Measures of Noncompactness in Metric Fixed Point Theory

By a metric fixed point theorem we mean an existence result for a fixed point of a mapping f under conditions which depend on a metric d, and which are not invariant when we replace d by an equivalent metric. The best known metric fixed point theorem is the Banach theorem, also called the contractive mapping principle: “every contraction from a complete metric space into itself has a (unique) fixed point”. It is clear that a contractive mapping can lose this property if d is replaced by an equivalent metric. The Banach theorem is a basic tool in functional analysis, nonlinear analysis and differential equations. If we relax the contractive condition, requiring only that the mapping be nonexpansive, that is, d(f (x), f (y)) ≤ d(x, y), then trivial examples show that the Banach theorem need no longer hold. This failure may have been the reason why no significant result about the existence of fixed points for nonexpansive mappings was obtained for many years. However, in 1965, Browder [Br1 and Br2], Göhde [Go] and Kirk [Ki1] proved the following results: “let X be a Banach space, C a closed, convex and bounded subset of X and T : C → C a nonexpansive mapping. If X is either a Hilbert space, or a uniformly convex Banach space or a reflexive Banach space with normal structure, then T has a fixed point”. This result is, in some sense, surprising because it uses convexity hypotheses (more usual in topological fixed point theory) and geometric properties of the Banach spaces (commonly used in linear functional analysis, but rarely considered in nonlinear analysis prior to this time). The above results were the starting point for a new mathematical field: the application of the geometric theory of Banach spaces to fixed point theory. The texts [GK1], [AK], [KZ] and [Z] constitute excellent surveys of this theory.

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 Poincaré [Po, 1886]).

As we have seen in Chapter I, 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 notion of a ø-minimal set for an MNC ø 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).

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.

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.

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, l^p,∞ 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 l^p,∞ and l^p is 2¹/^p and applying a stability result in [By3].

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.

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 l^p-spaces. We recall that the Lifshitz characteristic is only known in some renorming of Hilbert spaces.

The main purpose of this chapter is to study relationships between the ø-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.