M.A. Japón, T. Domínguez and G. López

METRIC FIXED POINT RESULTS CONCERNING

MEASURES OF NONCOMPACTNESS

Introduction

Fixed Point Theory has two main branches: on the one hand we can consider the results that are deduced from topological properties and on the other hand those which can be obtained using metric properties.

Among the former the most important result is Brouwer’s theorem, which assures that the euclidean unit ball \overline{B}_{n}(0, 1) in has the fixed point property for continuous functions, that is, if f : \overline{B}_{n}(0, 1) → \overline{B}_{n}(0, 1) is a continuous mapping, there exists a point x_{0} ∈ \overline{B}_{n}(0, 1) such that f(x_{0}) = x_{0}.

Schauder [S, 1930] generalized the Brouwer fixed point theorem to infinite dimensional Banach spaces for the case of compact operators. Compactness plays an essential role in the proof of the Schauder fixed point theorem. However, there are some important problems where the operators are not compact.

The first step to extend the Schauder theorem to the setting of noncompact operators was given by G. Darbo [D, 1955]. The main idea is to define a new class of operators which map any bounded set in a “more compact” set. In order to state the property: A set is mapped into a “more compact” set, we need to define some “measure of noncompactness”. The first such measure was defined by Kuratowski [Ku1, 1930] in connection with certain problems of General Topology.

Probably the best known and important metric fixed point theorem is the Banach contraction principle, which assures that every contraction from a complete metric space into itself has a unique fixed point. A mapping T from a metric space (X, d) into itself is said to be a contraction if there exists k ∈ [0, 1) such that d(T x, T y) ≤ kd(x, y) for every x, y ∈ X. Simple examples show that the Banach theorem does not hold letting k to be equal to 1. Such mappings are called nonexpansive. However in the sixties Browder, Gohde and Kirk proved that under suitable geometric conditions nonexpansive mappings have the fixed point property.

Although historically the two branches of the fixed point theory have had a separated development, in recent years measures of noncompactness have also been utilized to define new geometrical properties of Banach spaces which are interesting for fixed point theory for nonexpansive mappings(see for instance [KMP], [ADL1], [DL2], [Se2], [Pr5] and references therein).

Our goal in this chapter is to study the use of measures of noncompactness in metric fixed point theory. We have divided the chapter in five sections.

Typeset by AMS-TEX