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 :for all x, y ∈ C, where φ(x, y) = x 2 -2 x, Jy + y 2 for all x, y ∈ 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.

Locations

• Taiwanese Journal of Mathematics - View - PDF