Asymptotic Behaviour of Nonlinear Evolution Equations in Banach Spaces
arXiv:1312.2931
Abstract
We show how the approach of Yosida approximation of the derivative serves to obtain new results for evolution systems. Using this method we obtain multivalued time dependent perturbation results. Additionally, translation invariant subspaces of the bounded and uniformly continuous functions are considered, to obtain criteria for the existence of solutions to the equation $$ u^{\prime}(t)\in A(t)u(t)+ \om u(t) + f(t), t\in \re, $$ or of solutions asymptotically close to for the inhomogeneous differential equation \begin{eqnarray*} u^{\prime}(t)&\in& A(t)u(t) + \om u(t) + f(t), \ \ t > 0, u(0)&=&u_0, \end{eqnarray*} in general Banach spaces, where denotes a possibly nonlinear time dependent dissipative operator. Particular examples for the space are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior. Further, an application to functional differential equations is given.