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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)
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