On a generalized difference system. (English) Zbl 0894.39001

Let \(\overline T = \{t_0,t_1,\ldots \}\) denote the set of increasing time instances, and \(x: \overline T \to \mathbb R^n\) with \(x(k)=(x^1,x^2,\ldots,x^n)(t_k)\). Consider the difference system \[ x(k+1)=f_k(x(0),x(1),\ldots,x(k)), k \in \mathbb N = \{0,1,2,\ldots \} \tag{1} \] where \(f_k:\mathbb R^{n(k+1)} \to \mathbb R^n \), with the dependence of \(f_k\) at the time \(t_k\) annotated in the subscript. The system (1) is very general and in particular includes the prototype equation \(x(k+1)=f(k,x(k))\), equations with finite as well as infinite delays, equations of neutral type, and the discrete integral equations of Volterra type. The paper contains a survey of recent results for the above system obtained by the authors.


39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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