Functional differential equations and topological dynamics (Q1178377)

Let \(W\) denote the set of continuous functions from \([-r,0]\) to \(\mathbb{R}^ n\) equipped with the supremum norm, where \(r\) is some positive constant. The author considers the functional differential equation \(x'(t)=f(t,x_ t)\) where \(x_ t(\theta)=x(t+\theta)\) for \(-r\leq\theta\leq 0\) and where the functional \(f: \mathbb{R}^ +\times W\to\mathbb{R}^ n\) is continuous and uniformly continuous on each bounded and closed subset of \(\mathbb{R}^ +\times W\). Introducing a new kind of topology, called the bounded-closed topology he constructs for the above equation a dynamical system which is analogous to that constructed earlier by G. R. Sell for nonautonomous systems of ordinary differential equations.