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Asymptotic behavior of solutions for a class of delay difference equation. (English) Zbl 1076.39012

The authors study the convergence of the solutions and the existence of asymptotically stable periodic solutions for the delay difference equation

${x}_{n}=a{x}_{n-1}+\left(1-a\right)f\left({x}_{n-k}\right),\phantom{\rule{4.pt}{0ex}}n=1,2,\cdots$

Here $a\in \left(0,1\right),$ $k$ is a positive integer and $f:ℝ\to ℝ$ is a signal transmission function of the piecewise constant nonlinearity

$f\left(\xi \right)=\left\{\begin{array}{cc}1,\hfill & \xi \in \left(0,b\right],\hfill \\ 0,\hfill & \xi \in \left(-\infty ,0\right]\cup \left(b,\infty \right),\hfill \end{array}\right\$

for some constant $b>0·$

This equation can be regarded as the discrete analog of a delay differential equation with piecewise constant argument, which have wide application in biomedical models.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 92B20 General theory of neural networks (mathematical biology) 39A12 Discrete version of topics in analysis 34K13 Periodic solutions of functional differential equations