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 u ( t ) = U ( t , s ) u ( s ) + ∫ s t U ( t , ξ ) f ( ξ , u ( ξ ) ) d ξ . 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 , x 1 ) − f ( t , x 2 ) ‖ ⩽ φ ( t ) ‖ x 1 − x 2 ‖ 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 L p type, the Lorentz spaces L p , q and many other function spaces occurring in interpolation theory.