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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 $$ x_{n}=ax_{n-1}+\left( 1-a\right) f\left( x_{n-k}\right) ,\text{ }n=1,2,\dots $$ Here $a\in\left( 0,1\right) ,$ $k$ is a positive integer and $f:\Bbb R\rightarrow\Bbb R$ is a signal transmission function of the piecewise constant nonlinearity $$ f\left( \xi\right) =\cases 1, &\xi\in(0,b],\\ 0, &\xi\in(-\infty,0]\cup\left( b,\infty\right) , \endcases $$ 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.

39A11Stability of difference equations (MSC2000)
92B20General theory of neural networks (mathematical biology)
39A12Discrete version of topics in analysis
34K13Periodic solutions of functional differential equations