(Xn)

Applied Mathematics and Computation 109 (2000) 301±306 www.elsevier.nl/locate/amc

A note on the periodic cycle of Xn‡2 ˆ …1 ‡ Xn‡1†=…Xn† H.M. El-Owaidy, M.M. El-A®®

*

Mathematics Department, Faculty of Science Al-Azhar University, Nasr City, Cairo 11884, Egypt

Abstract n xn The behaviour of solutions of the di€erence equation xn‡1 ˆ anx‡b ; n ˆ 0; 1; . . . ; nÿ1 are investigated, where the sequences an and bn are periodic. Ó 2000 Elsevier Science Inc. All rights reserved.

AMS classi®cation: 39A10; 39A99; 34C99 Keywords: Di€erence equations; Oscillatory solutions

1. Introduction Kocic [1] has studied the behaviour of solutions of the di€erence equation xn‡1 ˆ

1 ‡ xn ‡ xnÿ1 ‡


‡ xnÿk‡2 ; xnÿk‡1

n ˆ 0; 1; 2; . . . ;

…1:1†

where k is a nonnegative integer and xÿk ; . . . ; x0 are arbitrary positive numbers. The special cases of Eq. (1.1) for k ˆ 0; 1 and 2 are:

*

xn‡1 ˆ

1 ; xn

xn‡1 ˆ

1 ‡ xn ; xnÿ1

n ˆ 0; 1; . . . ; n ˆ 0; 1; . . . ;

Corresponding author.

0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 0 3 1 - 4

…1:2† …1:3†

302

H.M. El-Owaidy, M.M. El-A®® / Appl. Math. Comput. 109 (2000) 301±306

xn‡1 ˆ

1 ‡ xn ‡ xnÿ1 ; xnÿ2

n ˆ 0; 1; . . . ;

…1:4†

respectively. They have the property that every solution of each equation is periodic with period 2, 5 and 8, respectively, that is, for k ˆ 0, 1 and 2 the positive solutions of Eq. (1.1) are all periodic with period (3k + 2). The 5-cycle Eq. (1.3) was discovered by Lyness [2,3], while he was working on a problem in number theory. This equation has also applications in geometry [4]. In this paper we shall investigate the behaviour of the di€erence equation an ‡ bn x n ; n ˆ 0; 1; . . . ; xnÿ1 where an and bn are periodic. xn‡1 ˆ

…1:5†

De®nition 1. A sequence fxn g is said to oscillate about zero or simply to oscillate if the terms xn are neither eventually all positive nor eventually all negative. Otherwise the sequence is called nonoscillatory. De®nition 2. A sequence fxn g is called strictly oscillatory if for every n0 P 0, there exists n1 ; n2 P n0 such that xn1 xn2 < 0. De®nition 3. A sequence fxn g is said to oscillate about x if the sequence xn ÿ x oscillates, and it is called strictly oscillatory about x if the sequence xn ÿ x is strictly oscillatory. De®nition 4. A positive semicycle of fxn g of Eq. (1.5) consists of `string' of terms fxl ; xl‡1 ; . . . ; xm g all greater than or equal to x, with l P ÿ 1 and m < 1 and such that either

l ˆ ÿ1 or l > ÿ1 and xlÿ1 < x

either

m ˆ 1 or m < 1 and xm‡1 < x:

and

De®nition 5. A negative semicycle of fxn g of Eq. (1.5) consists of `string' of terms fxl ; xl‡1 ; . . . ; xm g all less than x, with l P ÿ 1 and m < 1 and such that either

l ˆ ÿ1 or l > ÿ1 and xlÿ1 P x

either

m ˆ 1 or m < 1 and xm‡1 P x:

and The ®rst semicycle of a solution of Eq. (1.5) starts with the term xÿ1 and is positive if xÿ1 P x and negative if xÿ1 < x.

H.M. El-Owaidy, M.M. El-A®® / Appl. Math. Comput. 109 (2000) 301±306

303

A solution may have a ®nite number of semicycles or in®nitely many semicycles. De®nition 6. A sequence fxn g is said to be periodic with period p if xn‡p ˆ xn

for

n ˆ 0; 1; . . .

…1:6†

De®nition 7. A sequence fxn g is said to be periodic with prime period p, or with minimal period p, if it is periodic with period p, where p is the least positive integer for which (1.6) holds. De®nition 8. A di€erence equation is called a p-cycle, if every solution of it is periodic with period p. Consider the di€erence equation xn‡1 ˆ

a ‡ xn ‡


‡ xnÿk‡2 ; xnÿk‡1

n ˆ 0; 1; . . . ;

…1:7†

where a 2 ‰0; 1† and

k 2 f1; 2; . . .g:

…1:8†

The following theorems show that the solution of Eq. (1.7) is bounded from below and from above by positive constants. Theorem A [1]. Assume that (1.8) holds and that xÿk‡1 ; . . . ; x0 are arbitrary positive numbers then the soultion fxn g of equation (3.1) is such that " #   kÿ1 k ÿ1 Y X 1 xn…k‡1†‡j 1‡ ˆ constant; n ˆ 0; 1; . . . …1:9† a‡ xn…k‡1†‡j jˆ0 jˆ0 We note that, for a ˆ 1 and k ˆ 2, the invariance (1.9) appears in [3]. For a ˆ 1 and k > 2, (1.5) is due to [5]. Theorem A shows also, the solution xn is such that m 6 xn 6 M, where m and M are positive numbers. Theorem B [1]. Assume that a; b0 ; . . . ; bkÿ1 2 …0; 1† with a‡

kÿ1 X

bi > 0

and

k 2 f1; 2; . . .g

iˆ0

then every nontrivial solution of the equation, xn‡1 ˆ

a ‡ b0 xn ‡


‡ bkÿ1 xnÿk‡1 ; xnÿk

n ˆ 0; 1; . . .

