We give sufficient conditions under which every positive solu-tion of this equasolu-tion converges to a not necessarily prime 2-periodic solution, which extends and includes correspondi
Trang 1Volume 2008, Article ID 143723, 6 pages
doi:10.1155/2008/143723
Research Article
The Periodic Character of the Difference Equation
x n1 fx n−l1 , x n−2k1
Taixiang Sun 1 and Hongjian Xi 2
1 Department of Mathematics, College of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, China
2 Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003,
Guangxi, China
Received 3 February 2007; Revised 18 September 2007; Accepted 27 November 2007
Recommended by H Bevan Thompson
In this paper, we consider the nonlinear difference equation xn1 fx n−l1 , x n−2k1 , n 0, 1, , where k, l ∈ {1, 2, } with 2k / l and gcd 2k, l 1 and the initial values x −α , x −α 1, , x0 ∈
0, ∞ with α max{l − 1, 2k − 1} We give sufficient conditions under which every positive
solu-tion of this equasolu-tion converges to a not necessarily prime 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.
Copyright q 2008 T Sun and H Xi 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 consider a nonlinear difference equation and deal with the question of whether every positive solution of this equation converges to a periodic solution Recently, there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodic nature of nonlinear difference equations e.g., see 1,2 In 3 , Grove et al considered the following difference equation:
x n1 p x n−2m1
1 x n−2r , n 0, 1, , E1 where p ∈ 0, ∞ and the initial values x −α , x −α1 , , x0∈ 0, ∞ with α max {2r, 2m 1},
and proved that every positive solution of E1 converges to not necessarily prime a 2s-periodic solution with s gcd m 1, 2r 1 In 4 , Stevi´c investigated the periodic character
of positive solutions of the following difference equation:
x n1 1 x x n−2s1
n−2r1s1 , n 0, 1, , E2
Trang 2and proved that every positive solution of E2 converges to not necessarily prime a
2s-periodic solution, which generalized the main result of5 Furthermore, Stevi´c 6 studied the periodic character of positive solutions of the following difference equation:
x n 1
k
i1 α i x n−p i
m
j1 β j x n−q j , n 1, 2, , E3
where α i , i ∈ {1, , k}, and β j , j ∈ {1, , m}, are positive numbers such that Σ k i1 α i Σm j1 β j
1, and p i , i ∈ {1, , k}, and q j , j ∈ {1, , m}, are natural numbers such that p1< p2< · · · < p k
and q1< q2< · · · < q m For closely related results, see7,8
In this paper, we consider the more general equation
x n1 fx n−l1 , x n−2k1
, n 0, 1, 2, , 1.1
where k, l ∈ {1, 2, } with 2k / l and gcd 2k, l 1, the initial values x −α , x −α1 , , x0 ∈
0, ∞ with α max {l − 1, 2k − 1}, and f satisfies the following hypotheses:
H1 f ∈CE×E, 0,∞ with ainf u,v∈E×E fu, v∈E, where E∈{0,∞, 0,∞};
H2 fu, v is decreasing in u and increasing in v;
H3 there exists a decreasing function g ∈ Ca, ∞, a, ∞ such that
i for any x > a, ggx x and x fgx, x;
ii limx→agx ∞ and lim x→∞ gx a.
The main result of this paper is the following theorem
Theorem 1.1 Every positive solution of 1.1 converges to (not necessarily prime) a 2-periodic solu-tion.
2 Proof of Theorem 1.1
In this section, we will proveTheorem 1.1 Without loss of generality, we may assume l < 2k
the proof for the case l > 2k is similar; then
{l, 2l, 3l, , 2kl} {0, 1, 2, , 2k − 1} mod 2k. 2.1
Lemma 2.1 Let {x n}∞n−α be a positive solution of 1.1 Then there exists a real number L ∈ a, ∞ such that L ≤ x n ≤ gL for all n ≥ 1 Furthermore, let lim sup x n M and lim inf x n m, then
M gm and m gM.
Proof ByH1 and H2, we have
x i fx i−l , x i−2k
> f
x i−l 1, x i−2k
≥ a for every 1 ≤ i ≤ α 1. 2.2
Then there exists L ∈ a, ∞ with L < gL such that
L ≤ x i ≤ gL for every 1 ≤ i ≤ α 1. 2.3
Trang 3It follows from2.3 and H3 that
gL fL, gL≥ x α2 fx α2−l , x α2−2k
≥ fgL, L L. 2.4
Inductively, it follows that L ≤ x n ≤ gL for all n ≥ 1.
