On a Conjecture for a Higher-Order Rational Difference Equation

doi:10.1155/2009/394635

Research Article

On a Conjecture for a Higher-Order Rational Difference Equation

Maoxin Liao,

Xianhua Tang,

and Changjin Xu

1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

2School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

3College 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 _m

j1xn−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 xn−kxn−m

1x_n−kx_n−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 that 1≤k < l < mand that{x_n}satisfies

x_n x_n−kx_n−lx_n−mx_n−kx_n−lx_n−m

1x_n−kx_n−lx_n−lx_n−mx_n−mx_n−k, n0,1,2, . . . 1.2 withx_−m, x_−m1, . . . , x₋₁∈0,∞.Then, the sequence{x_n}converges to the unique equilibrium 1.

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

xn f1x_n−k1, x_n−k2, . . . , x_n−km

f2x_n−k1, x_n−k2, . . . , x_n−km, n0,1,2, . . . 1.3

withx−km, x−km1, . . . , x−1∈0,∞, where

f1

y1, y2, . . . , y_m

j∈{1,3,...,m}

{^t1,t2,...,tj}^⊂S;t1<t2<···<tj

y_t1y_t2· · ·y_tj,

f2

y1, y2, . . . , ym

1

j∈{2,4,...,m−1}

{^t1,t2,...,tj}^⊂S;t1<t2<···<tj

yt1yt2· · ·ytj. 1.4

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−m^pforp >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.

2. Preliminaries and Notations

Observe that

f1

y1, y2, . . . , y_m 1

2

⎡

⎣^m

j1

y_j1 ^m

j1

y_j−1⎤

⎦,

f2

y1, y2, . . . , y_m 1

2

⎡

⎣^m

j1

y_j1

−^m

j1

y_j−1⎤

⎦.

2.1

Define functionGas follows:

G

y1, y2, . . . , ym

_m

j1 y_j1

_m

j1 y_j−1 _m

j1 y_j1

−_m

j1

y_j−1, y1, y2, . . . , ym>0. 2.2

Then we can rewrite1.3as

xn _m

j1

x_n−kj1 _m

j1

x_n−kj−1 _m

j1

xn−kj1

−_m

j1

xn−kj−1, n0,1,2, . . . , 2.3

or

x_nGx_n−k1, x_n−k2, . . . , x_n−km, n0,1,2, . . . , 2.4

wheremis an odd integer andx−km, x−km1, . . . , x−1∈0,∞.

The following lemma can be obtained by simple calculations.

Lemma 2.1. LetGbe defined by2.2. Then

∂G

∂yi 4_m

j1,j /i

y_j²−1 _m

j1y_j1−_m

j1y_j−12

⎧⎪

⎪⎨

⎪⎪

⎩

>0, ^m

j1,j /i

y_j−1

>0,

<0, ^m

j1,j /i

yj−1

<0, 2.5

i1,2, . . . , m.

Lemma 2.2. Assume that 0< α <1< β <∞. Ifα≤y1, y2, . . . , ym≤β, then min{A1, A3, . . . , Am} ≤G

y1, y2, . . . , ym

≤max{B1, B3, . . . , Bm}, 2.6

where

A_i α1ⁱ

β1_m−i

α−1ⁱ

β−1_m−i α1ⁱ

β1_m−i

−α−1ⁱ

β−1_m−i, B_i α1^m−i

β1_i

α−1^m−i β−1_i α1^m−i

β1_i

−α−1^m−i β−1_i,

2.7

i1,3, . . . , m.

Proof. SinceGy1, y2, . . . , ymis symmetric iny1, y2, . . . , ym, we can assume, without loss of generality, thatα≤y1≤y2≤ · · · ≤y_m≤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≤ · · · ≤y_m≤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 that 0< α <1< β <∞.Then

Ai α1ⁱ

β1_m−i

α−1ⁱ

β−1_m−i α1ⁱ

β1_m−i

−α−1ⁱ

β−1_m−i ≥α, 2.8

Bi α1^m−i β1_i

α−1^m−i β−1_i α1^m−i

β1_i

−α−1^m−i

β−1_i ≤β, 2.9

i1,3, . . . , m.

