doi:10.1155/2009/463169
Research Article
On Boundedness of Solutions of the Difference
Equation
x
n
1
px
n
qx
n
−
1
/
1
x
n
for
q >
1
p >
1
Hongjian Xi,
1, 2Taixiang Sun,
1Weiyong Yu,
1and Jinfeng Zhao
11Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China 2Department of Mathematics, Guangxi College of Finance and Economics,
Nanning, Guangxi 530003, China
Correspondence should be addressed to Taixiang Sun,[email protected]
Received 4 February 2009; Revised 19 April 2009; Accepted 2 June 2009 Recommended by Agacik Zafer
We study the boundedness of the difference equationxn1 pxnqxn−1/1xn,n0,1, . . . ,
whereq >1p >1 and the initial valuesx−1, x0∈0,∞. We show that the solution{xn}∞n−1of this equation converges tox qp−1 ifxn ≥xorxn ≤xfor alln≥ −1; otherwise{xn}∞n−1is unbounded. Besides, we obtain the set of all initial valuesx−1, x0∈0,∞×0,∞such that the positive solutions{xn}∞n−1of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenovi´c and Ladas2002.
Copyrightq2009 Hongjian Xi 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 this paper, we study the following difference equation:
xn1
pxnqxn−1
1xn , n0,1, . . . , 1.1
wherep, q∈0,∞withq >1pand the initial valuesx−1, x0∈0,∞.
The global behavior of1.1for the casepq <1 is certainly folklore. It can be found, for example, in1 see also a precise result in2.
example, in papers4see also8–11. Some other properties of1.1have been also treated in4.
The caseq1pwas treated for the first time by Stevi´c’s in paper12. The main trick from12has been later used with a success for many times; see, for example,13–15.
Some existing results for1.1are summarized as follows16.
Theorem A. 1Ifpq≤1, then the zero equilibrium of 1.1is globally asymptotically stable. 2Ifq1, then the equilibriumxpof1.1is globally asymptotically stable.
3If1< q <1p, then every positive solution of 1.1converges to the positive equilibrium
xpq−1.
4Ifq1p, then every positive solution of 1.1converges to a period-two solution. 5Ifq >1p, then1.1has unbounded solutions.
In16, Kulenovi´c and Ladas proposed the following open problem.
Open problem B (see Open problem 6.10.12of [
16
])
Assume thatq∈1,∞.
aFind the set Bof all initial conditions x−1, x0 ∈ 0,∞×0,∞ such that the
solutions{xn}∞n−1of1.1are bounded.
bLetx−1, x0∈B. Investigate the asymptotic behavior of{xn}∞n−1.
In this paper, we will obtain the following results: letp, q∈0,∞withq >1p, and let{xn}∞n−1be a positive solution of1.1with the initial valuesx−1, x0∈0,∞×0,∞.
Ifxn ≥ xfor alln ≥ −1orxn ≤xfor alln ≥ −1, then{xn}∞n−1 converges tox qp−1.
Otherwise{xn}∞n−1is unbounded.
For closely related results see17–34.
2. Some Definitions and Lemmas
In this section, letq >1p >1 andxqp−1 be the positive equilibrium of1.1. Write D 0,∞×0,∞and definef:D → Dby, for allx, y∈D,
fx, yy,pyqx 1y
. 2.1
It is easy to see that if{xn}∞n−1is a solution of1.1, thenfnx−1, x0 xn−1, xnfor anyn≥0.
Let
A1 0, x×0, x, A2 x,∞×x,∞,
A3 0, x×x,∞, A4 x,∞×0, x,
R0{x} ×0, x, L0 {x} ×x,∞,
R1 0, x× {x}, L1 x,∞× {x}.
ThenD ∪4i1Ai∪L0∪L1∪R0∪R1∪ {x, x}. The proof ofLemma 2.1is quite similar to
that of Lemma 1 in35and hence is omitted.
Lemma 2.1. The following statements are true.
