## 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.

### MSC:

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

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