A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

Abstract

We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with state-dependent delay, implicitly defined as the time when the solution of a nonlinear ODE, that depends on the state of the DDE, reaches a threshold. For this application, previous results are restricted to initial histories belonging to the so-called solution manifold. We here generalize the results to a set of nonnegative Lipschitz initial histories which is much larger than the solution manifold and moreover convex. Additionally, we show that the solutions define a semiflow that is continuous in the state-component in the $C([-h,0],\R^2)$ topology, which is a variant of established differentiability of the semiflow in $C^1([-h,0],\R^2)$. For an associated system we show invariance of convex and compact sets under the semiflow for finite time.