Volume 2008, Article ID 678402,13pages doi:10.1155/2008/678402
Research Article
Dynamical Properties for a Class of Fourth-Order
Nonlinear Difference Equations
Dongsheng Li,1, 2Pingping Li,1and Xianyi Li3
1Ministry Education Key Laboratory of Modern Agricultural Equipment and Technology,
Jiangsu University, Jiangsu 212013, Zhenjiang, China
2School of Economics and Management, University of South China, Hunan 421001, Hengyang, China 3College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060,
Guangdong, China
Correspondence should be addressed to Xianyi Li,[email protected]
Received 6 May 2007; Revised 14 August 2007; Accepted 18 September 2007
Recommended by Jianshe Yu
We consider the dynamical properties for a kind of fourth-order rational difference equations. The key is for us to find that the successive lengths of positive and negative semicycles for non-trivial solutions of this equation periodically occur with same prime period 5. Although the pe-riod is same, the order for the successive lengths of positive and negative semicycles is com-pletely different. The rule is. . . ,3,2−,3,2−,3,2−,3,2−, . . . ,or. . . ,2,1−,1,1−,2,1−,1,1−, . . . ,or
. . . ,1,4−,1,4−,1,4−,1,4−, . . .. By the use of the rule, the positive equilibrium point of this equa-tion is proved to be globally asymptotically stable.
Copyrightq2008 Dongsheng Li 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 and preliminaries
Rational difference equation, as a kind of typical nonlinear difference equations, is always a subject studied in recent years. Especially, some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results of rational difference equations. For the systematical investiga-tions of this aspect, refer to the monographs1–3, the papers4–9, and the references cited therein.
Motivated by the work5–7, we consider in this paper the following fourth-order ratio-nal difference equation:
xn1
Fxn, xn−1, xn−2, xn−3
Gxn, xn−1, xn−2, xn−3
where
Fx, y, z, w xuyvxuzkxuwjyvzkyvwjzkwjxuyvzkwj1a,
Gx, y, z, w xuyvzkwj xuyvzkxuyvwjxuzkwjyvzkwja, 1.2
the parametera ∈0,∞,u∈ 0,1,v, k, j ∈0,∞, and the initial valuesx−3, x−2, x−1, x0 ∈ 0,∞.
Mainly, by analyzing the rule for the length of semicycle to occur successively, we clearly describe out the rule for the trajectory structure of its solutions. With the help of several key lemmas, we further derive the global asymptotic stability of positive equilibrium of1.1. To the best of our knowledge,1.1has not been investigated so far; therefore, it is theoretically meaningful to study its qualitative properties.
It is easy to see that the positive equilibriumxof1.1satisfies
x 1axuvxukxujxvkxvjxkjxuvkj
axuxvxkxj xuvkxuvjxukjxvkj. 1.3
From this, we see that1.1possesses a unique positive equilibriumx1.
It is essential in this note for us to obtain the general rule for the trajectory structure of solutions of1.1as follows.
Theorem 1.1. The rule for the trajectory structure of any solution of 1.1is as follows:
ithe solution is either eventually trivial,
iior the solution is eventually nontrivial,
1and further, either the solution is eventually positive nonoscillatory,
2or the solution is strictly oscillatory, and moreover, the successive lengths for posi-tive and negaposi-tive semicycles occur periodically with prime period 5, and the rule is . . . ,3,2−,3,2−,3,2−,3,2−, . . . ,or. . . ,2,1−,1,1−,2,1−,1,1−, . . . ,or. . . ,1,4−, 1,4−,1,4−,1,4−, . . . .
The positive equilibrium point of 1.1is a global attractor of all its solutions.
It follows from the results stated in the sequel thatTheorem 1.1is true. For the corresponding concepts in this paper, see3or the papers5–7.
2. Nontrivial solution
Theorem 2.1. A positive solution{xn}∞n−3of 1.1is eventually trivial if and only if
x−3−1
x−2−1
x−1−1
x0−1
0. 2.1
Proof. Sufficiency. Assume that2.1holds. Then, according to1.1, we know that the following conclusions are true: ifx−31, x−21, x−11,orx01,thenxn1 forn≥1.
