Fixed points, normal structure and moduli of semi-UKK, semi-NUC and semi-UKK∗UKK∗ spaces

Let X   and X∗X∗ be a Banach space and its dual. In this paper a new kind of separation measure, u-separation, is introduced. The new kind of Banach spaces: semi-UKK and semi-NUC, and modulus of semi-UKK and modulus of semi-NUC are introduced in terms of this u-separation measure. The relationships of these moduli with normal structure, reflexivity and other convex properties are studied respectively. Many known results for fixed points of nonexpansive mappings are improved upon. The u-separation measure and its properties in X∗X∗ are also defined and studied.