Uniform normal structure and related notions.

Kirk, William A. and Sims, Brailey (2001) Uniform normal structure and related notions. Journal of nonlinear and convex analysis, 2 (1). pp. 129-138. ISSN 1345-4773

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    Let X be a Banach space, let $\phi$ denote the usual Kuratowski measure of noncompactness, and let $k_X (\epsilon) = sup r (D)$ where $r (D)$ is the Chebyshev radius of $D$ and the supremum is taken over all closed convex subsets $D$ of $X$ for which $diam (D) = 1$ and $\phi(D) \ge \epsilon$. The space $X$ is said to have $\phi$-uniform normal structure if $k_X (\epsilon) < 1$ for each $\epsilon \in (0, 1)$. It is shown that this concept, which lies strictly between normal structure and uniform normal structure, implies reflexivity. Hence if $X$ has $\phi$-uniform normal structure then $X$ has the fixed point property for nonexpansive mappings. Related concepts in metric spaces are also discussed.

    Item Type: Article
    Additional Information: Special issue for Professor Ky Fan.
    Uncontrolled Keywords: Uniform normal structure, reflexivity, nonexpansive mappings, fixed points.
    Subjects: 46-xx Functional analysis > 46Bxx Normed linear spaces and Banach spaces; Banach lattices
    47-xx Operator theory > 47Hxx Nonlinear operators and their properties
    Faculty: UNSPECIFIED
    Depositing User: Mr Christopher Maitland
    Date Deposited: 07 Apr 2011 10:41
    Last Modified: 07 Apr 2011 10:41

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