Generation of semi-groups of nonlinear contractions
Isao Miyadera · Journal of the Mathematical Society of Japan · 1974
\S 1. Introduction.Let $X$ be a Banach space, and let $X_{0}$ be a subset of $X$ .By a contraction semi-group on $X_{0}$ we mean a family $\{T(t);t\geqq 0\}$ of operators $T(t):X_{0}\rightarrow X_{0}$ satisfying the following conditions:for $t\geqq 0$ and $x,$ $y\in X_{0}$ ,(1.3)We define the infinitesimal generator $A_{0}$ of $\{T(t);t\geqq 0\}$ (a contraction semi- group on $X_{0}$ ) by(1.4)whenever the limit exists.It is easy to see that $A_{0}$ is a dissipative operator.Recently, in the case when both $X$ and $X^{*}$ (the dual of $X$ ) are uniformly convex, Martin [5] has characterized the infinitesimal generator $A_{0}$ of a contraction semi-group on $X_{0}$ having the property that $\overline{D(A_{0})}=X_{0}$ .The purpose of this paper is to generalize his results to the case that $x*$ is uni- formly convex.To this end we introduce the following DEFINITION 1.1.Let $\{T(t);t\geqq 0\}$ be a contraction semi-group on $X_{0}$ , and let $A_{0}$ be the infinitesimal generator of $\{T(t);t\geqq 0\}$ .Define the set $\hat{D}$ by(1.5)$\hat{D}=$ { $x\in X_{0}$ ; $\Vert T(h)x-x\Vert=O(h)$ as $h\rightarrow 0+$ }.If $A$ is an extension of $A_{0}$ and maximal dissipative on $D$ , then $A$ is called a $(g)$ -operator of $\{T(t);t\geqq 0\}$ .If $\{T(t);t\geqq 0\}$ is a contraction semi-group on $X_{0}$ with $ D(A_{0}) eq\emptyset$ , then its $(g)$ -operator exists by the maximal principle.Our main results are stated as follows: Let $x*$ be uniformly convex.