Exponential dichotomies, invariant splitting and the existence of wave-front like solutions for a model of population genetics (Q1374842)

Consider the non-autonomous system \((*)\) \(dz / dt = g(t,z)\) under the assumption that \((*)\) has a non-trivial solution \(z=u(t)\) uniformly bounded such that \(dy/dt = g_z (t,u(t))y\) has an exponential dichotomy on \(\mathbb{R}\). The author derives conditions such that the perturbed system \(dz/dt=g(t,z) + \varepsilon h(t,z,\varepsilon)\) has a unique solution \(\overline{u} (t,\varepsilon)\) which is near \(u(t)\) for small \(\varepsilon\). He uses the theory of linear skew product dynamical systems and the splitting index introduced by Sacker. As an application, the existence of a solution of wave front type for a parabolic partial differential equation modelling a problem of population genetics is established.