Suppose for the sake of contradiction thatx0< x2, then
and by2.6we get
q−1−px02p2q−1−qx−1
x0pqx−1>0. 2.8
Claim 1. Ifx−1≥x, then
p2q−1−qx
−1
2
−4q−1−ppqx−1≥0. 2.9
Proof of
Claim 1
Letgx p2q−1−qx2−4q−1−ppqxx≥x, then we have
gx 2q1qx−p2−q−4pqq−1−p
≥2qq−12p2p1−qp
2qq−1q−p−1p2p
>0.
2.10
Sincex−1 ≥x, it follows
p2q−1−qx
−1
2
−4q−1−ppqx−1
≥q2qp−2q1−p22−4q−1−pqpqp−1
q2−2q1−p222qpq2−2q1−p2
qp2−4q2−2q1−p2pq
q2−2q1−p2−pq2
≥0.
2.11
This completes the proof ofClaim 1. By2.8, we have
x0> λ1
1qx−1−p2−q
1qx−1−p2−q
2−
4pqq−1−px−1
or
x0< λ2
1qx−1−p2−q
−1qx−1−p2−q
2−
4pqq−1−px−1
2q−1−p . 2.13
Claim 2. We have
λ1≥x, 2.14
λ2≤
x−1
q−1 x−1−p
. 2.15
Proof of
Claim 2
Since
1qqp−1−p2−q2−4pqq−1−ppq−1
q2−p2−2q1−qp
2qp−1q−1−p−1qqp−1−p2−q ,
2.16
we have
λ1
1qx−1−p2−q
1qx−1−p2−q
2−
4pqq−1−px−1
2q−1−p
≥
1qx−p2−q1qx−p2−q2−4pqq−1−px
2q−1−p
1qqp−1−p2−q1qqp−1−p2−q2−4pqq−1−ppq−1
2q−1−p
≥qp−1x.
2.17
The proof of2.14is completed. Now we show2.15. Let
hx pqx−p2−x−pq−
Note thatx0 < xsincex−1, x0∈A4. By2.12,2.13,2.14, and2.15, we seex0 <
x−1q−1/x−1−p,which contradicts to2.7. The proof ofLemma 2.3is completed.
3. Main Results
In this section, we investigate the boundedness of solutions of1.1. Letq > 1p > 1, and let{xn}∞n−1be a positive solution of1.1with the initial valuesx−1, x0∈0,∞×0,∞,
then we see thatxn1−xxn−x<0 for somen≥ −1 orxn≥xfor alln≥ −1 orxn ≤xfor
alln≥ −1.
Theorem 3.1. Letq >1p >1, and let{xn}∞n−1be a positive solution of 1.1such thatxn≥xfor
alln≥ −1orxn≤xfor alln≥ −1, then{xn}∞n−1converges toxqp−1.
Proof.
Case 1. 0< xn≤xfor anyn≥ −1. If 0< x2n≤q−1 for somen, then
x2n1−x2n−1
px2nqx2n−1−x2n−1−x2n−1x2n
1x2n >0. 3.1
Ifq−1< x2n≤xfor somen, then
px2n x2n−q1 ≥
px
x−q1 x≥x2n−1, 3.2
which implies thatpx2n≥x2n−1x2n−q1and
x2n1−x2n−1
px2nqx2n−1−x2n−1−x2n−1x2n
1x2n ≥0. 3.3
Thusx≥x2n1≥x2n−1for anyn≥0. In similar fashion, we can showx≥x2n2 ≥x2nfor any n≥0. Let limn→∞x2n1aand limn→∞x2nb, then
a pbqa
1b , b
paqb
1a , 3.4
Case 2. xn ≥ x pq−1 for any n ≥ −1. Sincefx, y pyqx/1y x > p/qis
decreasing iny, it follows that for anyn≥ −1,
xn2
pxn1qxn
1xn1
≤ pxqxn
1x ≤xn.
3.5
In similar fashion, we can show that limn→∞x2n1 limn→∞x2n x. This completes the
proof.
Lemma 3.2see20, Theorem 5. LetI be a set, and letF : I ×I → I be a functionFu, v which decreases inuand increases inv, then for every positive solution{xn}n∞−1of equationxn1
Fxn, xn−1,{x2n}n∞0and{x2n−1}∞n0do exactly one of the following.
