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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## Abstract

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 |
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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 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/866 |

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