Let lim sup x n M and lim inf x n m, then there exist A, B, C, D ∈ m, M and seque-nces t n ≥ 1 and r n≥ 1 such that
lim
n→∞ x t n M, lim
n→∞ x t n −l A, lim
n→∞ x t n −2k B,
lim
n→∞ x r n m, lim
n→∞ x r n −l C, lim
n→∞ x r n −2k D. 2.5
Thus by1.1, H2, and H3, we have
f
gM, M M fA, B ≤ fm, M,
f
gm, m m fC, D ≥ fM, m, 2.6 from which it follows that gM ≥ m and gm ≤ M Since g is decreasing, it follows that
m ggm≥ gM, M ggM≤ gm. 2.7
Therefore, M gm and m gM The proof is complete.
Proof of Theorem 1.1 Let {x n}∞n−α be a positive solution of 1.1 with the initial conditions
x0, x−1, , x −α ∈ 0, ∞ It follows fromLemma 2.1that
a < lim inf x n m gM ≤ lim sup x n M < ∞. 2.8 Obviously, every sequence
L, gL, L, gL, 2.9
is a 2-periodicnot necessarily prime solution of 1.1, where L ∈ {M, m}.
By taking a subsequence, we may assume that there exists a sequence t n ≥ 2kl 1 such
that
lim
n→∞ x t n M,
lim
n→∞ x t n −j A j ∈gM, M for j ∈ {1, 2, , 2kl}. 2.10
According to1.1, 2.10, and H3, we obtain
f
gM, M M fA l , A 2k
≤ fgM, M, 2.11 from which it follows that
Trang 4In a similar fashion, we can obtain
f
gM, M M A 2k fA 2kl , A 4k
≤ fgM, M,
f
M, gM gM A l fA 2l , A l2k
≥ fM, gM, 2.13 from which it follows that
A 4k A 2k A 2l M, A 2kl A l gM. 2.14 Inductively, we have
A j2k M for j ∈ {1, 2, , l},
A jl gM for j ∈ {1, 3, , 2k − 1},
A jl M for j ∈ {0, 2, , 2k},
A jlr2k A jl for j ∈ {0, 1, , 2k}, r ∈ {0, 1, , l}, jl r2k ≤ 2kl.
2.15
For every r ∈ {0, 1, 2, 3, , 2k − 1}, there exist j r ∈ {0, 1, 2, 3, , 2k − 1} and p r ∈
{0, 1, , l − 1} such that j r l 2kp r r, from which, with 2.15, it follows that
A 2kl−1r A j r l
M for r ∈ {0, 2, 4, , 2k − 2}, gM for r ∈ {1, 3, , 2k − 1}, 2.16
lim
n→∞ x t n −2kl−1−j M for j ∈ {0, 2, , 2k},
lim
n→∞ x t n −2kl−1−j gM for j ∈ {1, 3, , 2k − 1}. 2.17
In view of2.17, for any 0 < ε < M − a, there exists some t β ≥ 4kl such that
M − ε < x t β −2kl−1−j < M ε if j ∈ {0, 2, , 2k}, gM ε < x t β −2kl−1−j < gM − ε if j ∈ {1, 3, , 2k − 1}. 2.18
By1.1 and 2.18, we have
x t β −2kl−11 fx t β −2kl−1−l1 , x t β −2kl1
< f
M − ε, gM − ε gM − ε. 2.19 Also1.1, 2.18, and 2.19 imply that
x t β −2kl−12 fx t β −2kl−1−l2 , x t β −2kl2
> f
gM − ε, M − ε M − ε. 2.20 Inductively, it follows that
x t β −2kl−12n > M − ε ∀n ≥ 0,
Trang 5lim
n→∞ x 2n M, lim
or
lim
n→∞ x 2n gM, lim
The proof is complete
Remark 2.2. 1 The proofs ofLemma 2.1 andTheorem 1.1draw on ideas from the proofs of Theorems 2.1 and 2.2 in6
2 Consider the nonlinear difference equation
x n1 fx n−ls1 , x n−2ks1
, n 0, 1, , 2.24
where s, k, l ∈ {1, 2, } with 2k / l and gcd 2k, l 1, the initial values x −α , x −α1 , , x0 ∈
0, ∞ with α max {ls − 1, 2ks − 1}, and f satisfies H1–H3 Let y i
n1 x nsi1 for every
0≤ i ≤ s − 1 and n 0, 1, 2, , then 2.24 reduces to the equation
y i n1 fy i
n−l1 , y i n−2k1
, 0 ≤ i ≤ s − 1, n 0, 1, 2, 2.25
It follows fromTheorem 1.1that for any 0≤ i ≤ s − 1, every positive solution of the equation
y i
n1 fy i
n−l1 , y i
n−2k1 converges to not necessarily prime a 2-periodic solution Thus every positive solution of2.24 converges to not necessarily prime a 2s-periodic solution.
