Bratislav D. Iri ˇcanin

(1)

Volume 2010, Article ID 891564,6pages doi:10.1155/2010/891564

Research Article

On a Higher-Order Difference Equation

Bratislav D. Iri ˇcanin

¹

and Wanping Liu

²

1Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11120, Serbia

2College of Computer Science, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Bratislav D. Iriˇcanin,[email protected] Received 20 May 2010; Accepted 23 June 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 B. D. Iriˇcanin and W. Liu. 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.

We describe in an elegant and short way the behaviour of positive solutions of the higher-order diﬀerence equationxn cxn−pxn−p−q/xn−q,n ∈ N0, wherep, q ∈ Nandc > 0, extending some recent results in the literature.

1. Introduction

Studying diﬀerence equations has attracted a considerable interest recently, see, for example, 1–39 and the references listed therein. The study of positive solutions of the following higher-order diﬀerence equations:

xnmax

A, Bx^r_n−p¹ ₁x^r_n−p² ₂· · ·x^r_n−p^k _k x^s_n−q¹ ₁x^s_n−q² ₂· · ·x^s_n−q^l _l

, n∈N0, 1.1

and

xnABx^r_n−p¹ ₁x^r_n−p² ₂· · ·x_n−p^r^k _k

x^s_n−q¹ ₁x_n−q^s² ₂· · ·x_n−q^s^l _l, n∈N0, 1.2 whereA, B > 0, pi, qiare natural numbers such thatp1 < p2 < · · · < pk, q1 < q2 < · · · < ql, ri, si ∈R, andk∈Nwas proposed by Stevi´c in several talks, see, for example,21,26. For some results concerning equations related to1.1see, for example,6,7,10,29,31,32,34,38, while some results on equations related to1.2can be found, for example, in3,8,9,11–

14,18–20,22,25,29,32,33,35 see also related references cited therein.

(2)

CaseA 0 is of some less interest, since in this case positive solutions of1.1and 1.2, by using the changeyn lnxn, become solutions of a linear diﬀerence equation with constant coeﬃcients. However, some particular results for the case recently appeared in the literature, see16,17,39.

Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order diﬀerence equation

xn cx_n−px_n−p−q

x_n−q , n∈N0, 1.3

wherep, q∈Nandc >0,in, let us say, an elegant and short way.

Let us introduce the following.

Definition 1.1. A solutionxn^∞_n−pq of1.3is said to be eventually periodic with periodτif there isn0∈ {−pq, . . . ,−1,0,1, . . .}such thatxnτ xnfor alln≥n0.Ifn0−pq, then we say that the sequencexn^∞_n−pqis periodic with periodτ.

For some results on equations all solutions of which are eventually periodic see, for example,2,4,8,15,28,37and the references therein.

Definition 1.2. One says that a solutionxn^∞_nn₀of a diﬀerence equation converges geometrically tox^∗if there existL∈Randθ∈0,1such that

|xn−x^∗| ≤Lθⁿ, ∀n≥n0. 1.4

Now we return to1.3.

First, note that ifpq, then1.3becomes

xncx_n−2p, n∈N0, 1.5

from which easily follow the following results:

aifc1, then all positive solutions of1.5are periodic with period 2p;

bif c ∈ 0,1, then each positive solution of 1.5 converges to zero. Moreover, its subsequences x2pm−i_m∈N₀, i 1,2, . . . ,2p, converges decreasingly to zero as m → ∞;

cif c ∈ 1,∞, then each positive solution of 1.5 tends to infinity as n → ∞.

Moreover, its subsequencesx2pm−i_m∈N₀, i1,2, . . . ,2p,tend increasingly to infinity asm → ∞.

We may assume thatpandqare relatively prime integers, that is, gcdp, q 1the greatest common divisor of numberspandq. Namely, if gcdp, q r >1, then by using the changeszⁱm xmri, i0,1, . . . , r−1,1.3is reduced torcopies of the following equation:

zn czn−p1zn−p1−q1

zn−q1

, n∈N0, 1.6

wherep1p/r, q1q/r, c >0, and gcdp1, q1 1.

(3)

Further, note that from1.3, we have that

xnx_n−q cx_n−px_n−p−q, n∈N0, 1.7

which implies that the sequenceunxnx_n−q, n≥ −p,satisfies the following simple diﬀerence equation:

uncun−p, n∈N0. 1.8

2. Main Results

Here we formulate and prove our main results.

Theorem 2.1. Assume thatc1, gcdp, q 1, andpis odd, then all positive solutions of 1.3are eventually periodic with periodτ 2pq.

