Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))t⩾s⩾0 on a half-line R+ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))t⩾s⩾0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x1)−f(t,x2)‖⩽φ(t)‖x1−x2‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.