 𝐿=𝑀, from which it follows that limπ‘›β†’βˆžπ‘¦π‘›=1, which implies that 𝑦=1 is a global attractor.
Case (iii). 𝑝+π‘Ÿ<π‘ž<𝑝+(π‘ž/𝑝)π‘Ÿ.
By Theorem 4.3 (ii), we know that all solutions of (1.1) lies eventually in the invariant interval [π‘Ÿ/(π‘žβˆ’π‘),1]. Furthermore, the function 𝑓(π‘₯,𝑦) decreases in π‘₯ and increases in 𝑦 in the interval [π‘Ÿ/(π‘žβˆ’π‘),1]. Thus, applying Lemma 1.3, every solution of converges to 𝑦, that is, 𝑦 is a global attractor.
Case (iv). π‘žβ‰₯𝑝+(π‘ž/𝑝)π‘Ÿ.
In this case, we note that π‘Ÿ/(π‘žβˆ’π‘)≀𝑝/π‘ž<1 holds. From Theorem 4.3 (ii) and Lemma 3.3 (i), we know that all solutions of (1.1) eventually enter the invariant interval [𝑝/π‘ž,1]. Hence, by using the same argument in (iii), 𝑦 is a global attractor. The proof is complete.

4.3. The Case 𝑝>π‘ž

In this subsection, we discuss the global behavior of (1.1) when 𝑝>π‘ž.

The following three results are the direct consequences of equations (3.4), (3.5), (3.6), and (3.8).

Lemma 4.5. Assume that βˆšπ‘ž<𝑝<π‘ž(1+1+4π‘Ÿ)/2, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)If for some 𝑁β‰₯0, 𝑦𝑁<𝑝/π‘ž, then 𝑦𝑁<𝑦𝑁+2(π‘˜+1)<π‘žπ‘Ÿ/(π‘βˆ’π‘ž);(ii)If for some 𝑁β‰₯0, 𝑦𝑁>π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑝/π‘ž<𝑦𝑁+2(π‘˜+1)<𝑦𝑁;(iii)If for some 𝑁β‰₯0, 𝑝/π‘žβ‰€π‘¦π‘β‰€π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑝/π‘žβ‰€π‘¦π‘+2(π‘˜+1)β‰€π‘žπ‘Ÿ/(π‘βˆ’π‘ž).

Lemma 4.6. Assume that βˆšπ‘ž(1+1+4π‘Ÿ)/2<𝑝<π‘ž+π‘žπ‘Ÿ, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)If for some 𝑁β‰₯0, 𝑦𝑁<π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then 𝑦𝑁<𝑦𝑁+2(π‘˜+1)<𝑝/π‘ž;(ii)If for some 𝑁β‰₯0, 𝑦𝑁>𝑝/π‘ž, then π‘žπ‘Ÿ/(π‘βˆ’π‘ž)<𝑦𝑁+2(π‘˜+1)<𝑦𝑁;(iii)If for some 𝑁β‰₯0, π‘žπ‘Ÿ/(π‘βˆ’π‘ž)≀𝑦𝑁≀𝑝/π‘ž, then π‘žπ‘Ÿ/(π‘βˆ’π‘ž)≀𝑦𝑁+2(π‘˜+1)≀𝑝/π‘ž.

Lemma 4.7. Assume that 𝑝β‰₯π‘ž+π‘žπ‘Ÿ, and {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ is a positive solution of (1.1). Then the following statements are true: (i)𝑦𝑛>1 for 𝑛β‰₯1;(ii)If for some 𝑁β‰₯0, 𝑦𝑁>𝑝/π‘ž, then 1<𝑦𝑁+2(π‘˜+1)<𝑝/π‘ž and 𝑦𝑁+2(π‘˜+1)<𝑦𝑁;(iii)If for some 𝑁β‰₯0, 1<𝑦𝑁≀𝑝/π‘ž, then 1<𝑦𝑁+2(π‘˜+1)≀𝑝/π‘ž.