Necessity. Conversely, assume that
x−3−1
x−2−1
x−1−1
x0−1
/
Then, we can show that xn/1 for any n ≥ 1. For the sake of contradiction, assume that for
someN≥1,
xN 1, xn/1 for any −3≤n≤N−1. 2.3
Clearly,
1xN F
xN−1, xN−2, xN−3, xN−4
GxN−1, xN−2, xN−3, xN−4
. 2.4
From this, we can know that
0xN−1
xu
N−1−1
xv
N−2−1
xk
N−3−1
xjN−4−1
GxN−1, xN−2, xN−3, xN−4
, 2.5
which implies thatxN−11, xN−21, xN−31,orxN−41.This contradicts2.3.
Remark 2.2. Theorem 2.1actually demonstrates that a positive solution{xn}∞n−3of1.1is even-tually nontrivial ifx−3−1x−2−1x−1−1x0−1/0. So, if a solution is a nontrivial one, then
xn/1 for anyn≥ −3.
3. Several key lemmas
We state several key lemmas in this section, which will be important in the proofs of the sequel. DenoteNk {k, k1, . . .}for any integerk.
Lemma 3.1. If the integeri∈N−3, then
xn1−xi K
xn, xn−1, xn−2, xn−3, xi
Gxn, xn−1, xn−2, xn−3
, n0,1, . . . , 3.1
where
Kx, y, z, w, p
1−xup1yvzkyvwjzkwja1−xixu−pyvzkwjyvzkwj.
3.2
Lemma 3.2. If the integeri∈N−3,t1,2, . . . ,then
1−xn1x1/u t
i
Mxn, xn−1, xn−2, xn−3, xi Gxn, xn−1, xn−2, xn−3
, n0,1, . . . , 3.3
where
Mx, y, z, w, p xu−p1/ut
1yvzkyvwj zkwja1−p1/ut
1−xup1/utyvzk wjyvzkwj.
By virtue of3.26,3.29,3.33, and3.34, as well as1−xn1/u−431−xn−4 > 0 foru∈ 0,1,
Equation3.3instructs us that
Utilizing3.38,3.41,3.42, and3.43, and adding1−xn1/u−41−xn−4> 0 whenu∈0,1,
we know that the following is true:1−xnxn1/u−41−xn−4>0.So,
1−xunxn−4
1−xn−4
>0. 3.44
Similar to3.44, by virtue of3.5,3.38,3.41, and1−xn1/u−41−xn−4> 0 whenu∈0,1,
we know thatxn−xn1/u−41−xn−4>0 is true. So,
xu
n−xn−4
1−xn−4
>0. 3.45
From3.21,3.22,3.44, and3.45, one knows that the following is true:
xn−xn−4
1−xn−4
>0. 3.46
This shows thatLemma 3.5is true.
4. Oscillation and nonoscillation
Theorem 4.1. There exist nonoscillatory solutions of 1.1with x−3,x−2, x−1, x0 ∈ 1,∞, which
must be eventually positive. There are not eventually negative nonoscillatory solutions of 1.1.
Proof. Consider a solution of 1.1 with x−3, x−2, x−1, x0 ∈ 1,∞. We then know from
Lemma 3.4athatxn > 1 forn ∈ N−3. So, this solution is just a nonoscillatory solution and it is, furthermore, eventually positive.
Suppose that there exist eventually negative nonoscillatory solutions of1.1. Then, there exists a positive integerN such thatxn <1 forn≥N. Thereout, forn≥N3,xn1−1xn−
1xn−2−1xn−3−10.This contradictsLemma 3.4a. So, there are not eventually negative nonoscillatory solutions of1.1, as desired.
5. Rule of cycle length
Theorem 5.1. Let{xn}∞−3be a strictly oscillatory solution of 1.1, then the rule for the lengths of
posi-tive and negaposi-tive semicycles of this solution to occur successively is. . . ,3,2−,3,2−,3,2−,3,2−, . . . ,
or. . . ,2,1−,1,1−,2,1−,1,1−, . . . ,or. . . ,1,4−,1,4−,1,4−,1,4−, . . . .
