J. M. Ayerbe Toledano

Measures of Noncompactness in Metric Fixed Point Theory

The fixed point theorems of Brouwer and Schauder measures of noncompactness minimal sets for a measure of noncompactness convexity and smoothness near uniform convexity and near uniform smoothness fixed point for nonexpansive mappings and normal structure fixed point theorems in the absence of normal structure uniformly Lipschitzian mappings asymptotically regular mappings packing rates and 0-contractiveness constants.

Asymptotically Regular Mappings

We shall study in this chapter the existence of fixed points for a different class of mappings, called asymptotically regular mappings. The concept of asymptotically regular mappings is due to Browder and Petryshyn [BP]. Some fixed point theorems for this class of mappings can be found in [Gr1], [Gr2] and references therein. The fixed point theorems which we shall study are based upon results in [DX]. As we shall see, there is a strong connection between these results and those in Chapter VIII. In particular, in some of them the role of the Clarkson modulus of convexity will be played by the moduli of near uniform convexity. In Section 1 we define a new geometric coefficient in Banach spaces which plays the role of the Lifshitz characteristic for asymptotically regular mappings, and we prove the corresponding version for these mappings of Theorem VIII.1.4. In Section 2 we study some relationships between the new coefficient and either the modulus of NUC or the weakly convergent sequence coefficient. We also find a simpler expression for the new coefficient in Banach spaces with the uniform Opial property. Moreover we prove that, in contrast to the Lifshitz characteristic, the new coefficient is easy to compute in lp-spaces. We recall that the Lifshitz characteristic is only known in some renorming of Hilbert spaces.

The Fixed Point Theorems of Brouwer and Schauder

We are going to dedicate the first chapter to the study of the fixed point theorem of Schauder [S, 1930]. We have divided the chapter into two parts: In the first part we give the finite dimensional version of Schauder’s fixed point theorem (usually known as Brouwer’s theorem [Br, 1912], though an equivalent form had been proved by Poincare [Po, 1886]).

Minimal Sets for a Measure of Noncompactness

The notion of a o-minimal set for an MNC o was introduced in [Do1] in order to study the relationships between condensing mappings for Kuratowski and Haus-dorff’s measures of noncompactness (see Chapter X).

Fixed Points for Nonexpansive Mappings and Normal Structure

The most known and important metric fixed point theorem is the Banach fixed point theorem, also called the contractive mapping principle, which assures that every contraction from a complete metric space into itself has a unique fixed point. We recall that 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(Tx, Ty) ≤ kd(x, y) for every x, y ∈ X. This theorem appeared in explicit form in Banach’s Thesis in 1922 [Bn] where it was used to establish the existence of a solution for an integral equation. The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations have made this theorem a very useful tool in Analysis and in Applied Mathematics.

Packing Rates and ø-Contractiveness Constants

The main purpose of this chapter is to study relationships between the o-con-tractiveness constants of an operator when different measures of noncompactness are considered. The first results in this direction were obtained by Nussbaum [N, 1970], Petryshyn [Pe, 1972] and Webb [W1, 1973] for linear mappings.

Nearly Uniform Convexity and Nearly Uniform Smoothness

Reflexivity and the uniform Kadec-Klee property are among the most important properties of k-uniformly convex spaces. The study of spaces satisfying both properties was initiated by Huff in 1980 [Hu] who called these spaces nearly uniformly convex.

Convexity and Smoothness

In this and the following chapters we are going to study some important metric properties in the framework of Banach spaces. We call metric properties those which are invariant under isometries, in contrast to topological properties which are invariant with respect to homeomorphisms. Schauder’s fixed point theorem for continuous mappings is the most celebrated topological fixed point theorem.

Fixed Point Theorems in the Absence of Normal Structure

In Chapter VI we studied the f.p.p. as a property which is implied by normal structure. However, there are Banach spaces without normal structure which have the f.p.p. For instance, lp,∞ does not have normal structure (Example VI.2) but this space has the f.p.p. This fact can be proved, for instance, checking that the Banach-Mazur distance between lp,∞ and lp is 21/p and applying a stability result in [By3].

Uniformly Lipschitzian Mappings

Assume that M is a metric space and T : M → M is nonexpansive. Clearly T and all iterate mappings Tn are Lipschitzian with constant k = 1.