Mohamed A. Khamsi

Nonstandard Methods in Fixed Point Theory

A unified account of the major new developments inspired by Maureys application of Banach space ultra-products to the fixed point theory for non-expansive mappings is given in this text. The first third of the book is devoted to laying a foundation for the actual fixed point theoretic results which follow. Set theoretic and Banach space ultra-products constructions are studied in detail in the second part of the book, while the remainder of the book gives an introduction to the classical fixed point theory in addition to a discussion of normal structure.

Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings

We discuss the newly introduced concept of cone metric spaces. We also discuss the fixed point existence results of contractive mappings defined on such metric spaces. In particular, we show that most of the new results are merely copies of the classical ones.

INTRODUCTION TO HYPERCONVEX SPACES

The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.

Fixed point and selection theorems in hyperconvex spaces

It is shown that a set valued mapping T ∗ of a hyperconvex metric space M which takes values in the space of nonempty externally hyperconvex subsets of M always has a lipschitzian single valued selection T which satisfies d(T (x), T (y)) ≤ dH(T ∗(x), T ∗(y)) for all x, y ∈M . (Here dH denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded λ-lipschitzian self-mappings of M is itself hyperconvex. Several related results are also obtained.

On metric spaces with uniform normal structure

In this work, we prove that metric spaces with uniform normal structure have a kind of intersection property, which is equivalent to reflexivity in Banach spaces.

On asymptotically nonexpansive mappings in hyperconvex metric spaces

Since bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings, it is natural to extend such a powerful result to asymptotically nonexpansive mappings. Our main result states that the approximate fixed point property holds in this case. The proof is based on the use, for the first time, of the ultrapower of a metric space.

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.

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping T* of a metric tree M with convex values has a selection T: M → M for which d(T(x),T(y)) < d H (T*(x),T*(y)) for each x,y ∈ M. Here by d H we mean the Hausdroff distance. Many applications of this result are given.

Quasicontraction Mappings in Modular Spaces without Δ2-Condition

As a generalization to Banach contraction principle, Ćirić introduced the concept of quasi-contraction mappings. In this paper, we investigate these kinds of mappings in modular function spaces without the 2-condition. In particular, we prove the existence of fixed points and discuss their uniqueness.

Some geometrical properties and fixed point theorems in Orlicz spaces

Abstract Let (G, ∑, μ) be a finite, atomless measure space and let Lφ be an Orlicz space of measurable functions on G. We consider some geometrical properties of the functional ρ(ƒ) = ∝ G φ(ƒ(t)) dμ(t) , called the Orlicz modular. These properties, like strict convexity, uniform convexity or uniform convexity in every direction, can be equivalently expressed in terms of the properties of the corresponding Orlicz function φ. We use these properties in order to prove some fixed point results for mappings T: B → B, B ⊂ Lφ, that are nonexpansive with respect to the Orlicz modular ρ, i.e., ρ(Tƒ − Tg) ⩽ ρ(ƒ − g) for all ƒ and g in B. We prove also existence and uniqueness in Lφ of the best approximant with respect to ρ and some convex subsets of Lφ. Our results are valid also in the case when the Orlicz function φ does not satisfy the Δ2-condition. This demonstrates the advantage of our method because, in the latter case, both Luxemburgs and Orliczs norms cannot possess suitable convexity properties.