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 $u\in Y$
to the equation $u^{\prime}(t)+ A(u(t))+\omega u(t)\ni f(t),t\in\mathbb{R}$ , or of solutions $u$ asymptotically close to $Y$ for the inhomogeneous differential equation $u^{\prime}(t)+ A(u(t))+\omega u(t)\ni f(t),t > 0, u(0)= u_0$ , 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.