1They are both monotonically increasing. 2They are both monotonically decreasing.
3Eventually, one of them is monotonically increasing, and the other is monotonically decreasing.
Remark 3.3. Using arguments similar to ones in the proof of Lemma 3.2, Stevi´c proved Theorem 2 in25. Beside this, this trick have been used by Stevi´c in18,28,29.
Theorem 3.4. Letq > 1p >1, and let{xn}n∞−1 be a positive solution of 1.1such thatxn1−
xxn−x<0for somen≥ −1, then{xn}∞n−1is unbounded.
Proof. We may assume without loss of generality thatx0−xx−1−x<0 andx−1, x0∈A4
the proof for x−1, x0 ∈ A3 is similar. FromLemma 2.1 we seex2n−1, x2n ∈ A4 for all
n ≥ 0.If {x2n}∞n0 is eventually increasing, then it follows fromLemma 2.3that{x2n−1}∞n0 is
eventually increasing. Thus limn→∞x2n−1 b > xand limn→∞x2n a≤x, it follows from
Lemma 2.2thatb∞.
If{x2n}∞n0is not eventually increasing, then there exists someN≥0 such that
x2N≥x2N2
px2N1qx2N
1x2N1
, 3.6
from which we obtainx2N≥px2N1/1x2N1−q≥p, sincex2N1 ≥xpq−1 andq >1.
Sincefy, x pyqx/1y p qx−p/1y x ≥p, y ≥ pis increasing inxand is decreasing iny, we have thatx2n ≥ pfor anyn≥N. It follows fromLemma 3.2
that{x2n}∞n0is eventually decreasing. Thus limn→∞x2n a < xand limn→∞x2n−1b≥x.
It follows fromLemma 2.2thatb∞. This completes the proof.
By Theorems3.1and3.4we have the following.
Corollary 3.5. Letq > 1p > 1, and let {xn}∞n−1 be a positive bounded solution of 1.1, then
Now one can find out the set of all initial valuesx−1, x0∈0,∞×0,∞such that
R1, then it follows from Lemma 2.1 that
f2 x
implies that{xn}is unbounded.
Ifx−1, x0∈A1−
∞
n1Qn, then there existsn≥0 such thatx−1, x0∈Qn−Qn1Qn− f−1Q
nandfnx−1, x0 xn−1, xn∈A1−f−1A1. Again byLemma 2.1andTheorem 3.4,
we have that{xn}is unbounded. This completes the proof.
Acknowledgment
Project Supported by NNSF of China10861002and NSF of Guangxi0640205, 0728002.
References
1 M. R. Taskovi´c, Nonlinear Functional Analysis. Vol. I: Fundamental Elements of Theory, Zavod, za udˇzbenike i nastavna sredstva, Beograd, Serbia, 1993.
2 S. Stevi´c, “Behavior of the positive solutions of the generalized Beddington-Holt equation,”
PanAmerican Mathematical Journal, vol. 10, no. 4, pp. 77–85, 2000.
3 E. T. Copson, “On a generalisation of monotonic sequences,”Proceedings of the Edinburgh Mathematical Society. Series 2, vol. 17, pp. 159–164, 1970.
5 S. Stevi´c, “A note on bounded sequences satisfying linear inequalities,”Indian Journal of Mathematics, vol. 43, no. 2, pp. 223–230, 2001.
6 S. Stevi´c, “A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality,”Indian Journal of Mathematics, vol. 43, no. 3, pp. 277–282, 2001.
7 S. Stevi´c, “A global convergence result,”Indian Journal of Mathematics, vol. 44, no. 3, pp. 361–368, 2002.
8 S. Stevi´c, “A note on the recursive sequence xn1 pkxn pk−1xn−1· · ·p1xn−k1,”Ukrainian
Mathematical Journal, vol. 55, no. 4, pp. 691–697, 2003.
9 S. Stevi´c, “On the recursive sequencexn1xn−1/gxn,”Taiwanese Journal of Mathematics, vol. 6, no.
3, pp. 405–414, 2002.
