On a Conjecture for a Higher Order Rational Difference Equation

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doi:10.1155/2009/394635

Research Article

On a Conjecture for a Higher-Order Rational

Difference Equation

Maoxin Liao,

1, 2

_{Xianhua Tang,}

1

_{and Changjin Xu}

1, 3

1_{School of Mathematical Sciences and Computing Technology, Central South University, Changsha,}

Hunan 410083, China

2_{School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China} 3_{College of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China}

Correspondence should be addressed to Maoxin Liao,[email protected]

Received 30 December 2008; Revised 11 March 2009; Accepted 14 March 2009

Recommended by Jianshe Yu

This paper studies the global asymptotic stability for positive solutions to the higher order rational diﬀerence equationxn mj1xn−kj1

m

j1xn−kj−1/

m

j1xn−kj1−

m

j1xn−kj−1, n

0,1,2, . . ., wheremis odd andx−km, x−km1, . . . , x−1 ∈0,∞. Our main result generalizes several

others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.

Copyrightq2009 Maoxin Liao 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.

1. Introduction

In 2007, Berenhaut et al.1proved that every solution of the following rational diﬀerence equation

xn ₁xn−_xkxn−m

n−kxn−m, n0,1,2, . . . 1.1

converges to its unique equilibrium 1, where x−m, x−m1, . . . , x−1 ∈ 0,∞ and 1 ≤ k < m.

Based on this fact, they put forward the following two conjectures.

Conjecture 1.1. Suppose that1≤k < l < mand that{x_n}satisfies

xn ₁xn−_xkxn−lxn−mxn−kxn−lxn−m

n−kxn−lxn−lxn−mxn−mxn−k, n0,1,2, . . . 1.2

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Conjecture 1.2. Suppose thatmis odd and1≤k1 < k2 <· · ·< km, and defineS{1,2, . . . , m}. If

Then the sequence{x_n}converges to the unique equilibrium 1.

Motivated by 2, Berenhaut et al. started with the investigation of the following diﬀerence equationyn A yn−k/yn−mpforp >0see,3,4. Among others, in3they

used a transformation method, which has turned out to be very useful in studying1.1and

1.2as well as in confirmingConjecture 1.1; see5.

Some particular cases of1.2had been studied previously by Li in 6,7, by using semicycle analysis similar to that in8. The problem concerning periodicity of semicycles of diﬀerence equations was solved in very general settings by Berg and Stevi´c in9, partially motivated also by10.

In the meantime, it turned out that the method used in11by C¸ inar et al. can be used in confirmingConjecture 1.2see also12. More precisely11,12use Corollary 3 from13

in solving similar problems. For example, C¸ inar et al. has shown, in an elegant way, that the main result in14is a consequence of Corollary 3 in13. With some calculations it can be also shown thatConjecture 1.2can be confirmed in this waysee15.

Some other related results can be found in16–24.

In this paper, we will prove that Conjecture 1.2 is correct by using a new method. Obviously, our results generalize the corresponding works in1,5–7and other literature.

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Define functionGas follows:

The following lemma can be obtained by simple calculations.

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Proof. SinceGy1, y2, . . . , ymis symmetric iny1, y2, . . . , ym, we can assume, without loss of

generality, thatα≤y1≤y2≤ · · · ≤ym≤β. Then there arem1 possible cases:

1α≤1≤y1≤y2≤ · · · ≤ym≤β;

2α≤y1≤1≤y2≤ · · · ≤ym≤β;

3α≤y1≤y2≤1≤ · · · ≤ym≤β;

4α≤y1≤y2≤y3 ≤1≤ · · · ≤ym≤β;

.. .

m1α≤y1≤y2≤ · · · ≤ym≤1≤β.

And, for the above cases1–m1, by the monotonicity ofGy1, y2, . . . , ym, in turn, we may

get

11≤Gy1, y2, . . . , ym≤Bm;

2A1≤Gy1, y2, . . . , ym≤1;

31≤Gy1, y2, . . . , ym≤Bm−2;

4A3≤Gy1, y2, . . . , ym≤1;

.. .

m1 Am≤Gy1, y2, . . . , ym≤1.

From the above inequalities, it follows that2.6holds. The proof is complete.

Lemma 2.3. Assume that0< α <1< β <∞.Then

Ai α1

i_β₁m−i _α₋

1iβ−1m−i

α1iβ1m−i−α−1iβ−1m−i ≥α, 2.8

Bi α1

m−i_β₁i _α₋₁m−i_β₋₁i

α1m−iβ1i−α−1m−iβ−1i ≤β, 2.9

i1,3, . . . , m.

Proof. Fori1,3, . . . , m, it is easy to see that

α−1i−1β−1m−i≤α1i−1β1m−i, 2.10

which yields

α1α−1iβ−1m−i≥α−1α1iβ1m−i, 2.11

and so

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It follows that2.8holds. Similarly, fori1,3, . . . , m, it is easy to see that

α−1m−iβ−1i−1≤α1m−iβ1i−1, 2.13

which yields

β1α−1m−iβ−1i≤β−1α1m−iβ1i. 2.14

It follows that2.9holds. The proof is complete.

