Bounded solutions of nonlinear Cauchy problems

Abstract

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