T. Domínguez Benavides

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.

Uniformly Lipschitzian mappings in modular function spaces

The theory of modular spaces was initiated by Nakano [14] in 1950 in connection with the theory of order spaces and rede8ned and generalized by Musielak and Orlicz [13] in 1959. De8ning a norm, particular Banach spaces of functions can be considered. Metric 8xed theory for these Banach spaces of functions has been widely studied (see, for instance, [15]). Another direction is based on considering an abstractly given functional which controls the growth of the functions. Even though a metric is not de8ned, many problems in 8xed point theory for nonexpansive mappings can be reformulated in modular spaces (see, for instance, [8] and references therein). In this paper, we study the existence of 8xed points for a more general class of mappings: uniformly Lipschitzian mappings. Fixed point theorems for this class of mappings in Banach spaces have been studied in [2,3] and in metric spaces in [11,12] (for further information about this subject, see [1, Chapter VIII] and references therein). The main tool in our approach is the coeAcient of normal structure Ñ(L ). We prove that under suitable conditions a k-uniformly Lipschitzian mapping has a 8xed point if k ¡ ( Ñ(L ))−1=2. In the last section we show a class of modular spaces where Ñ(L )¡ 1 and so, the above theorem can be successfully applied.

Fixed-point theorems for multivalued non-expansive mappings without uniform convexity

Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C, and T a nonexpansive mapping from C into KC(C). We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is a 1-χ-contractive mapping.

Normal structure coefficients of L p (Ω)

Let X be a uniformly convex Banach space, and N ( X ) the normal structure coefficient of X . In this paper it is proved that N ( X ) can be calculated by considering only sets whose points are equidistant from their Chebyshev centre. This result is applied to prove that N ( L P (Ω)) = min {2 1−1/ p , 2 1/ p }, Ω being a σ-finite measure space. The computation of N ( L p ) lets us also calculate some other coefficients related to the normal structure.

Does Kirk’s theorem hold for multivalued nonexpansive mappings?

Fixed Point Theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings. Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. However, many other questions remain still open, for instance, the possibility of extending the well-known Kirks Theorem, that is: do Banach spaces with weak normal structure have the fixed point property (FPP) for multivalued nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for single-valued mappings (for example, uniform convexity, nearly uniform convexity, uniform smoothness,…). Thus, it is natural to consider the following problem: do these properties also imply the FPP for multivalued mappings? In this way, some partial answers to the problem of extending Kirks Theorem have appeared, proving that those properties imply the existence of fixed point for multivalued nonexpansive mappings. Here we present the main known results and current research directions in this subject. This paper can be considered as a survey, but some new results are also shown.

Set-Contractions and Ball-Contractions in LP-Spaces

Abstract Consider the space L p (Ω), 1 ⩽ p , where Ω is a σ-finite measure space. Defined δ and γ by: δ = 2 max { 1 p , (p − 1) p } and γ = 2 ¦p − 2¦ p . The following relationship between set-contractions and ball-contractions in separable L p (Ω) spaces is proved: If T: L p (Ω) → L p (Ω) is a k-set-contraction (respectively set-condensing mapping) then T γ is a k-ball-contraction (respectively ball-condensing mapping). If T: L p (Ω) → L p (Ω) is a k-ball-contraction, then T δ is a k-set-contraction. Furthermore these constants γ and δ are the best possible.

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.

Fixed Points of Asymptotically Contractive Mappings

Abstract Let X and Y be metric spaces. We say that a continuous mapping T : X → Y is asymptotically- k -contractive if lim n , m ; n ≠ m d ( Tx n , Tx m ) ⩽ k lim n , m ; n ≠ m d ( x n , x m ) for every sequence { x n } such that both limits exist. It is proved that every k -set-contractive mapping T : X → Y is asymptotically k -contractive and that every asymptotically k -contractive mapping T : C → C has a fixed point when C is a bounded closed convex set of a Banach space.

Some questions in metric fixed point theory, by A. W. Kirk, revisited

In this survey, we comment on the current status of several questions in Metric Fixed Point Theory which were raised by W. A. Kirk in 1995.

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]).