Lemma 2.4. Let

αj1min

A1j, A3j, . . . , Amj,

βj1max

B1j, B3j, . . . , Bmj,

2.15

where

Aij

αj1iβj1m−iαj−1iβj−1m−i

αj1iβj1m−i−αj−1iβj−1m−i

,

Bij

αj1m−iβj1iαj−1m−iβj−1i

αj1m−iβj1i−αj−1m−iβj−1i

,

2.16

i1,3, . . . , m;j0,1,2, . . . .Assume that0< α0<1< β0<∞.Then

lim

j→ ∞αjjlim→ ∞βj1. 2.17

Proof. By induction, we easily show that

0< αj<1< βj <∞, j0,1,2, . . . . 2.18

It follows fromLemma 2.3that

Aij

αj1iβj1m−iαj−1iβj−1m−i

αj1iβj1m−i−αj−1iβj−1m−i ≥αj,

Bij

αj1m−iβj1iαj−1m−iβj−1i

αj1m−iβj1i−αj−1m−iβj−1i ≤βj,

2.19

i1,3, . . . , m;j0,1,2, . . . .Hence, by2.15and2.18, we have

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Equation2.20implies that the limits limj→ ∞αjand limj→ ∞βjexist, and

α∗ _lim

j→ ∞αj∈α0,1, β ∗ _lim

j→ ∞βj ∈

1, β0

. 2.21

It follows from2.16that

A∗

i :_jlim_{→ ∞}Aij α

∗₁i_β∗₁m−i _α∗₋₁i_β∗₋₁m−i

α∗₁i_β∗₁m−i₋_α∗₋₁i_β∗₋₁m−i,

B∗

i :_jlim_{→ ∞}Bij α

∗₁m−i_β∗₁i _α∗₋₁m−i_β∗₋₁i

α∗₁m−i_β∗₁i₋_α∗₋₁m−i_β∗₋₁i,

2.22

i1,3, . . . , m. Letj → ∞in2.15, we have

α∗_min_A∗

1, A∗3, . . . , A∗m

, β∗_max_B∗

1, B∗3, . . . , B∗m

. 2.23

It follows that there existi, j∈ {1,3, . . . , m}such that

α∗ α∗1i

β∗₁m−i _α∗₋₁i_β∗₋₁m−i

α∗₁i_β∗₁m−i₋_α∗₁i_β∗₁m−i,

β∗ α∗1m−j

β∗₁j _α∗₋₁m−j_β∗₋₁j

α∗₁m−j_β∗₁j₋_α∗₋₁m−j_β∗₋₁j.

2.24

From2.24, we have

α∗₋₁_α∗₁i−1_β∗₁m−i₋_α∗₋₁i−1_β∗₋₁m−i₀_,

β∗₋₁_α∗₁m−j_β∗₁j−1₋_α_∗₋

1m−jβ∗−1j−10.

2.25

Since

α∗₁i−1_β∗₁m−i₋_α∗₋₁i−1_β∗₋₁m−i_>₀_,

α∗₁m−j_β∗₁j−1₋_α_∗₋

1m−jβ∗−1j−1>0,

2.26

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3. Proof of

Conjecture 1.2

Theorem 3.1. Suppose that0< α <1< β <∞and that

x−km, x−km1, . . . , x−1∈

α, β. 3.1

Then the solution{x_n}of1.3satisfies

xn∈α, β, for n0,1,2, . . . . 3.2

Theorem 3.1is a direct corollary of Lemmas2.2and2.3.

Proof ofConjecture 1.2. Let{xn}be a solution of 1.3 withx−km, x−km1, . . . , x−1 ∈ 0,∞. We need to prove that

lim

n→ ∞xn1. 3.3

Chooseα0∈0,1andβ0 ∈1,∞such that

x−km, x−km1, . . . , x−1 ∈

α0, β0

. 3.4

In view ofTheorem 3.1, we have

xn∈α0, β0

, n−km,−km1,−km2, . . . . 3.5

Letαj, βj, Aij, andBijbe defined as inLemma 2.4. Then by3.5andLemma 2.2, we have

min{A10, A30, . . . , Am0} ≤Gxn−k1, xn−k2, . . . , xn−km

≤max{B10, B30, . . . , Bm0}, n0,1,2, . . . .

3.6

That is

xn∈α1, β1

, n0,1,2, . . . . 3.7

By3.7andLemma 2.2, we obtain

min{A11, A31, . . . , Am1} ≤Gxn−k1, xn−k2, . . . , xn−km

≤max{B11, B31, . . . , Bm1}, nkm, km1, km2, . . . .

3.8

That is

xn∈α2, β2

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Repeating the above procedure, in general, we can obtain

xn∈αj1, βj1

, njkm, jkm1, jkm2, . . . , j 0,1,2, . . . . 3.10

ByLemma 2.4, we have

lim

n→ ∞xnjlim→ ∞αj1 jlim→ ∞βj11, 3.11

which implies that3.3holds. The proof ofConjecture 1.2is complete.

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and many valuable comments and suggestions that greatly improved the presentation of this work. This work is supported partly by NNSF of ChinaGrant: 10771215, 10771094, Project of Hunan Provincial Youth Key Teacher and Project of Hunan Provincial Education DepartmentGrant: 07C639.

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