# zbMATH — the first resource for mathematics

On a difference equation with min-max response. (English) Zbl 1070.39013
The authors study the global behavior of the positive solutions of the difference equation $x_{n+1}=\alpha _{n}+F(x_{n},\dots,x_{n-k}),\quad n=0,1,2,\dots,$ where the initial conditions $$x_{-k},\dots,x_{0}$$ are positive real numbers, $$\{\alpha _{n}\}$$ is a sequence of positive real numbers, and $$F$$ is a min-max function. A min-max function is defined by the authors as follows: A function $$F:\mathbb{R}_{+}^{k+1}\rightarrow \mathbb{R}_{+}$$ is called a min-max function if it satisfies the inequality $\frac{\min_{j}u_{j}}{\max_{j}u_{j}}\leq F(u_{1},u_{2},\dots,u_{k+1})\leq \frac{\max_{j}u_{j}}{\min_{j}u_{j}}$ for all $$u_{j}>0,~j=1,\dots,k+1.$$ An exact characterization of min-max functions is given. One of the main results that is obtained is the following
Theorem: Consider the equation $x_{n+1}=\alpha_n+\frac{\sum_{i=0}^ma_ix_{n-2i-1}}{\sum_{i=0}^mb_ix_{n-2i}}~\;,\tag{$$*$$}$ where $$m\in \mathbb{N}$$, $$\{\alpha_n\}$$ is a sequence of positive real numbers such that $$\lim_{n\to\infty}\alpha_n=:A\in [ 0,1),$$ and where the coefficients $$a_j$$ and $$b_j$$, $$j=0,\dots,m$$ are nonnegative constants which satisfy the conditions $$\sum_{i=0}^ma_i=\sum_{i=0}^mb_i$$. Then there exist unbounded solutions of ($$*$$).

##### MSC:
 39A11 Stability of difference equations (MSC2000)
Full Text: