Reflexivity of a Banach space with a uniformly normal structure

Abstract

Abstract.In this note we prove that any Banach space with a uniformly normal structure is reflexive.

Modulus of convexity, characteristic of convexity and fixed point theorems

“…Maluta [17] and Bae [1] proved that if N(E)<1 then E is reflexive. We can prove the following: THEOREM 3.2.…”

“…We may assume that C is bounded. Let {C n \ be an arbitrary decreasing sequence of nonempty closed convex subsets of C. COROLLARY 3.1 (Maluta [17] and Bae [1]) Let E be a real Banach space with N(E)<1. Then E is reflexive and has normal structure.…”

“…On the other hand, Goebel [7] defined the characteristic ε 0 of convexity of E and showed that E is uniformly convex if and only if ε o =O, if ε o <l then E has normal structure and if ε o <2 then E is reflexive. Also, Bynum [3] defined the normal structure coefficient N(E) of E, and then Maluta [17] and Bae [1] proved that if NiE)' 1^ then E is reflexive and has normal structure. Using these coefficients, Goebel and Kirk [8], Goebel, Kirk and Thele [9] and Casini and Maluta [4] proved the fixed point theorems for uniformly ^-lipschitzian mappings.…”