Recent Advances and Applications of Fuzzy Metric Fixed Point Theory 9781032544496, 9781032548555, 9781003427797

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory This book not only presents essential material to understand fuzzy metric fixed point theory, but also enables the readers to appreciate the recent advancements made in this direction. It contains seven chapters on different topics in fuzzy metric fixed point theory. These chapters cover a good range of interesting topics such as convergence problems in fuzzy metrics, fixed figure problems, and applications of fuzzy metrics. The main focus is to unpack a number of diverse aspects of fuzzy metric fixed point theory and its applications in an understandable way so that it could help and motivate young graduates to explore new avenues of research to extend this flourishing area in different directions. The discussion on fixed figure problems and fuzzy contractive fixed point theorems and their different generalizations invites active researchers in this field to develop a new branch of fixed point theory. Features: • Explore the latest research and developments in fuzzy metric fixed point theory. • Describes applications of fuzzy metrics to color image processing. • Covers new topics on fuzzy fixed figure problems. • Filled with examples and open problems. This book serves as a reference book for scientific investigators who want to analyze a simple and direct presentation of the fundamentals of the theory of fuzzy metric fixed point and its applications. It may also be used as a textbook for postgraduate and research students who try to derive future research scope in this area.

Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

Dhananjay Gopal and Juan Martinez Moreno

Designed cover image: Shutterstock First edition published 2024 by CRC Press 2385 NW Executive Center Drive Suite 320 Boca Raton, FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Dhananjay Gopal and Juan Martinez Moreno Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-54449-6 (hbk) ISBN: 978-1-032-54855-5 (pbk) ISBN: 978-1-003-42779-7 (ebk) DOI: 10.1201/9781003427797 Typeset in Times by codeMantra

Contents Foreword .................................................................................................................viii Preface.......................................................................................................................ix Authors......................................................................................................................xi Chapter 1

Fuzzy Set and Basic Operation........................................................ 1 1.1 1.2

Introduction............................................................................. 1 Operation on Fuzzy Sets......................................................... 3 1.2.1 Fuzzy Complements................................................... 4 1.2.2 Fuzzy Intersection: t-Norm ........................................ 5 1.2.3 Fuzzy Union: t-conorms ............................................ 7 1.2.4 Characterization Theorem of t-conorm...................... 8 1.3 Conclusion .............................................................................. 8 References........................................................................................ 8

Chapter 2

Origin and Motivation of Fuzzy Metric........................................... 9 2.1 2.2 2.3

Introduction............................................................................. 9 Kramosil and Michalek Fuzzy Metric Spaces ....................... 9 George and Veeramani Fuzzy Metric Spaces ...................... 11 2.3.1 Generalizing Well-Known Fuzzy Metrics ............... 12 2.3.2 Fuzzy Metrics Defined by Means of a Metric ......... 13 2.3.3 Discrete Fuzzy Metric.............................................. 14 2.3.4 Fuzzy Metrics Deduced from Metrics ..................... 15 2.3.5 Fuzzy Metrics Deduced from Partition.................... 17 2.4 Conclusions........................................................................... 19 References...................................................................................... 20 Chapter 3

Convergence in Fuzzy Metric Spaces............................................ 21 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Introduction........................................................................... 21 p-Convergent Sequences ...................................................... 23 p-Fuzzy Diameter ................................................................. 32 p-Accumulation .................................................................... 33 p-Completeness .................................................................... 37 Only for the Standard Fuzzy Metric ..................................... 39 A Classification of Fuzzy Metric Spaces.............................. 42 3.7.1 p-Convergence and p-Cauchyness ........................... 46 3.7.2 s-Convergence and s-Cauchyness ............................ 47 3.7.3 Strong Convergence and Strong Cauchyness .......... 49 v

vi

Contents

3.8

Relating the Concepts ........................................................... 50 3.8.1 Relating Weak Concepts .......................................... 50 3.8.2 Relating Strong Concepts ........................................ 51 3.9 Compactness and Completeness........................................... 52 3.10 Conclusion ............................................................................ 53 References...................................................................................... 53

Chapter 4

Theory of Fuzzy Contractive Mappings and Fixed Points ............ 55 4.1 4.2 4.3 4.4 4.5 4.6

