Asymptotic behavior of nonlinear evolution equations in Banach spaces

In this study, we show how Yosida's approach for the approximation of the derivative obtains new results for evolution systems. Using this method, we obtain multivalued time-dependent perturbation results. In addition, we consider translation-invariant subspaces Y of the bounded and uniformly continuous functions to obtain criteria for the existence of solutions u∈Y to the equationu′(t)∈A(t)u(t)+ωu(t)+f(t),t∈R, or of solutions u, which are asymptotically close to Y for the inhomogeneous differential equationu′(t)∈A(t)u(t)+ωu(t)+f(t),t>0,u(0)=u0, in general Banach spaces, where A(t) denotes a possibly nonlinear time-dependent dissipative operator. Particular examples of the space Y are spaces of functions with various properties of almost periodicity and more general types of asymptotic behavior. Furthermore, an application to functional differential equations is described.