NONEXPANSIVE RETRACTIONS ONTO CLOSED CONVEX CONES IN BANACH SPACES

Abstract

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $C^*$ be a closed convex subset of the dual space $E^*$ of $E$ and let $\Pi_{C^*}$ be the generalized projection of $E^*$ onto $C^*$. Then the mapping $R_{C^*}$ defined by $R_{C^*}=J^{-1}\Pi_{C^*}J$ is a sunny generalized nonexpansive retraction of $E$ onto $J^{-1}C^{*}$, where $J$ is the normalized duality mapping on $E$. In this paper, we first prove that if $K$ is a closed convex cone in $E$ and $P$ is the nonexpansive retaction of $E$ onto $K$, then $P$ a sunny generalized nonexpansive retraction of $E$ onto $K$. Using this result, we obtain an equivalent condition for a closed half-space of $E$ to be a nonexpansive retract of $E$.