Property($K^*$) Implies $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)}$

TL;DR

The paper demonstrates that if a Banach space's dual satisfies Property(K*), then the space has a bounded fixed point property, with explicit bounds involving Opial's modulus, extending understanding of fixed point theory in Banach spaces.

Contribution

It establishes a new link between Property(K*) in dual Banach spaces and the weak fixed point property, providing explicit bounds involving Opial's modulus.

Findings

If the dual satisfies Property(K*), then R(X) is bounded by 1 + 1/(1 + r_{X^*}(1)).

This bound is less than 2, implying the space has the weak fixed point property.

The result connects geometric properties of the dual space to fixed point properties of the original space.

Abstract

It is shown that if the dual of a Banach space satisfies Property($K^*$) then $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)} < 2$ where $r_{X^*}(c)$ is Opial's modulus for $X^*.$ Thus $X$ has the weak fixed point property.