A global convergence result with applications to periodic solutions. (English) Zbl 0971.39004

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.


39A11 Stability of difference equations (MSC2000)
39B05 General theory of functional equations and inequalities
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