Theorem 4.8. Assume that 𝑝>π‘ž holds, and let {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ be a positive solution of (1.1). Then the following statements hold true: (i)If βˆšπ‘<π‘ž(1+1+4π‘Ÿ)/2, then every positive solution of (1.1) lies eventually in the interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)].(ii)If βˆšπ‘ž(1+1+4π‘Ÿ)/2<𝑝<π‘ž+π‘žπ‘Ÿ, then every positive solution of (1.1) lies eventually in the interval [π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž].(iii)If 𝑝β‰₯π‘ž+π‘žπ‘Ÿ, then every positive solution of (1.1) lies eventually in the interval [1,𝑝/π‘ž].

Proof. We only give the proof of (i), the proofs of (ii) and (iii) are similar and will be omitted.
When βˆšπ‘ž<𝑝<π‘ž(1+1+4π‘Ÿ)/2, recall that from Lemma 3.5, [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)] is an invariant interval and so it follows that every solution of (1.1) with π‘˜+1 consecutive values in [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)], lies eventually in this interval. If the solution is not eventually in [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)], there are three cases to be considered.
Case (i). If for some 𝑁β‰₯0, 𝑦𝑁>π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then there are two cases to be considered. If 𝑦𝑁+2(π‘˜+1)𝑛β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž) for every π‘›βˆˆπ‘, then by Lemma 4.5, we have
𝑦𝑁+2(π‘˜+1)(π‘›βˆ’1)>𝑦𝑁+2(π‘˜+1)𝑛>π‘π‘ž,(4.13) hence, the subsequence {𝑦𝑁+2(π‘˜+1)𝑛} is strictly monotonically decreasing convergent and its limit 𝑆 satisfies 𝑆β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž). Taking limit on both sides of (3.8), we obtain a contradiction. If for some 𝑛0, 𝑦𝑁+2(π‘˜+1)𝑛0<π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then by Lemma 4.5 we obtain that {𝑦𝑁+2(π‘˜+1)𝑛} is eventually in the interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)].
Case (ii). If for some 𝑁β‰₯0, 𝑦𝑁<𝑝/π‘ž, then there are two cases to be considered. If 𝑦𝑁+2(π‘˜+1)𝑛<𝑝/π‘ž for every π‘›βˆˆπ‘, then by Lemma 4.5 we obtain
𝑦𝑁+2(π‘˜+1)(π‘›βˆ’1)<𝑦𝑁+2(π‘˜+1)𝑛<π‘žπ‘Ÿ,π‘βˆ’π‘ž(4.14) which implies that the subsequence {𝑦𝑁+2(π‘˜+1)𝑛} is convergent. Then as the same argument in case (i), obtain a contradiction. If for some 𝑛0, 𝑦𝑁+2(π‘˜+1)𝑛0>𝑝/π‘ž, then by Lemma 4.5 we have that {𝑦𝑁+2(π‘˜+1)𝑛} is eventually in the interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)].
Case (iii). If for some 𝑁β‰₯0,𝑝/π‘žβ‰€π‘¦π‘β‰€π‘žπ‘Ÿ/(π‘βˆ’π‘ž), then by Lemma 4.5 it follows that 𝑝/π‘žβ‰€π‘¦π‘+2(π‘˜+1)π‘›β‰€π‘žπ‘Ÿ/(π‘βˆ’π‘ž). Assume that there is a subsequence {𝑦𝑁0+2(π‘˜+1)𝑛} such that 𝑦𝑁0+2(π‘˜+1)𝑛β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž), or 𝑦𝑁0+2(π‘˜+1)𝑛≀𝑝/π‘ž, for every π‘›βˆˆπ‘. Then its limit 𝑆 satisfies 𝑆β‰₯π‘žπ‘Ÿ/(π‘βˆ’π‘ž), or 𝑆≀𝑝/π‘ž. Taking limit on both sides of (3.8), obtain a contradiction. Hence, for all π‘βˆˆ{1,2,…,2(π‘˜+1)} the subsequences {𝑦𝑁+2(π‘˜+1)𝑛} are eventually in the interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)].

