For a given closed and translation invariant subspace Y
of the bounded and uniformly continuous functions, we will give criteria for the existence of solutions u∈Y
to 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 A
denotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space Y
are spaces of functions with various almost periodicity
properties and more general types of asymptotic behavior.