Consider an evolution family U = (U (t, s)) t s 0 on a half-line R + and an integral equationWe characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.

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Consider an evolution family U = (U (t, s)) t s 0 on a half-line R + and a semi-linear integral equationWe prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p,q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U (t, s)) t s 0 has an exponential dichotomy and the nonlinear forcing term f (t, x) satisfies the nonuniform Lipschitz conditions: f (t, x 1 ) − f (t, x 2 ) ϕ(t) x 1 − x 2 for ϕ being a real and positive function which belongs to certain classes of admissible function spaces.

We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0 -semigroup on C 0 (R − ,X) yielding the solutions of the equation.