Weakly continuous nonlinear accretive operators in reflexive Banach spaces

Abstract

Let $X$ be a reflexive space and $A$ be a weakly continuous (possibly nonlinear) operator which maps $X$ to $X$. We are concerned with the autonomous differential equation \[ (1.1)\quad u’(t) + Au(t) = 0,\quad u(0) = X.\] We first provide a local solution to (1.1) and then we apply the additional hypothesis that $A$ is accretive to extend the local solution of (1.1) to a global solution. If $A$ is weakly continuous and accretive then $A$ is shown to be $m$-accretive, i.e. $R(I + \lambda A) = X$ for all $\lambda \geqq 0$. The $m$-accretiveness of $A$ enables us to provide a semigroup representation of solutions to (1.1), \[ (1.2)\quad u(t) = T(t)x = \lim \limits _{n \to \infty } (I + (t/n)A){x^n}\quad {\text {for }}t \in [0,\infty ).\] We then investigate properties of semigroups which have weakly continuous infinitesimal generators.