Proof. Fori1,3, . . . , m, it is easy to see that α−1ⁱ⁻¹

β−1_m−i

≤α1ⁱ⁻¹

β1_m−i

, 2.10

which yields

α1α−1ⁱ

β−1_m−i

≥α−1α1ⁱ

β1_m−i

, 2.11

and so α

α1ⁱ

β1_m−i

−α−1ⁱ

β−1_m−i

≤α1ⁱ

β1_m−i

α−1ⁱ

β−1_m−i

. 2.12

It follows that2.8holds. Similarly, fori1,3, . . . , m, it is easy to see that α−1^m−i

β−1_i−1

≤α1^m−i

β1_i−1

, 2.13

which yields

β1

α−1^m−i β−1_i

≤ β−1

α1^m−i β1_i

. 2.14

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

Lemma 2.4. Let

α_j1min

A_1j, A_3j, . . . , A_mj , β_j1max

B_1j, B_3j, . . . , B_mj

, 2.15

where

Aij

αj1_i

βj1_m−i

αj−1_i

βj−1_m−i α_j1_i

β_j1_m−i

−

α_j−1_i

β_j−1_m−i, B_ij

α_j1_m−i β_j1_i

α_j−1_m−i β_j−1_i αj1_m−i

βj1_i

−

αj−1_m−i

βj−1_i,

2.16

i1,3, . . . , m;j0,1,2, . . . .Assume that 0< α0<1< β0<∞.Then

jlim→ ∞αj lim

j→ ∞βj1. 2.17

Proof. By induction, we easily show that

0< α_j<1< β_j <∞, j0,1,2, . . . . 2.18 It follows fromLemma 2.3that

Aij

αj1_i

βj1_m−i

αj−1_i

βj−1_m−i α_j1_i

β_j1_m−i

−

α_j−1_i

β_j−1_m−i ≥αj, B_ij

α_j1_m−i β_j1_i

α_j−1_m−i β_j−1_i αj1_m−i

βj1_i

−

αj−1_m−i

βj−1_i ≤β_j,

2.19

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

αj≤αj1<1< βj1≤βj, j 0,1,2, . . . . 2.20

Equation2.20implies that the limits lim_{j→ ∞}αjand lim_{j→ ∞}βjexist, and α^∗ lim

j→ ∞αj∈α0,1, β^∗ lim

j→ ∞βj ∈ 1, β0

. 2.21

It follows from2.16that

A^∗_i : lim

j→ ∞Aij α^∗1ⁱ

β^∗1_m−i

α^∗−1ⁱ

β^∗−1_m−i α^∗1ⁱ

β^∗1_m−i

−α^∗−1ⁱ

β^∗−1_m−i, B_i^∗: lim

j→ ∞Bij α^∗1^m−i β^∗1_i

α^∗−1^m−i β^∗−1_i α^∗1^m−i

β^∗1_i

−α^∗−1^m−i

β^∗−1_i,

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^∗1_m−i

α^∗−1ⁱ

β^∗−1_m−i α^∗1ⁱ

β^∗1_m−i

−α^∗1ⁱ

β^∗1_m−i, β^∗ α^∗1^m−j

β^∗1_j

α^∗−1^m−j β^∗−1_j α^∗1^m−j

β^∗1_j

−α^∗−1^m−j

β^∗−1_j.

2.24

From2.24, we have α^∗−1

α^∗1ⁱ⁻¹

β^∗1_m−i

−α^∗−1ⁱ⁻¹

β^∗−1_m−i 0, β^∗−1

α^∗1^m−j

β^∗1_j−1

−α^∗−1^m−j

β^∗−1_j−1 0.

2.25

Since

α^∗1ⁱ⁻¹

β^∗1_m−i

−α^∗−1ⁱ⁻¹

β^∗−1_m−i

>0, α^∗1^m−j

β^∗1_j−1

−α^∗−1^m−j

β^∗−1_j−1

>0, 2.26

it follows from2.25and2.18thatα^∗β^∗1. The proof is complete.

