Chapter 1

EXAMPLES OF FIXED-POINT-FREE MAPPINGS

Brailey Sims

bsims@maths.newcastle.edu.au

Mathematics, School of Mathematical and Physical Sciences The University of Newcastle NSW, 2308, Australia

1. Introduction

In this short chapter we collect together examples of fixed point free nonexpansive mappings defined on nonempty closed convex subsets of a Banach space X, that are bounded, weak compact, or, in the case when X = E^{∗}, weak^{∗} compact; that is, σ(X, E) compact. These example help delineate the class of spaces enjoying the fpp, the w-fpp, or the w^{∗}-fpp. Unfortunately, known examples are rather sparse. With the exception of Alspach’s example (or modifications of it, see section ???), the mappings concerned are adaptations of affine maps (indeed, modified shifts), or minor variants thereof. This dearth of examples is a major impediment to a fuller understanding of metric fixed point theory and the discovery of informative new examples would be an important step forward. EXAMPLE 1: c_{0} fails the FPP. Let C = B_{c_{0}}^{+} := {(x_{n}) ∈ c_{0} : 0 ≤ x_{n} ≤ 1, all n} and define T by

T (x_{n}) := (1 − x_{1}, x_{1}, x_{2}, …).

Then for any x⃗, y⃗ ∈ c_{0} we have ∥T x⃗ − T y⃗∥_{∞} = ∥x⃗ − y⃗∥_{∞}, so T is nonexpansive, indeed a metric isometry, and T maps C into C. On the other hand the only possible fixed point for T is (\frac{1,2}{,} \frac{1,2}{,} \frac{1,2}{,} …) ∉ c_{0}. EXAMPLE 2: c_{0} fails the E_{a}-FPP for a contraction.

Let a = (a(n)) ∈ l_{∞} be a strictly decreasing sequence of ‘weights’satisfying α ≤ a(n) ≤ β, for some 0 < α ≤ β < ∞, and take

a_{0} := 0
a_{n} := (a(1), …, a(n), 0, 0, …) for n = 1, 2, 3, ….

Now consider the set

K := \overline{co}\{a_{n}\}_{n=0}^{∞}

in c_{0} consisting of vectors of the form

∑_{n=0}^{∞} λ_{n}a_{n} = (a(1)(1 − λ_{0}), a(2)(1 − (λ_{0} + λ_{1})), a(3)(1 − (λ_{0} + λ_{1} + λ_{2})), …),

where λ_{n} ≥ 0 and ∑_{n=0}^{∞} λ_{n} = 1, that admits fixed point free affine contraction T_{a} defined by

T_{a}(a(1)(1−λ_{0}), a(2)(1−(λ_{0}+λ_{1})), …) := (a(1), a(2)(1−λ_{0}), a(3)(1−(λ_{0}+λ_{1})), …).