Introduction........................................................................... 55 Fuzzy Contractive Mappings ................................................ 55 Fuzzy Ψ-Contractive Mappings............................................ 59 α-φ -Fuzzy Contractive Mappings ........................................ 61 β -ψ-Fuzzy Contractive Mappings........................................ 67 Fuzzy H -Contractive Mappings and α Type Fuzzy H -Contractive Mappings .................................................... 71 4.7 Fuzzy Z -Contractive Mappings .......................................... 79 4.8 Suzuki Type Fuzzy Z -Contractive Mappings and Fixed Point............................................................................ 87 4.9 Observations ......................................................................... 95 ´ c Operators and Unified Fixed 4.9.1 Fuzzy-Preˇsi´c-Ciri´ Point Theorems ........................................................ 96 4.10 Caristi Type Mappings and Fixed Point ............................. 112 4.11 Fuzzy Meir-Keeler Contractive Mappings and Fixed Point ................................................................... 119 4.12 Conclusions......................................................................... 120 References.................................................................................... 121

Chapter 5

Common Fixed-Point Theorems in Fuzzy Metric Spaces........... 124 5.1 Introduction and Preliminaries ........................................... 124 5.2 Some Important Results...................................................... 124 5.3 Conclusion .......................................................................... 161 5.4 Acknowledgments .............................................................. 161 References.................................................................................... 161

Chapter 6

Introduction to Fixed Figure Problems in Fuzzy Metric Spaces .............................................................................. 164 6.1 6.2 6.3

Introduction......................................................................... 164 The Fixed-Circle Problem on Fuzzy Metric Spaces........... 166 The Fixed-Cassini Curve Problem on Fuzzy Metric Spaces...................................................................... 173 6.4 Fixed Point Sets of Fuzzy Quasi-Nonexpansive Maps ...... 175 6.5 Conclusion and Future Scope ............................................. 179 References.................................................................................... 180

vii

Contents

Chapter 7

Applications of Fuzzy Metrics and Fixed-Point Theorems......... 183 7.1 7.2 7.3

Introduction......................................................................... 183 Image Filtering Using Fuzzy Metrics ................................. 183 Application of the Fuzzy Metric M 0 to Measure Perceptual Color Differences.............................................. 188 7.4 Applications to Fuzzy Fixed Point Theorems .................... 191 7.4.1 Applications to Differential Equations .................. 191 7.5 Conclusion .......................................................................... 198 References.................................................................................... 198 Index...................................................................................................................... 201

Foreword This book, Recent Advances and Applications of Fuzzy Metric Fixed Point Theory, is an attempt to present an attractive research area on fuzzy metric fixed point theory in a simple way with focus on clarity of arguments in the proofs of classical and recent results in this area. A collection of examples illustrate a variety of essential concepts. Fixed point theory makes this book an easily accessible source of knowledge. This book touches on several research directions within the fuzzy metric structure and fuzzy fixed point theory and opens new avenues of investigation to extend and explore it further. Written by well-known fixed point theorists with vast teaching experience, this book is particularly suitable for young mathematicians who want to study fuzzy metric fixed point theory and to pursue their research careers in that area. The presentation of fixed point theory of fuzzy contractive mappings and its generalizations provides a handy account of development and progress on the subject. This book also includes an interesting concept of fuzzy fixed circle. A thorough discussion on various types of convergence in fuzzy metric setting and related results invites active researchers to contribute and to extend this theory further. The last chapter of this book highlights the significance of fuzzy metric and fixed point theory in connection with image processing, integral equations, and difference equations. We believe that this effort will inspire young mathematicians to explore more applications of fuzzy metric fixed point theory and extend the boundaries of this theory further. Sompong Dhompongsa Chiang Mai University Chiang Mai,Thailand Yeol Je Cho Gyeongsang National University Jinju, Korea