Proof. ByLemma 3.4a, one can see that the length of a negative semicycle is at most 4, and that of a positive semicycle is at most 3. On the basis of the strictly oscillatory character of the solution, we see, for some integerp≥0, that one of the following sixteen cases must occur:
8xp >1, xp1<1, xp2<1, xp3<1; 9xp <1, xp1>1, xp2>1, xp3>1; 10xp <1, xp1>1, xp2>1, xp3<1; 11xp <1, xp1>1, xp2<1, xp3>1; 12xp <1, xp1>1, xp2<1, xp3<1; 13xp <1, xp1<1, xp2>1, xp3>1; 14xp <1, xp1<1, xp2>1, xp3<1; 15xp <1, xp1<1, xp2<1, xp3>1; 16xp <1, xp1<1, xp2<1, xp3<1.
If case1occurs, of course, it will be a nonoscillatory solution of1.1.
If case2occurs, it follows fromLemma 3.4athatxp4 < 1, xp5 > 1,xp6 > 1,xp7 > 1, xp8 < 1, xp9 < 1, xp10 > 1, xp11 > 1, xp12 > 1, xp13 < 1, xp14 < 1, xp15 > 1, xp16 > 1, xp17>1,xp18<1, xp19<1, . . . .
This means that the rule for the lengths of positive and negative semicycles of the solu-tion of1.1to occur successively is
. . . ,3,2−,3,2−,3,2−,3,2−,3,2−,3,2−,3,2−,3,2−. . . . 5.1
If case3occurs, it follows fromLemma 3.4athatxp4<1,xp5>1,xp6>1,xp7 <1,
xp8 > 1, xp9 < 14,xp10 > 1, xp11 > 1, xp12 < 1, xp13 > 1, xp14 < 1, xp15 > 1, xp16> 1,
xp17 < 1, xp18 > 1, xp19 < 1, . . . ,which means that the rule for the lengths of positive and negative semicycles of the solution of1.1to occur successively is
. . . ,2,1−,1,1−,2,1−,1,1−,2,1−,1,1−,2,1−,1,1−, . . . . 5.2
If case 8 is reached, Lemma 3.4a tells us thatxp4 < 1, xp5 > 1, xp6 < 1, xp7 < 1,
xp8<1, xp9<1,xp10>1,xp11<1, xp12<1,xp13<1, xp14<1, xp15>1,xp16<1, xp17< 1,xp18<1,xp19<1, . . . .
This implies that the rule for the lengths of positive and negative semicycles of the solu-tion of1.1to occur successively is
. . . ,1,4−,1,4−,1,4−,1,4−,1,4−,1,4−,1,4−,1,4−, . . . . 5.3
Moreover, the rule for the cases2.3,3.5,3.13,3.15, and3.17is the same as that of case2.1. And cases3.1,3.3, and3.15are completely similar to case3except possibly for the first semicycle. And cases3.16,3.18,3.19, and3.20are like case8with a possible exception for the first semicycle.
Up to now, the proof ofTheorem 5.1is complete.
6. Global asymptotic stability
Theorem 6.1. The positive equilibrium point of 1.1is locally asymptotically stable.
Proof. The linearized equation of1.1about thepositive equilibrium pointxis
yn10·yn0·yn−10·yn−20·yn−3, n0,1, . . . , 6.1
and so it is clear from3, Remark 1.3.7that the positive equilibrium pointxof1.1is locally asymptotically stable. The proof is complete.
We are now in a position to study the global asymptotic stability of positive equilibrium pointx.
Theorem 6.2. The positive equilibrium point of 1.1is globally asymptotically stable.
Proof. We must prove that the positive equilibrium pointx of 1.1is both locally asymptot-ically stable and globally attractive. Theorem 6.1has shown the local asymptotic stability of
x. Hence, it remains to verify that every positive solution{xn}∞n−3of 1.1converges toxas
n→∞.Namely, we want to prove that
lim
n→∞xnx1. 6.2
We can divide the solutions into two types:
itrivial solutions; iinontrivial solutions.
If a solution is a trivial one, then it is obvious for 6.2 to hold becausexn 1 holds
eventually.
If the solution is a nontrivial one, then we can further divide the solution into two cases:
anonoscillatory solution; boscillatory solution.
Consider now {xn} to be nonoscillatory about the positive equilibrium point x of
1.1. By virtue ofLemma 3.4b, it follows that the solution is monotonic and bounded. So, limn→∞xnexists and is finite. Taking limits on both sides of1.1, one can easily see that6.2
holds.
