Admissible, bounded and periodic solutions of semilinear evolution equations on the line

In this paper, we investigate semi-linear evolution equations having the following form u(t) = U(t,s)u(s) + ∫ t s U(t,ξ ) f (ξ ,u(ξ ))dξ for t ≥ s and s ∈ R in Banach space X . Under the assumptions that the evolution family (U(t,s))t≥s has the exponential dichotomy and the function f : R×X → X has the Carathéodory property, we show that the semi-linear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function f satisfies the condition φ-Lipschitz and exists a periodic solution when the function f satisfies the condition ‖ f (t,x)‖ ≤ φ(t)(1+‖x‖) for all x ∈ X and almost everywhere t ∈ R.