Fixed Point Theory in Probabilistic Metric Spaces
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
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Mathematics and Its Applications
Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 536
Fixed Point Theory in Probabilistic Metric Spaces by Olga Hadii6 and Endre Pap Institute of Mathematics, University of Novi Sad, Yugoslavia
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SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5875-1 ISBN 978-94-017-1560-7 (eBook) DOI 10.1007/978-94-017-1560-7
Printed on acid-fr-ee paper
All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents Introduction
VB
1 Triangular norms 1.1 Triangular norms and conorms . 1.2 Properties of t-norms . . . . . . 1.3 Ordinal sums . . . . . . . . . . 1.4 Representation of continuous t-norms 1.4.1 Pseudo-inverse . . . . . 1.4.2 Additive generators . . . . . . 1.4.3 Multiplicative generators . . . 1.4.4 Isomorphism of continuous Archimedean t-norms with either T p or TL . . . . . . . . . . . . . 1.4.5 General continuous t-norms . . 1.5 t-norms with left-continuous diagonals 1.6 Triangular norms of H-type . . . . . . 1. 7 Comparison of t-norms . . . . . . . . . 1.7.1 Comparison of continuous Archimedean t-norms 1.7.2 Comparison of continuous t-norms 1.7.3 Domination of t-norms . 1.8 Countable extension of t-norms 2 Probabilistic metric spaces 2.1 Copulas and triangle functions . 2.1.1 Copulas . . . . . . . . . 2.1.2 Triangle functions . . . . 2.2 Definitions of probabilistic metric spaces 2.3 Some classes of probabilistic metric spaces 2.3.1 Menger and Wald spaces . . . . . 2.3.2 Transformation-generated spaces 2.3.3 E-processes and Markov chains. . 2.4 Topology,uniformity, metries and semi-metrics on probabilistic metric . . . . . . . . . . . . . . . . . . . . spaces . . . . . . . . . . . . . v
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1 5 10 13 13 15 19 21 22 24 26 29 29 33 35 38
47 47 47 50 53 55 56 59 60 62
CONTENTS
VI
2.5 2.6 2.7 2.8
3
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Random normed and para-normed spaces . Fuzzy metric spaces . . . . . . . . . . . . . Functions of non-compactness . . . . . . . Probabilistic metric spaces related to decomposable measure 2.8.1 Decomposable measures . . . . . . 2.8.2 Related probabilistic metric spaces ., . . . . . . . .
Probabilistic B-contraction principles for single-valued 3.1 Probabilistic B-contraction principles . . . . . . . . . . . 3.2 Two special classes of probabilistic q-contractions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Generalizations of probabilistic B-contractions principles valued mappings . . . . . . . . . . . . 3.4 Fixed point theorems of Caristi's type. 3.5 Common fixed point theorems. . . . .
75 85 85 91
mappings . . . . ..
95 96
. . . . . . . for single. . .
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