Fixed Point Theory in Probabilistic Metric Spaces

by Hadzic, Olga; Pap, Endre

• ISBN13: 9781402001291
• ISBN10: 1402001290
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: Kluwer Academic Pub

Summary

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.

Table of Contents

Introduction	vii
Triangular norms	1	(46)
Triangular norms and conorms	1	(4)
Properties of t-norms	5	(5)
Ordinal sums	10	(3)
Representation of continuous t-norms	13	(11)
Pseudo-inverse	13	(2)
Additive generators	15	(4)
Multiplicative generators	19	(2)
Isomorphism of continuous Archimedean t-norms with either TP or TL	21	(1)
General continuous t-norms	22	(2)
t-norms with left-continuous diagonals	24	(2)
Triangular norms of H-type	26	(3)
Comparison of t-norms	29	(9)
Comparison of continuous Archimedean t-norms	29	(4)
Comparison of continuous t-norms	33	(2)
Domination of t-norms	35	(3)
Countable extension of t-norms	38	(9)
Probabilistic metric spaces	47	(48)
Copulas and triangle functions	47	(6)
Copulas	47	(3)
Triangle functions	50	(3)
Definitions of probabilistic metric spaces	53	(2)
Some classes of probabilistic metric spaces	55	(7)
Menger and Wald spaces	56	(3)
Transformation-generated spaces	59	(1)
E-processes and Markov chains	60	(2)
Topology, uniformity, metrics and semi-metrics on probabilistic metric spaces	62	(3)
Random normed and para-normed spaces	65	(5)
Fuzzy metric spaces	70	(5)
Functions of non-compactness	75	(10)
Probabilistic metric spaces related to decomposable measure	85	(10)
Decomposable measures	85	(6)
Related probabilistic metric spaces	91	(4)
Probabilistic B-contraction principles for single-valued mappings	95	(60)
Probabilistic B-contraction principles	96	(15)
Two special classes of probabilistic q-contractions	111	(5)
Generalizations of probabilistic B-contractions principles for single-valued mappings	116	(16)
Fixed point theorems of Caristi's type	132	(8)
Common fixed point theorems	140	(15)
Probabilistic B-contraction principles for multi-valued mappings	155	(30)
Multi-valued contractions of Mihet's type	155	(3)
Multi-valued probabilistic Ψ-contrctions	158	(4)
Probabilistic Nadler q-contraction	162	(6)
A fixed point theorem of Itoh's type	168	(6)
Fixed point theorems in probabilistic metric spaces with convex structures	174	(7)
A common fixed point theorem for sequence of mappings	181	(4)
Hicks' contraction principle	185	(20)
Hicks' contraction principle for single-valued mappings	185	(10)
Multi-valued generalizations of Hicks' contraction principle	195	(10)
Fixed point theorems in topological vector spaces and applications to random normed spaces	205	(40)
Tychonoff's and Browder's fixed point theorems	206	(7)
Admissible subsets of topological vector spaces and their application on the fixed point theory	213	(12)
Fixed point theorems of Krasnoselski's type	225	(8)
Continuous dependence of the fixed points on the parameters of (α, g)- condensing mappings	233	(6)
A degree theory in topological vector spaces	239	(6)
Bibliography	245	(26)
Index	271