NORMAL STRUCTURE, FIXED POINTS AND MODULUS OF n-DIMENSIONAL U-CONVEXITY IN BANACH SPACES X AND X*

Abstract

Let X and X* be a Banach space and its dual, respectively, and let B(X) and S(X) be the unit ball and unit sphere of X, respectively. In this paper, we study the relation between Modulus of n-dimensional U-convexity in X* and normal structure in X. Some new results about fixed points of nonexpansive mapping are obtained, and some existing results are improved. Among other results, we proved: if X is a Banach space with U_{X*}^{n}(n+1)>1− 1/(n+1) where n∈N, then X has weak normal structure.