an:01449169
Zbl 0971.39004
El-Metwally, H.; Grove, E. A.; Ladas, G.
A global convergence result with applications to periodic solutions
EN
J. Math. Anal. Appl. 245, No. 1, 161-170 (2000).
00065223
2000
j
39A11 39B05
rational recurrence relation; periodic solution; global convergence
Suppose that \(I\) is an interval of \(\mathbb{R}\), that \(F=F(z_1, \dots, z_{k+1}): I^{k+1}\to I\) is continuous, nondecreasing in each of its arguments, and that it is strictly increasing in at least two of its arguments \(z_i\) and \(z_j\) where \(i\) and \(j\) are relatively prime. Suppose further that \(F(x,x, \dots,x) =x\) for every \(x\in I\). Then it is shown that every solution of
\[
x_{n+1}= F(x_n,x_{n-1}, \dots,x_{n-k}),\;n=0,1, \dots
\]
tends to a finite limit in \(I\).
This result is applied to the rational recursive relation
\[
x_{n+1}= {A_0\over x_n} +{A_1\over x_{n-2}}+ \cdots+ {A_m\over x_{n-2m}},\;n=0,1, \dots,\tag{1}
\]
where \(m\) is a positive integer, \(A_0,\dots,A_m\) are nonnegative and at least two of them are positive. It is shown that if \(0\leq i<j\leq m\), if \(2i+1\) and \(2j+1\) are relatively prime, and if \(A_i\) and \(A_j\) are positive, then every positive solution of (1) converges to a period two solution.
Sui Sun Cheng (Hsinchu)