Theorem 4.9. Assume that 𝑝>π‘ž and 𝑝≀1 hold. Then the unique positive equilibrium 𝑦 of (1.1) is a global attractor.

Proof. The proof will be accomplished by considering the following four cases.
Case (i). βˆšπ‘<π‘ž(1+1+4π‘Ÿ)/2.
By part (i) of Theorem 4.8, we know that all positive solutions of (1.1) lie eventually in the invariant interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)]. Furthermore, the function 𝑓(π‘₯,𝑦) is nonincreasing in each of its arguments in the interval [𝑝/π‘ž,π‘žπ‘Ÿ/(π‘βˆ’π‘ž)]. Thus, applying Lemma 1.3, every solution of (1.1) converges to 𝑦, that is, 𝑦 is a global attractor.
Case (ii). βˆšπ‘=π‘ž(1+1+4π‘Ÿ)/2.
In this case, the only positive equilibrium of (1.1) is 𝑦=𝑝/π‘ž. From Lemma 3.6 and (3.11), we know that each of the 2(π‘˜+1) subsequences
𝑦2(π‘˜+1)𝑛+π‘–ξ€Ύβˆžπ‘›=0for𝑖=1,2,…,2(π‘˜+1)(4.15) of any solution {𝑦𝑛}βˆžπ‘›=βˆ’π‘˜ of (1.1) is either identically equal to 𝑝/π‘ž or strictly monotonically convergent and its limit is greater than zero. Set 𝐿𝑖=limπ‘›β†’βˆžπ‘¦2(π‘˜+1)𝑛+𝑖for𝑖=1,2,…,2(π‘˜+1).(4.16) Then, clearly, …,𝐿1,𝐿2,…,𝐿2(π‘˜+1),…(4.17) is a period solution of (1.1) with period 2(π‘˜+1). By applying (3.11) to the solution (4.17) and using the fact 𝐿𝑖>0 for 𝑖=1,2,…,2(π‘˜+1), we see that 𝐿𝑖=π‘π‘žfor𝑖=1,2,…,2(π‘˜+1),(4.18) and so limπ‘›β†’βˆžπ‘¦π‘›=π‘π‘ž,(4.19) which implies that 𝑦=𝑝/π‘ž is a global attractor.
Case (iii). βˆšπ‘ž(1+1+4π‘Ÿ)/2<𝑝<π‘ž+π‘žπ‘Ÿ.
By Theorem 4.8 (ii), all positive solutions of (1.1) eventually enter the invariant interval [π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž]. Furthermore, the function 𝑓(π‘₯,𝑦) increases in π‘₯ and decreases in 𝑦 in the interval [π‘žπ‘Ÿ/(π‘βˆ’π‘ž),𝑝/π‘ž]. Thus, applying Lemma 1.3 and assumption 𝑝≀1, every solution of (1.1) converges to 𝑦. So, 𝑦 is a global attractor.
Case (iv). 𝑝β‰₯π‘ž+π‘žπ‘Ÿ.
In this case, we note that π‘žπ‘Ÿ/(π‘βˆ’π‘ž)≀1<𝑝/π‘ž holds. In view of Theorem 4.8 (iii), we obtain that all solutions of (1.1) eventually enter the invariant interval [1,𝑝/π‘ž]. Furthermore, the function 𝑓(π‘₯,𝑦) increases in π‘₯ and decreases in 𝑦 in the interval [1,𝑝/π‘ž]. Then using the same argument in case (iii), every solution of (1.1) converges to 𝑦. Thus the equilibrium 𝑦 is a global attractor. The proof is complete.

