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A new convexity property that implies a fixed point property for $L_{1}$

100%

Lennard C.

Studia Mathematica

1991

tom 100

nr 2

95-108

In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result holds. The Chebyshev centre of any norm bounded, convergence locally in measure compact subset of L₁ must be norm compact. Immediately from normal structure, we get a new proof of a fixed point theorem for L₁ due to Lami Dozo and Turpin.

A contractive fixed point free mapping on a weakly compact convex set

51%

Burns J. , Lennard C. , Sivek J.

Studia Mathematica

2014

tom 223

nr 3

275-283

We prove the existence of a contractive mapping on a weakly compact convex set in a Banach space that is fixed point free. This answers a long-standing open question.