Proof. By using repeatedly relation1.7p-times, we obtain

xn un

xn−q un

un−qxn−2q · · · un

un−q

un−2q

un−3q· · ·un−2qp−1

un−q2p−1xn−2pq. 2.1

Now, note that from1.8, it follows that in this caseunis periodic with periodp. On the other hand, since gcdp, q 1 for eachi, j∈ {0,1, . . . , p−1}, i /j, we have that

n−2i1q

− n−

2j1 q

j−i

2q /≡0

modp , n−2i2q

− n−

2j2 q

j−i

2q /≡0

modp

. 2.2

Hence, the indicesn−2i1q, i∈ {0,1, . . . , p−1}, andn−2i2q, i∈ {0,1, . . . , p−1}, belong topdiﬀerent subsequences. From this and the periodicity ofun, it follows that

unun−2q· · ·un−2qp−1 un−qun−3q· · ·un−q2p−1, 2.3

from which the theorem follows.

By taking the logarithm of1.3and using the changevnlnxn, we get

vnv_n−q−v_n−p−v_n−p−q lnc, n∈N0. 2.4

The characteristic polynomial of the homogeneous part of2.4is

λ^pqλ^p−λ^q−1 λ^q1λ^p−1 0, 2.5

(4)

from which it follows that all its roots are expressed by

exp

2k1πi q

, k0,1, . . . , q−1, exp 2lπi

p

, l0,1, . . . , p−1. 2.6

These roots are simple if and only if 2k1

q /2l

p, for eachk, l∈N0. 2.7

Clearly, ifp is odd, inequality 2.7 holds. Ifpis even, that is, p 2^sr, for some s, r ∈ N, then, since gcdp, q 1, qmust be odd. Then, fork q−1/2 andl r, we will get that inequality2.7does not hold.

From the above consideration andTheorem 2.1, we get the next corollary.

Corollary 2.2. Assume that c 1 and gcdp, q 1. Then all positive solutions of 1.3 are eventually periodic if and only ifpis odd. Moreover, ifpis odd, then the period isτ2pq.

Since the root λ 1 of characteristic polynomial 2.5 is a simple one, a particular solution of nonhomogeneous2.4has the form

v^P_n An, 2.8

from which, by a direct calculation, we easily get thatAlnc/2p.

Hence, ifpis odd, the general solution of1.3is

xne^vⁿ c^n/2pexp _q−1

k0

ck,1cos2k1πn

q ck,2sin2k1πn q

^p−1

l1

dk,1cos2lπn

p dk,2sin2lπn p

.

2.9

Note that from2.9, it follows that

xnc^n/2pxn, 2.10

and thatxnis a positive solution of1.3withc1.

From2.9,2.10, andTheorem 2.1the following results directly follow.

Theorem 2.3. Assume thatc ∈0,1, gcdp, q 1, andpis odd, then every positive solution of 1.3converges geometrically to zero. Moreover, for each s ∈ {0,1, . . . ,2pq−1}, the subsequence x_2pqms_m∈N₀converges monotonically to zero asm → ∞.

Theorem 2.4. Assume thatc >1, gcdp, q 1, andpis odd, then every positive solution of 1.3 tends to infinity. Moreover, for eachs∈ {0,1, . . . ,2pq−1}, the subsequencex2pqms_m∈N₀converges increasingly to infinity asm → ∞.

(5)

Finally, there are two concluding interesting remarks.

Remark 2.5. Note that, since the functions cos2k1πn/qand sin2k1πn/qare periodic with period 2qand the functions cos2lπn/pand sin2lπn/pare periodic with periodp, from the representation2.9we also obtainTheorem 2.1.

Remark 2.6. The results in papers 16, 17, 39, which are obtained in much complicated ways, are particular cases of our results. Namely, in16Ozban studied a system which is¨ transformed into1.3withp1, qmk1 andc1, in17he studied a system which is transformed into 1.3with p 3, andc b/a, while in39 the authors considered a system which is transformed into1.3withcb/a, but they only considered the case when p≤q.

Acknowledgments

The authors are indebted to the anonymous referees for their advice resulting in numerous improvements of the text. The research of the first author was partly supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

References

1 M. Aloqeili, “Global stability of a rational symmetric diﬀerence equation,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 950–953, 2009.

2 F. Balibrea, A. Linero, S. L ´opez, and S. Stevi´c, “Global periodicity ofxnk1 fkxnk· · ·f1xn1,”

Journal of Diﬀerence Equations and Applications, vol. 13, no. 10, pp. 901–910, 2007.