viii

Preface Fuzzy metric fixed point theory is one of the notable generalizations of classical metric fixed point theory. Indeed, fixed point theory of fuzzy metric spaces is more diverse than classical metric setting. This is due to the pliability exhibited in the concept of fuzzy metric. But at the same time, due to complexity involved in the nature of fuzzy metrics, one might need to use or develop new fuzzy mathematical tools to establish new results in this field (for examples [11–14]). This pursuance of need makes the new fuzzy metric results worthwhile. This intrigued several researchers to work on this area, which amounted a large literature devoted to this topic in the last five decades. However, most of the materials on this topic are scattered either in the form of research papers or in the form of general articles. Also there are many excellent books and monographs on fixed point theory, but to the best of our knowledge, there is not a single book on fixed point results in fuzzy metric spaces. Considering these aspects, this monograph aims to provide a text on this subject. The technique of using this theory which would be of great use to specialists in the field of nonlinear analysis and decision-making has not yet been sufficiently developed. The purpose of this monograph is to fill the gap. Especially, we present many new topics, applications, and recent results that are not mentioned in general articles and papers. Also, these notes are addressed to non-experts who want to learn something about fuzzy metric and corresponding fixed point theory. Our emphasis is on the applicability to many problems including applications of fuzzy metric to color image processing. This book contains seven chapters with numerous illustrative examples, remarks, and some open problems concerning the related topics. This book is organized as follows: Chapter 1 provides some fundamental concepts, definitions, and results on fuzzy sets and corresponding set theoretical operations, which are necessary for understanding the main topics such as fuzzy metrics and related theorems, propositions, and other ideas discussed in the remaining chapters. Chapter 2 introduces the motivation of fuzzy distances and some basic results of fuzzy metrics. Several examples and counter examples are given to distinguish various classes of fuzzy metric spaces including open problems. Chapter 3 deals with the problem of convergence and completeness of fuzzy metric spaces. Comparative diagrams are presented to simplify various types of convergence inclusion aspects. Chapter 4 presents a concise study of fixed point problems concerning various classes of fuzzy contractive mappings including characterization of complete fuzzy metric spaces. We hope that the results presented in this chapter illustrate the direction of research over the last five decades up to the most recent contributions on the topic. Chapter 5 studies some common fixed point results. The main results in this chapter were obtained by Refs. [2, 14, 15, 18-20, 4, 8 and 31]. ix

x

Preface

Chapter 6 presents most recent findings on the problem of fixed figure in the setting of GV-fuzzy metric spaces. The first section of this chapter deals with fixed circle concept. Other sections studies Cassini curves and fuzzy quasi non-expansive mappings. Chapter 7 is devoted to study and explore various applications of fuzzy metric and corresponding fixed point results in image processing, integral and functional equations problems. We greatly admire and are deeply indebted to our friends and colleagues working in Nonlinear analysis and Fixed point theory for their encouragement and support, especially Professor Sompong Dhompongsa, Chiang Mai University, Thailand, Professor Yeol-Je Cho, Gyeongsang National University, Jinju, Korea, Professor Ismat Beg, Centre for Mathematics and Statisical Sciences, Lahore School of Economics, Lahore, Pakistan and Professor S. Radenovic, Faculty of Mathematics and Statistics ¨ ur, Department of Mathematics, Balıkesir, Turkey. In particular, we wish Nihal Ozg¨ to express our deepest thanks to our colleagues who contributed their recent research work in the form of chapters for inclusion in this book. This work was completed while the first author (Dr. Gopal) was working as visiting Professor at the Department of Mathematics, University of Jean Spain during February 20 to July 10, 2023. He thanks administration of Guru Ghasidas Vishawavidyala for granting extraordinary leave for the mentioned period. He also thank University of Jean for their kind hospitality and support. Our deepest gratitude and thanks are also due to our family members who always encouraged us and refreshed our energies with their sweet words while we were busy in accomplishment of this project. The authors are very thankful to Isha Singh, Rajesh Dey, and the staff from the CRC Publishers for their unfailing support cooperation and patience in publishing this book. Dhananjay Gopal Juan Martinez Moreno