3. Proof of Conjecture 1.2

Theorem 3.1. Suppose that 0< α <1< β <∞and that x_−km, x_−km1, . . . , x₋₁∈

α, β

. 3.1

Then the solution{x_n}of1.3satisfies x_n∈

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

nlim→ ∞x_n1. 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, . . . , B_m0}, 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, . . . , B_m1}, nk_m, k_m1, k_m2, . . . . 3.8

That is

xn∈ α2, β2

, nkm, km1, km2, . . . . 3.9

Repeating the above procedure, in general, we can obtain x_n∈

α_j1, β_j1

, njk_m, jk_m1, jk_m2, . . . , j 0,1,2, . . . . 3.10

ByLemma 2.4, we have

nlim→ ∞x_n lim

j→ ∞α_j1 lim

j→ ∞β_j11, 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.

References

1 K. S. Berenhaut, J. D. Foley, and S. Stevi´c, “The global attractivity of the rational difference equation yn yn−kyn−m/1yn−kyn−m,” Applied Mathematics Letters, vol. 20, no. 1, pp. 54–58, 2007.

2 S. Stevi´c, “On the recursive sequencexn1αx^p_n−1/x^pn,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229–234, 2005.

3 K. S. Berenhaut, J. D. Foley, and S. Stevi´c, “The global attractivity of the rational difference equation yn1yn−k/yn−m,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1133–1140, 2007.

4 K. S. Berenhaut, J. D. Foley, and S. Stevi´c, “The global attractivity of the rational difference equation yn A yn−k/yn−m^p,” Proceedings of the American Mathematical Society, vol. 136, no. 1, pp. 103–110, 2008.

5 K. S. Berenhaut and S. Stevi´c, “The global attractivity of a higher order rational difference equation,”

Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 940–944, 2007.

6 X. Li, “Qualitative properties for a fourth-order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 103–111, 2005.

7 X. Li, “Global behavior for a fourth-order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 555–563, 2005.

8 A. M. Amleh, N. Kruse, and G. Ladas, “On a class of difference equations with strong negative feedback,” Journal of Difference Equations and Applications, vol. 5, no. 6, pp. 497–515, 1999.

9 L. Berg and S. Stevi´c, “Linear difference equations mod 2 with applications to nonlinear difference equations,” Journal of Difference Equations and Applications, vol. 14, no. 7, pp. 693–704, 2008.

10 L. Berg and S. Stevi´c, “Periodicity of some classes of holomorphic difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 827–835, 2006.

11 C. C¸ inar, S. Stevi´c, and I. Yalc¸inkaya, “A note on global asymptotic stability of a family of rational equations,” Rostocker Mathematisches Kolloquium, no. 59, pp. 41–49, 2005.

12 S. Stevi´c, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006.

13 N. Kruse and T. Nesemann, “Global asymptotic stability in some discrete dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 151–158, 1999.

14 X. Li and D. Zhu, “Global asymptotic stability in a rational equation,” Journal of Difference Equations and Applications, vol. 9, no. 9, pp. 833–839, 2003.

15 M. Aloqeily, “Global stability of a rational symmetric difference equation,” preprint, 2008.

16 L. Gutnik and S. Stevi´c, “On the behaviour of the solutions of a second-order difference equation,”

Discrete Dynamics in Nature and Society, vol. 2007, Article ID 27562, 14 pages, 2007.

17 G. Ladas, “A problem from the Putnam Exam,” Journal of Difference Equations and Applications, vol. 4, no. 5, pp. 497–499, 1998.

18 “Putnam Exam,” The American Mathematical Monthly, pp. 734–736, 1965.

19 S. Stevi´c, “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 56813, 20 pages, 2007.

20 S. Stevi´c, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007.

21 S. Stevi´c, “Nontrivial solutions of a higher-order rational difference equation,” Matematicheskie Zametki, vol. 84, no. 5, pp. 772–780, 2008.

22 T. Sun and H. Xi, “Global asymptotic stability of a higher order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 462–466, 2007.

23 X. Yang, F. Sun, and Y. Y. Tang, “A new part-metric-related inequality chain and an application,”

Discrete Dynamics in Nature and Society, vol. 2008, Article ID 193872, 7 pages, 2008.

24 X. Yang, Y. Y. Tang, and J. Cao, “Global asymptotic stability of a family of difference equations,”

Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2643–2649, 2008.