Now, let {xn} be strictly oscillatory about the positive equilibrium point of 1.1. By
virtue of Theorem 5.1, one understands that the rule for the lengths of positive and negative semicycles occurring successively is
i. . . ,3,2−,3,2−,3,2−,3,2−, . . . , ii. . . ,2,1−,1,1−,2,1−,1,1−, . . . ,or iii. . . ,1,4−,1,4−,1,4−,1,4−, . . . .
Now, we consider the case i. For simplicity, for some nonnegative integer p,we denote by
semicycle, and so on. Namely, the rule for the lengths of positive and negative semicycles to occur successively can be periodically expressed as follows:
{xp5n, xp5n1, xp5n2}, {xp5n3, xp5n4}−, n0,1,2, . . . . 6.3
Lemma 3.4b,c,d,eandLemma 3.5teach us that the following results are true: axp5n> xp5n1> xp5n2> xp5n5, n0,1,2, . . .;
bxp5n3< xp5n4< xp5n8, n0,1,2, . . . .
So, by virtue ofa, one can see that{xp5n}∞n0is decreasing with lower bound 1. So, its
limit exists and is finite, denoted byS. Moreover, the limits of{xp5n1}∞n0and{xp5n2}∞n0are all equal to that of{xp5n}∞n0.
Similarly, usingb, one can see that{xp5n3}∞n0is increasing with upper bound 1. So, its limit exists and is finite too. Furthermore, the limits of {xp5n4}∞n0 are equal to that of
{xp5n3}∞n0, and one can assume the limit of it to be T. It is easy to see that S ≥ 1 ≥ T. It suffices to show thatS1T.
Noting that
xp5n5
Fxp5n4, xp5n3, xp5n2, xp5n1
Gxp5n4, xp5n3, xp5n2, xp5n1
, 6.4
xp5n4
Fxp5n3, xp5n2, xp5n1, xp5n
Gxp5n3, xp5n2, xp5n1, xp5n, 6.5
and taking limits on both sides of6.4and6.5, respectively, we get
S TuvTuSkTuSjTvSkTvSjSkjTuvSkj1a
TuTvSkSjTuvSkTuvSjTuSkjTvSkja ,
T TuSvTuSkTuSjSvkSvjSkjTuSvkj 1a
TuSvSkSj TuSvkTuSvj TuSkjSvkja .
6.6
From6.6, we can show thatST 1. Otherwise, assume that
S >1. 6.7
From3.1, with bothnandibeing replaced by 5n, we get
x5n1−x5n K
x5n, x5n−1, x5n−2, x5n−3, x5n
Gx5n, x5n−1, x5n−2, x5n−3
. 6.8
Taking limits on both sides of the above equation, we can obtain
a1−S 1−Su11TvkTvSjTkSjSu−STvTkSj0. 6.9
By virtue of6.7, one has 1−S < 0,1−Su1 <0, andSu−S ≤0 whenu∈ 0,1; and so, one will get
a1−S 1−Su11TvkTvSjTkSjSu−STvTkSj<0. 6.10
When caseiioriiioccurs, using the method similar to that proving casei, we can prove that6.2is also true. Thus, the proof of Theorem is complete.
Acknowledgments
This work was partly supported by NNSF of ChinaGrant no. 10771094, and the Foundations for “New Century Engineering of 121 Talents in Hunan Province” and “Chief Professor of Mathematical Discipline in Hunan Province.”
References
1 M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
2 R. P. Agarwal,Difference Equations and Inequalities. Theory, Methods, and Applications, vol. 155 of Mono-graphs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992. 3 V. L. Koci´c and G. Ladas,Global Behavior of Nonlinear Difference Equations of Higher Order with
Applica-tions, vol. 256 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Nether-lands, 1993.
4 A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequencexn1 α xn−1/xn,”Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790–798, 1999.
5 L. Xianyi and Z. Deming, “Global asymptotic stability in a rational equation,”Journal of Difference Equa-tions and ApplicaEqua-tions, vol. 9, no. 9, pp. 833–839, 2003.
6 X. Li and D. Zhu, “Global asymptotic stability of a nonlinear recursive sequence,”Applied Mathematics Letters, vol. 17, no. 7, pp. 833–838, 2004.
7 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.
8 H. Xi and T. Sun, “Global behavior of a higher-order rational difference equation,”Advances in Difference Equations, vol. 2006, Article ID 27637, 7 pages, 2006.