10 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.
11 S. Stevi´c, “Existence of nontrivial solutions of a rational difference equation,”Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007.
12 S. Stevi´c, “On the recursive sequencexn1gxn, xn−1/Axn,”Applied Mathematics Letters, vol. 15,
pp. 305–308, 2002.
13 K. S. Berenhaut, J. E. Dice, J. D. Foley, B. Iriˇcanin, and S. Stevi´c, “Periodic solutions of the rational difference equationyn yn−3yn−4/yn−1,”Journal of Difference Equations and Applications, vol. 12,
no. 2, pp. 183–189, 2006.
14 S. Stevi´c, “Periodic character of a class of difference equation,”Journal of Difference Equations and Applications, vol. 10, no. 6, pp. 615–619, 2004.
15 S. Stevi´c, “On the difference equationxn1 αβxn−1γxn−2fxn−1, xn−2/xn,”Dynamics of
Continuous, Discrete & Impulsive Systems, vol. 14, no. 3, pp. 459–463, 2007.
16 M. R. S. Kulenovi´c and G. Ladas,Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002.
17 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.
18 K. S. Berenhaut and S. Stevi´c, “The behaviour of the positive solutions of the difference equation
xnA xn−2/xn−1p,”Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909–918, 2006.
19 E. Camouzis and G. Ladas, “When does periodicity destroy boundedness in rational equations?”
Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 961–979, 2006.
20 E. Camouzis and G. Ladas, “When does local asymptotic stability imply global attractivity in rational equations?”Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 863–885, 2006.
21 R. Devault, V. L. Kocic, and D. Stutson, “Global behavior of solutions of the nonlinear difference equationxn1pnxn−1/xn,”Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 707–
719, 2005.
22 J. Feuer, “On the behavior of solutions ofxn1pxn−1/xn,”Applicable Analysis, vol. 83, no. 6, pp. 599–606, 2004.
23 M. R. S. Kulenovi´c, G. Ladas, and N. R. Prokup, “On the recursive sequencexn1 αxnβxn−1/1
xn,”Journal of Difference Equations and Applications, vol. 6, no. 5, pp. 563–576, 2000.
24 S. Stevi´c, “Asymptotic behavior of a nonlinear difference equation,”Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681–1687, 2003.
25 S. Stevi´c, “On the recursive sequencexn1 αnxn−1/xn. II,”Dynamics of Continuous, Discrete &
Impulsive Systems. Series A, vol. 10, no. 6, pp. 911–916, 2003.
26 S. Stevi´c, “On the recursive sequencexn1αxpn−1/x
p
n,”Journal of Applied Mathematics & Computing,
vol. 18, no. 1-2, pp. 229–234, 2005.
27 S. Stevi´c, “On the recursive sequencexn1 αki1αixn−pi/1
m
j1βjxn−qj,”Journal of Difference
Equations and Applications, vol. 13, no. 1, pp. 41–46, 2007.
28 S. Stevi´c, “On the difference equationxn1αnxn−1/xn,”Computers & Mathematics with Applications,
vol. 56, no. 5, pp. 1159–1171, 2008.
29 S. Stevi´c and K. S. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation,”Abstract and Applied Analysis, vol. 2008, Article ID 653243, 8 pages, 2008.
30 T. Sun and H. Xi, “Global asymptotic stability of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 724–728, 2005.
31 T. Sun, H. Xi, and Z. Chen, “Global asymptotic stability of a family of nonlinear recursive sequences,”
32 T. Sun and H. Xi, “On the system of rational difference equations xn1 fyn−q, xn−s, yn1
gxn−t, yn−p,”Advances in Difference Equations, vol. 2006, Article ID 51520, 8 pages, 2006.
33 T. Sun, H. Xi, and L. Hong, “On the system of rational difference equationsxn1fyn, xn−k, yn1
fxn, yn−k,”Advances in Difference Equations, vol. 2006, Article ID 16949, 7 pages, 2006.
34 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.
35 T. Sun and H. Xi, “On the basin of attraction of the two cycle of the difference equationxn1xn−1/p
