On super fixed point property and super weak compactness of convex subsets in Banach spaces

For a nonempty convex set C of a Banach space X, a self-mapping on C is said to a linear (respectively, affine) isometry if it is the restriction of a linear (respectively, affine) isometry defined on the whole space X. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a Banach space has super fixed point property for affine (or, equivalently, linear) isometrics if and only if it is super weakly compact; and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a Banach space under equivalent reforming sense. With the application of Fabian Montesinos Zizler's renorming theorem, we finally show that every strongly super weakly compact generated Banach space can be renormcd so that every weakly compact convex set has super fixed point property. (C) 2015 Elsevier Inc. All rights reserved.