Periodic Solutions of Generalized Difference Systems

Let $$\bar T = \left\{ {{t_0},{t_1}, \cdots } \right\}$$ denote the set of increasing time instances, and $$x:\bar T \to {\mathbb{R}^n}$$ with x(k) = (x1, x2,..., xn)(t _k ). Later in several sections we shall study a variety of problems for the difference system 2.1 $$x\left( {k + 1} \right) = {f_k}\left( {x\left( 0 \right),x\left( 1 \right), \cdots x\left( k \right)} \right),\,k \in N$$ where f _k : ℝn(k+1) → ℝn, with the dependence of f _k at the time t _k annotated in the subscript.