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Zbl 1076.39012
Zhu, Huiyan; Huang, Lihong
Asymptotic behavior of solutions for a class of delay difference equation. (English)
[J] Ann. Differ. Equations 21, No. 1, 99-105 (2005). ISSN 1002-0942

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 $b>0.$ \par 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.
[N. C. Apreutesei (Iaşi)]

MSC 2000:: *39A11 Stability of difference equations
92B20 General theory of neural networks
39A12 Discrete version of topics in analysis
34K13 Periodic solutions of functional differential equations

Keywords: delay difference equation; piecewise constant nonlinearity; asymptotic stability; periodic solutions; convergence; biomedical models

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