Convergence and periodicity of solutions for a class of delay difference equations
The authors investigate convergence and asymptotic stability of solutions of the \(k^{th}\) order scalar difference equation: \[ x_n=a ~x_{n-1} + (1-a) f (x_{n-k}), \quad n=0,1,2,\dots \] where \(k\) is a positive integer, \(a\in (0,1)\), \(b,c \in [0,\infty)\) with \(b<c\), and \(f:\mathbb{R}\rightarrow \mathbb{R}\) is a piecewise constant function given by: \[ f(\xi)= \begin{cases} 1 &\text{if } \xi\in(b,c] \\ 0 &\text{if } \xi\in(-\infty,b]\cup (c,\infty).\end{cases} \] By imposing conditions on the magnitudes of the parameters \(a,~b,~c\), and restrictions on the initial conditions \(x_{-1},\dots,x_{-k}\) (such as all less than or equal to \(b\), all bigger than \(b\) and less than or equal to \(c\), or all greater than \(c\)), they establish two main results. One is concerned with the convergence of solutions to either 0 or 1. The other one is concerned with the existence of asymptotically stable periodic solution of a prescribed minimal period.
scientific article; zbMATH DE number 2173419
scientific article