Global behavior of a higher-order rational difference equation
- PDF / 831,897 Bytes
- 8 Pages / 467.717 x 680.315 pts Page_size
- 61 Downloads / 214 Views
We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk (xn i0 ,xn i1 ,...,xn i2k ) + b)/(Qk (xn i0 ,xn i1 ,...,xn i2k ) + b), n = 0,1,..., under appropriate assumptions, where b [0, ), k 1, i0 ,i1 ,...,i2k 0,1,... with i0 < i1 < < i2k , the initial conditions xi 2k ,xi 2k +1 ,...,x0 (0, ). We prove that unique equilibrium x = 1 of that equation is globally asymptotically stable. Copyright © 2006 H. Xi and T. Sun. 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 For some difference equations, although their forms (or expressions) look very simple, it is extremely difficult to understand thoroughly the global behaviors of their solutions. Accordingly, one is often motivated to investigate the qualitative behaviors of difference equations (e.g., see [2, 3, 6, 9, 10]). In [6], Ladas investigated the global asymptotic stability of the following rational difference equation: (E1) xn+1 =
xn + xn 1 xn xn xn 1 + xn
2
,
2
n = 0,1,...,
(1.1)
where the initial values x 2 ,x 1 ,x0 R+ (0,+). In [9], Nesemann utilized the strong negative feedback property of [1] to study the following difference equation: (E2) xn+1 =
xn 1 + xn xn xn xn 1 + xn
where the initial values x 2 ,x 1 ,x0 Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 27637, Pages 1–7 DOI 10.1155/ADE/2006/27637
R+ .
2 2
,
n = 0,1,...,
(1.2)
2
Global behavior of a difference equation
In [10], Papaschinopoulos and Schinas investigated the global asymptotic stability of the following nonlinear difference equation: (E3)
xn+1 =
iZk
j
1, j xn i + xn j xn j+1 + 1
iZk xn i
n = 0,1,...,
,
(1.3)
where k 1,2,3,..., j, j 1 Zk 0,1,...,k, and the initial values x k ,x k+1 ,..., x 0 R+ . Recently, Li [7, 8] studied the global asymptotic stability of the following two nonlinear difference equations: (E4) xn+1 =
xn 1 xn 2 xn 3 + xn 1 + xn 2 + xn 3 + a , xn 1 xn 2 + xn 1 xn 3 + xn 2 xn 3 + 1 + a
n = 0,1,...
(1.4)
(E5) xn+1 =
xn xn 1 xn 3 + xn + xn 1 + xn 3 + a , xn xn 1 + xn xn 3 + xn 1 xn 3 + 1 + a
n = 0,1,...,
(1.5)
where a [0,+) and the initial values x 3 ,x 2 ,x 1 ,x0 R+ . Let k 1 and i0 ,i1 ,...,i2k 0,1,... with i0 < i1 < < i2k . Let P0 (xn Q0 (xn i0 ) = 1, for any 1 j k, let
P j xn
i0 ,...,xn i2 j
= xn
i2 j x n i2 j
+ xn
Q j xn
i0 ,...,xn i2 j
= xn
i2 j
+ xn
i2 j x n i2 j
+ xn
i2 j
1
+ 1 Pj i2 j
xn
1
Qj
1
1
+ xn
+ 1 Qj i2 j
1
1
Pj
xn
1
xn
1
xn
i0 ,...,xn i2 j
i0 )
i0 ,...,xn i2 j
i0
and
2
i0 ,...,xn i2 j i0 ,...,xn i2 j
= xn
,
2
(1.6)
2
2
.
In this paper, we consider the following difference equation:
Pk xn i ,xn i1 ,...,xn xn+1 = 0 Qk xn i0 ,xn i1 ,...,xn
i2k i2k
+b ,
+b
n = 0,1,...,
(1.7)
where b [0, ) and the initial conditions x i2k ,x i2k +1 ,
Data Loading...
