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2001 | Buch

Fixed Point Theory in Probabilistic Metric Spaces

verfasst von: Olga Hadžić, Endre Pap

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

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Über dieses Buch

Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory.
Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces.
In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces.
Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Triangular norms
Abstract
Triangular norms first appeared in the framework of probabilistic metric spaces in the work K. Menger [186] (see [164, 268]). It turns also out that this is a crucial operation in several fields, e.g., in fuzzy sets, fuzzy logics (see [164]) and their applications, but also, among other fields, in the theory of generalized measures [164, 168, 201, 217, 321] and in nonlinear differential and difference equations [217].
Olga Hadžić, Endre Pap
Chapter 2. Probabilistic metric spaces
Abstract
In 1942 K. Menger introduced the notion of a statistical metric space as a natural generalization of the notion of a metric space (M, d) in which the distance d(p, q) (p, qM) between p and q is replaced by a distribution function F p, q ∈ Δ+. F p,q (x) can be interpreted as the probability that the distance between p and q is less than x.
Olga Hadžić, Endre Pap
Chapter 3. Probabilistic B-contraction principles for single-valued mappings
Abstract
The first fixed point theorem in probabilistic metric spaces was proved by Sehgal and Barucha-Reid [272] for mappings f : SS, where (S, F, T M) is a Menger space. Further development of the fixed point theory in a more general Menger space (S, F, T) was connected with investigations of the structure of the t-norm T. Very soon the problem was in some sense completely solved. Namely, if we restrict ourselves to complete Menger spaces (S, F, T), where T is a continuous t-norm, then any probabilistic q-contraction f : SS has a fixed point if and only if the t-norm is of H-type.
Olga Hadžić, Endre Pap
Chapter 4. Probabilistic B-contraction principles for multi-valued mappings
Abstract
The inequality F fx,fy (qs) ≥ F x,y (s) (s ≥ 0), where q ∈ (0, 1), is generalized for multi-valued mappings in many directions. In this chapter we consider three generalizations of the above inequality for multi-valued mappings, and for such a kind of mappings some fixed point theorems are proved. In section 4.1 a fixed point theorem is proved for multi-valued mappings which satisfy a multi-valued version of the strict probabilistic (b n ,)-contraction condition introduced in section 3.3. We introduce in section 4.2 the notion of a multi-valued probabilistic Ψ-contraction, and by using the notion of the function of non-compactness a fixed point theorem is proved. Using Hausdorff distance S.B. Nadler obtained in [205] a generalization of the Banach contraction principle in metric spaces, and in section 4.3 a probabilistic version of Nadler’s fixed point theorem is proved. As a corollary a multi-valued version of Tardiff’s fixed point theorem is obtained. In section 4.4 a probabilistic version of Itoh’s fixed point theorem from [146] is given, and section 4.5 contains a fixed point result for probabilistic non-expansive multi-valued mappings of Nadler’s type, defined on probabilistic metric spaces with convex structures.
Olga Hadžić, Endre Pap
Chapter 5. Hicks’ contraction principle
Abstract
T. Hicks in [128] considered another notion of probabilistic contraction mapping than probabilistic q-contraction, which is incomparable with probabilistic q-contraction, [262]. In section 5.1 two types of generalizations of Hick’s notion of C-contraction is introduced and some fixed point theorems for these new classes of mappings are proved. A multi-valued generalization of the notion of C-contraction is given in section 5.2 and two fixed point theorems for multi-valued mappings are proved, where some results on infinitary operations from section 1.8 are used.
Olga Hadžić, Endre Pap
Chapter 6. Fixed point theorems in topological vector spaces and applications to random normed spaces
Abstract
This chapter contains some results from the fixed point theory in topological vector spaces which are of special interest for the fixed point theory in random normed spaces. Namely, a random normed space (S, F, T) with a continuous t-norm T is a topological vector space which is not necessarily a locally convex space. It is known that a random normed space (S, F, T) is a locally convex space when T is a continuous t-norm of H-type. In the fixed point theory in a not necessarily locally convex topological vector spaces a very useful notion is that of an admissible subset which was introduced by Klee. Many important function spaces are admissible.
Olga Hadžić, Endre Pap
Backmatter
Metadaten
Titel
Fixed Point Theory in Probabilistic Metric Spaces
verfasst von
Olga Hadžić
Endre Pap
Copyright-Jahr
2001
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-1560-7
Print ISBN
978-90-481-5875-1
DOI
https://doi.org/10.1007/978-94-017-1560-7