3 L. Berg, “On the asymptotics of nonlinear diﬀerence equations,” Zeitschrift f ¨ur Analysis und ihre Anwendungen, vol. 21, no. 4, pp. 1061–1074, 2002.

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

5 L. Berg and S. Stević, “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.

6 E. M. Elsayed, B. Iriˇcanin, and S. Stevi´c, “On the max-type equationxn1max{An/xn, xn−1},” Ars Combinatoria, vol. 95, no. 2, pp. 187–192, 2010.

7 E. M. Elsayed and S. Stevi´c, “On the max-type equationxn1max{A/xn, xn−2},” Nonlinear Analysis:

Theory, Methods & Applications, vol. 71, no. 3-4, pp. 910–922, 2009.

8 E. A. Grove and G. Ladas, Periodicities in Nonlinear Diﬀerence Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2005.

9 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.

10 B. D. Iriˇcanin and E. M. Elsayed, “On the max-type diﬀerence equationxn1 max{A/xn, xn−3},”

Discrete Dynamics in Nature and Society, vol. 2010, Article ID 675413, 13 pages, 2010.

11 B. Iriˇcanin and S. Stevi´c, “On a class of third-order nonlinear diﬀerence equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479–483, 2009.

12 G. L. Karakostas and S. Stevi´c, “On a diﬀerence equation with min-max response,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 53–56, pp. 2915–2926, 2004.

13 G. L. Karakostas and S. Stevi´c, “On the recursive sequencexn1Bxn−k/α0xn· · ·αk−1xn−k1γ,”

Journal of Diﬀerence Equations and Applications, vol. 10, no. 9, pp. 809–815, 2004.

14 G. L. Karakostas and S. Stevi´c, “On the recursive sequence xn1 Afxn, . . . , xn−k1/xn−k,”

Communications on Applied Nonlinear Analysis, vol. 11, no. 3, pp. 87–99, 2004.

15 R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics, vol. 26, pp. 1356–1371, 1974.

(6)

16 A. Y. ¨Ozban, “On the positive solutions of the system of rational diﬀerence equationsxn11/yn−k, yn1 yn/xn−myn−m−k,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 26–32, 2006.

17 A. Y. ¨Ozban, “On the system of rational diﬀerence equationsxn a/yn−3,yn byn−3/xn−qyn−q,”

Applied Mathematics and Computation, vol. 188, no. 1, pp. 833–837, 2007.

18 S. Stevi´c, “On the recursive sequence xn1 A/_k

i0xn−i1/2k1

jk2xn−j,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249–259, 2003.

19 S. Stevi´c, “On the recursive sequencexn1 αn xn−1/xn. II,” Dynamics of Continuous, Discrete &

Impulsive Systems. Series A, vol. 10, no. 6, pp. 911–917, 2003.

20 S. Stević, “A note on periodic character of a difference equation,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 929–932, 2004.

21 S. Stevi´c, “Some open problems and conjectures on diﬀerence equations,”

http://www.mi.sanu.ac.rs/colloquiums/mathcoll programs/mathcoll.apr2004.htm.

22 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.

23 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.

24 S. Stevi´c, “On positive solutions of ak1-th order diﬀerence equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006.

25 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.

26 S. Stevi´c, “Boundedness character of a max-type diﬀerence equation,” in Conference in Honour of Allan Peterson, p. 28, Novacella, Italy, July-August 2007.

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

28 S. Stevi´c, “On global periodicity of a class of diﬀerence equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 23503, 10 pages, 2007.

29 S. Stevi´c, “On the recursive sequencexn1A x^pn/x^r_n−1,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007.

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

31 S. Stevi´c, “On the recursive sequencexn1max{c, x^pn/x_n−1^p },” Applied Mathematics Letters, vol. 21, no.

8, pp. 791–796, 2008.

32 S. Stevi´c, “Boundedness character of a class of diﬀerence equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 839–848, 2009.

33 S. Stevi´c, “Boundedness character of a fourth order nonlinear diﬀerence equation,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2364–2369, 2009.

34 S. Stevi´c, “Global stability of a diﬀerence equation with maximum,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 525–529, 2009.

35 S. Stevi´c, “On a class of higher-order diﬀerence equations,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 138–145, 2009.

36 S. Stevi´c, “Global stability of some symmetric diﬀerence equations,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 179–186, 2010.

37 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.

38 I. Yalçinkaya, B. D. Iriˇcanin, and C. Çinar, “On a max-type difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 47264, 10 pages, 2007.

39 X. Yang, Y. Liu, and S. Bai, “On the system of high order rational diﬀerence equationsxn a/yn−p, ynbyn−p/xn−qyn−q,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 853–856, 2005.