3 Examples
To illustrate the applicability ofTheorem 1.1, we present the following examples
Example 3.1 Consider the equation
x n1 p
m1
i1 x i n−2k1
m
i0 x i n−2k1 x n−l1 , n 0, 1, , 3.1
where m, k, l ∈ {1, 2, } with 2k / l and gcd 2k, l 1 and the initial values x −α , x −α1 , ,
x0∈ 0, ∞ with α max {l − 1, 2k − 1}, 0 < p ≤ 1 Let E 0, ∞ and
fx, y p
m1
i1 y i
m
i0 y i x x ≥ 0, y ≥ 0, gx
p
x x > 0. 3.2
Trang 6It is easy to verify thatH1–H3 hold for 3.1 It follows fromTheorem 1.1that every solution
of3.1 converges to not necessarily prime a 2-periodic solution
Example 3.2 Consider the equation
x n1 1 x
m1 n−2k1
m
i1 x i n−2k1 x n−l1 , n 0, 1, , 3.3
where m, k, l ∈ {1, 2, } with 2k / l and gcd 2k, l 1 and the initial values x −α , x −α1 , ,
x0∈ 0, ∞ with α max {l − 1, 2k − 1} Let E 0, ∞ and
fx, y 1 m y m1
i1 y i x x > 0, y > 0, gx
x
x − 1 x > 1. 3.4
It is easy to verify thatH1–H3 hold for 3.3 It follows fromTheorem 1.1that every solution
of3.3 converges to not necessarily prime a 2-periodic solution
Acknowledgments
The authors would like to thank the referees for some valuable and constructive comments and suggestions The project is supported by NNSF of China10461001 and NSF of Guangxi
0640205, 0728002
References
1 S Stevo, “The recursive sequence x n1 gx n , x n−1 /Ax n ,” Applied Mathematics Letters, vol 15, no 3,
pp 305–308, 2002.
2 R DeVault, C Kent, and W Kosmala, “On the recursive sequence x n1 p x n−k /x n ,” Journal of Di ffer-ence Equations and Applications, vol 9, no 8, pp 721–730, 2003.
3 E A Grove, G Ladas, and M Predescu, “On the periodic character of the difference equation x n1
px n−2l x n−2k1 /1 x n−2l ,” Mathematical Sciences Research Journal, vol 6, no 5, pp 221–233, 2002.
4 S Stevi´c, “A note on periodic character of a difference equation,” Journal of Difference Equations and
Applications, vol 10, no 10, pp 929–932, 2004.
5 W T Patula and H D Voulov, “On the oscillation and periodic character of a third order rational difference equation,” Proceedings of the American Mathematical Society, vol 131, no 3, pp 905–909, 2003.
6 S Stevi´c, “On the recursive sequence x n 1 k i1 α i x n−p i /m
j1 β j x n−q j ,” Discrete Dynamics in Nature
and Society, vol 2007, Article ID 39404, 7 pages, 2007.
fx n , x n−k ,” Computers & Mathematics with Applications, vol 51, no 9-10, pp 1431–1436, 2006.
8 K S Berenhaut, J D Foley, and S Stevi´c, “The global attractivity of the rational difference equation
y n 1 y n−k /y n−m ,” Proceedings of the American Mathematical Society, vol 135, no 4, pp 1133–1140,
2007.