Finally, we summarize our results and obtain the following theorem, which shows that 𝑦 is a global attractor in three cases.

Theorem 4.10. The unique positive equilibrium 𝑦 of (1.1) is a global attractor, when one of the following three cases holds: (i)𝑝=π‘ž;(ii)𝑝<1 and 𝑝<π‘žβ‰€π‘π‘ž+1+3𝑝;(iii)π‘ž<𝑝≀1.

References

  1. S. Basu and O. Merino, β€œGlobal behavior of solutions to two classes of second-order rational difference equations,” Advances in Difference Equations, vol. 2009, Article ID 128602, 27 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problem and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at MathSciNet
  3. V. L. Kocić 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. View at MathSciNet
  4. R. P. Agarwal, W.-T. Li, and P. Y. H. Pang, β€œAsymptotic behavior of a class of nonlinear delay difference equations,” Journal of Difference Equations and Applications, vol. 8, no. 8, pp. 719–728, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. A. M. Amleh, E. Camouzis, and G. Ladas, β€œOn the boundedness character of rational equations. II,” Journal of Difference Equations and Applications, vol. 12, no. 6, pp. 637–650, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. K. Cunningham, M. R. S. Kulenović, G. Ladas, and S. V. Valicenti, β€œOn the recursive sequence xn+1=(α+βxn)/(Bxn+Cxn1),” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4603–4614, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. R. DeVault, W. Kosmala, G. Ladas, and S. W. Schultz, β€œGlobal behavior of yn+1=(p+ynk)/(qyn+ynk),” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4743–4751, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. M. M. El-Afifi, β€œOn the recursive sequence xn+1=(α+βxn+γxn-1)/(Bxn+Cxn-1),” Applied Mathematics and Computation, vol. 147, no. 3, pp. 617–628, 2004. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  9. G. H. Gibbons, M. R. S. Kulenović, and G. Ladas, β€œOn the dynamics of xn+1=(α+βxn+γxn-1)/(A+Bxn),” in New Trends in Difference Equaations, G. Ladas and M. Pinto, Eds., pp. 141–158, Taylor & Francis, London, UK, 2002. View at Google Scholar
  10. L.-X. Hu, W.-T. Li, and S. Stević, β€œGlobal asymptotic stability of a second order rational difference equation,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 779–797, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. Y. S. Huang and P. M. Knopf, β€œBoundedness of positive solutions of second-order rational difference equations,” Journal of Difference Equations and Applications, vol. 10, no. 11, pp. 935–940, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  12. V. L. Kocić, G. Ladas, and I. W. Rodrigues, β€œOn rational recursive sequences,” Journal of Mathematical Analysis and Applications, vol. 173, no. 1, pp. 127–157, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. M. R. S. Kulenović, G. Ladas, L. F. Martins, and I. W. Rodrigues, β€œThe dynamics of xn+1=(α+βxn)/(A+Bxn+Cxn-1) facts and conjectures,” Computers and Mathematics with Applications, vol. 45, no. 6–9, pp. 1087–1099, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet Β· View at Scopus
  14. M. R. S. Kulenović and O. Merino, β€œA note on unbounded solutions of a class of second order rational difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 7, pp. 777–781, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  15. M. R. S. Kulenović and O. Merino, β€œGlobal attractivity of the equilibrium of xn+1=(pxn+xn1)/(qxn+xn1) for q<p,” Journal of Difference Equations and Applications, vol. 12, no. 1, pp. 101–108, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  16. R. Mazrooei-Sebdani and M. Dehghan, β€œDynamics of a non-linear difference equation,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 250–261, 2006. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  17. D. Simsek, B. Demir, and C. Cinar, β€œOn the solutions of the system of difference equations xn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn},” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 325296, 11 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  18. Y.-H. Su, W.-T. Li, and S. Stević, β€œDynamics of a higher order nonlinear rational difference equation,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 133–150, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet