Continuity in functional differential equations with infinite delay (Q1180501)

A continuity approach is presented for the equation (1) \(\dot x=f(t,x_ t)\) with infinite delay, where the function \(f\) satisfies a so-called fading memory condition. The space \(X\) of the initial functions is endowed with two topologies (the sup-norm topology and a \(g\)-norm topology). The authors investigate when (i) continuity of the solution \(x(t,0,\varphi)\) in \(\varphi\) (with respect to the \(g\)-topology) is equivalent to uniqueness, and (ii) \(x(t,0,\varphi)\) is jointly continuous in \((t,\varphi)\). An application is given for the Poincaré mapping generated from the solutions of equation (1).