THE FIXED POINT PROPERTY AND UNBOUNDED SETS IN BANACH SPACES

Abstract

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $J$ be the duality mapping of $E$ and let $C$ be a nonempty closed convex subset of $E$. Then, a mapping $S: C \to C$ is said to be nonspreading [23] if \[ \phi(Sx,Sy) + \phi(Sy,Sx) \le \phi(Sx,y) + \phi(Sy,x) \] for all $x,y \in C$, where $\phi(x,y) = \|x\|^2 - 2\langle x,Jy \rangle + \|y\|^2$ for all $x,y \in E$. In this paper, we prove that every nonspreading mapping of $C$ into itself has a fixed point in $C$ if and only if $C$ is bounded. This theorem extends Ray's theorem [27] in a Hilbert space to that in a Banach space.