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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 +1-afx n-k ,n=1,2,

Here a0,1, k is a positive integer and f: is a signal transmission function of the piecewise constant nonlinearity

fξ=1,ξ(0,b],0,ξ(-,0]b,,

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