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For a given closed and translation invariant subspace Yof the bounded and uniformly continuous functions, we will give criteria for the existence of solutions u∈Yto the equation u′(t)+A(u(t))+ωu(t)∍f(t),t∈ℝ, or of solutions u asymptotically close to Y for the inhomogeneous differential equation u′(t)+A(u(t))+ωu(t)∍f(t),t>0,u(0)=u0, in general Banach spaces, where Adenotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space Yare spaces of functions with various almost periodicityproperties and more general types of asymptotic behavior

Bounded solutions of nonlinear Cauchy problems