Mathematics and Its Applications

By: O. Hadzic, Endre Pap

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Paperback | 30 November 2001

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292 Pages

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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.

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Published: 8th December 2010

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

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