Nonexpansive retractions onto closed convex cones in Banach spaces (Q614491)

This paper studies the relation between nonexpansive retractions and sunny generalized nonexpansive retractions in a smooth, strictly convex and reflexive Banach space \(E\). Theorem 3.5, the first important result of the paper, states that, given a closed convex cone \(K\) in \(E\), any quasi-nonexpansive mapping \(T:K\to K\), with its set of fixed points \(F(T)\) being a cone, is generalized nonexpansive in the sense of \textit{T. Ibaraki} and \textit{W. Takahashi} [Contemporary Mathematics 513, 169--180 (2010; Zbl 1218.47076)]. As a corollary of Theorem 3.5 and results of \textit{F. Kohsaka} and \textit{W. Takahashi} [J. Nonlinear Convex Anal. 8, No. 2, 197--209 (2007; Zbl 1132.47051)], the authors prove that if a cone \(K\) is a nonexpansive retract of \(E\), then \(K\) and \(JK\) are closed convex cones in \(E\), resp.\ \(E^*\), and, furthermore, \(K\) is a sunny generalized nonexpansive retract of \(E\) (Corollary 3.2), where \(J\) is the normalized duality mapping. In the last section of the paper, the authors obtain that a closed half-space \(H=\{x\in E\mid \langle x,z^*\rangle \leq 0\}, \, z^*\in E^*\setminus \{0\}\), is a nonexpansive retract of \(E\) if and only if \(JH\) is a closed half-space in \(E^*\) (Theorem 4.1).