Existence of an almost periodic solution in a difference equation by Liapunov functions (Q2783626)

An existence theorem for an almost periodic solution in the case of an almost periodic difference equation NEWLINE\[NEWLINEx_{n+1}=f(n,x_n)NEWLINE\]NEWLINE is obtained by assuming the existence of a Lyapunov function. The function \(f:\mathbb{Z}\times D\to \mathbb{R}^m: (n,x)\mapsto f(n,x)\) is continuous in \(x\), and almost periodic in \(n\) uniformly for \(x\) belonging to the open set \(D\subset \mathbb{R}^m\). This means that for any \(\varepsilon >0\) and any compact set \(K\) in \(D\) there exists a positive integer \(L\) such that any interval of length \(L\) contains an integer \(\tau \) such that NEWLINE\[NEWLINE\|f(n+\tau ,x)-f(n,x)\|\leq \varepsilonNEWLINE\]NEWLINE for all \(n\in \mathbb{Z}\) and all \(x\in K\).