Abstract
We investigate semilinear evolution equations of the form u(t)=U(t,s)u(s)+∫ stU(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 semilinear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function f satisfies the condition ϕ-Lipschitz and there 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.
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