…1:10†

304

H.M. El-Owaidy, M.M. El-A®® / Appl. Math. Comput. 109 (2000) 301±306

is strictly oscillatory about the positive equilibrium x of this equation. Furthermore, every semicycle of a nontrivial solution contains no more than (2k + 1) terms. For the di€erence equation a ‡ bxn ; n ˆ 0; 1; . . . …1:11† xnÿ1 when a ˆ 0 and b 2 …0; 1†, every solution of Eq. (1.11) is periodic with period six. When a 2 …0; 1† and b ˆ 0, every solution of Eq. (1.11) is periodic with period four. if a; b 2 …0; 1†, by change of variables, xn ˆ byn reduces Eq. (1.11) to xn‡1 ˆ

a ‡ xn ; xnÿ1 where a ˆ a=b2 . xn‡1 ˆ

n ˆ 0; 1; . . .

…1:12†

Theorem C [1]. If a; b 2 …0; 1†, then every positive solution of Eq. (1.11) is bounded away from zero and in®nity by positive constants, and every nontrivial solution of this equation is strictly oscillatory about the positive equilibrium x of Eq. (1.11). Furthermore, every semicycle of nontrivial solution contains at most three terms. Theorem D [1]. If a; b 2 …0; 1†, then the following statements about the positive solutions of Eq. (1.11) are true: (a) The absolute extreme in a semicycle occurs in the ®rst or in the second term. (b) Every nontrivial semicycle, after the ®rst one, contains at least two and at most three terms. Theorem E [1]. Assume that a P 0 and let fxn g be a positive solution of Eq. (1.12) then for n ˆ 0; 1; . . . ; the following statments are true: …1 †, (i) yn‡5 ÿ yn ˆ …1 ÿ a†……1=yn‡2 † ÿ yn‡3 † (ii) yn‡6 ÿ yn ˆ a…1 ‡ …1=yn‡3 ††……1=yn‡4 † ÿ …1=yn‡2 ††: This theorem shows that, the solutions of Eq. (1.12) are periodic with period ®ve i€ a ˆ 1, and are periodic with period six i€ a ˆ 0. 2. Main result In this section we investigate the behaviour of solutions of the di€erence equation xn‡1 ˆ

an ‡ bn x n ; xnÿ1

n ˆ 0; 1; . . .

…2:1†

H.M. El-Owaidy, M.M. El-A®® / Appl. Math. Comput. 109 (2000) 301±306

305

with 1 X 1 < 1; b nˆ0 n

…2:2†

where fan g and fbn g are periodic sequences, Lemma 2.1. If the sequence fbn g satis®es bn‡1 bnÿ1 ˆ 1; n ˆ 0; 1; . . . ; then fbn g is periodic with period four. Proof. Since bn‡1 bnÿ1 ˆ 1, then bn‡1 ˆ

1 bnÿ1

and also bn‡2 ˆ

1 bn

and

bn‡3 ˆ

1 bn‡1

which gives bn‡4 ˆ

1 1 ˆ ˆ bn bn‡2 1=bn

and hence fbn g is periodic with period four. Theorem 2.1. If the sequence fbn g is periodic with period four and fan g is periodic with period one, then all results in Theorems D and E are true for the solution of Eq. (2.1). Proof. Since fan g is periodic with period one, then an‡1 ˆ an , which has a solution an ˆ constant, say a. The change of variables xn ˆ bn yn reduces Eq. (2.1) to yn‡1 ˆ bn‡1 bnÿ1

a ‡ yn : ynÿ1

…2:3†

Since fbn g is periodic with period four, then by using Lemma 2.1, Eq. (2.3) takes the form yn‡1 ˆ

a ‡ yn ynÿ1

…2:4†

which has the same form of Eq. (1.12) and has the same results of Theorems D and E.

306

H.M. El-Owaidy, M.M. El-A®® / Appl. Math. Comput. 109 (2000) 301±306

References [1] V.L. Kocic, G. Ladas, I.W. Rodrigues, On rational recursive sequences, J. Math. Anal. Appl. 173 (1993) 127±157. [2] R.C. Lyness, Note 1581, Math. Gaz 26 (1942) 62. [3] R.C. Lyness, Note 1847, Math. Gaz 29 (1945) 231. [4] J. Leech, The rational cubiod revisited, Amer. Math. Monthly 84 (1977) 518±533. [5] Y. Jianslle, H. Xuli, Private communication.