Authors Dhananjay Gopal has a doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India, and is currently an Associate Professor of Mathematics at Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur (C.G.) India. He also serves as visiting Professor at the Department of Mathematics, University of Jaen, Spain. He was an Assistant Professor of Applied Mathematics at S.V. National Institute of Technology, Surat, Gujarat from 2009 to 2020. His research interests include nonlinear analysis and fuzzy metric fixed point theory. He is the author and co-author of more than 110 papers in journals, proceedings, and three books in the field of metric spaces and fixed-point theory. He is an editorial board member of more than three international journals and a regular reviewer of more than 50 journals published by Springer, Elsevier, Taylor & Francis, Wiley, IOS Press, World Scientific, American Mathematical Society, and De Gruyter. He was the guest editor of the special issue “Fixed point theory in abstract metric spaces with generalized contractive conditions; new methods, algorithms, and Applications,” in the Journal of Mathematics and the Special Issue on “Nonlinear operator theory and its Applications” in the Journal of function spaces. Dr Gopal has active research collaborations with KMUTT, Bangkok, Thammasat University Bangkok, and Jaen University Spain, and in his research pursuits, he has visited South Africa, Thailand, Japan, and Iran. Juan Martinez Moreno is Full Professor at the Department of Mathematics, University of Jaen, Spain. His research focuses on topology, fuzzy mathematics, fixed point theory, and their applications. Dr. Juan work has been published in several journals in the areas of general and applied mathematics and computer science. He also serves as an editor and a referee for several mathematics journals. For full list of publication, please see https://www.ujaen.es/departamentos/matema/ contactos/martinez-moreno-juan

xi

Set and Basic 1 Fuzzy Operation 1.1

INTRODUCTION

Among the various paradigmatic changes in science and mathematics in this century, one such change concerns the concept of uncertainty. In science, this change has been manifested by a gradual transition from a traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it. According to the traditional view, science should strive for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); hence, uncertainty (impressions, nonspecificity, vagueness, inconsistency, etc.) is regarded as unscientific. According to the alternative (or modern) view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility. It is generally agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by Lotfi A. Zadeh in 1965 [1], even though some ideas presented in the paper were envisioned some 30 years earlier by the American philosopher Max Black. In this paper, Zadeh introduced a theory whose objects - Fuzzy Set- are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter a degree. Let us recall one of the basic methods to define a set : A set is defined by a function, usually called a characteristic function, that declares which elements of X are members of the set and which are not. Set A is defined by its characteristic function, χA , as follows:  1 : for x ∈ A χA = 0 : for x ∈ /A i.e. characteristic function maps elements of X to elements of the set 0, 1, which is formally expressed as χA : X → {0, 1} . for each x ∈ X,when χA = 1, x is declared to be a member of A; when χA = 0, x is declared as a non-member of A. The characteristic function of a crisp set assigns a value of either 1 or 0 to each individual in the universal set, thereby discriminating between members and nonmembers of the crisp set under consideration. This function can be generalized such that the values assigned to the elements of the universal set fall within a specified DOI: 10.1201/9781003427797-1

1

2

Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

range and indicate the membership grade of these elements in the set question. Larger values denote higher degrees of set membership. Such a function is a membership function, and the set defined by it is a fuzzy set. The most commonly used range of values of membership functions is the unit interval [0, 1]. In this case, each membership function maps the elements of a given universal set X into real numbers in [0, 1]. A fuzzy set A on X in membership function i.e; A : X → [0, 1]. Note that fuzzy sets allow us to represent vague concepts expressed in natural language. The representation depends not only on the concept but also on the context in which it is used. For example, applying the concept of high temperature in one context to weather and in another context to a nuclear reactor would necessarily be represented by very different fuzzy sets. Even for similar contexts, fuzzy sets representing the same concept may vary considerably. In this case, however, they also have to be similar in some key features. For example, the general conception of a class of real numbers that are close to 2. In spite of their differences, the four fuzzy sets are similar in the sense that the following properties are possessed by each Ai (i ∈ (1, 2, 3, 4)): 1. Ai (2) = 1 and Ai (x) < 1 for all x ̸= 2 2. Ai is symmetric with respect to x = 2 i.e. Ai (2 + x) = Ai (2 − x) for all x ∈ R 3. Ai (x) decreases monotonically from 1 to 0 with the increasing difference |(x − 2)|. The following diagrams illustrate about fact: 1

A2(x)

A1(x)

1

5

0

1

2

0

3

2

3

A4(x)

1

A3(x)

1

1

0

1

2

3

0

1

2

3

3

Fuzzy Set and Basic Operation

Now let us consider a simple example to understand the idea of fuzzy set. Consider three fuzzy sets that represent the concepts of a young, middle-aged and old person. A reasonable expression of these concepts is illustrated by the following figure: Middle A2

Young A1

Old A3

1

Ai(x) i=1,2,3

0

10

20

30

40

50

60

70

80

Age: x

The membership functions for these fuzzy sets can be described by the following functions:  when x ≤ 20  1 35−x A1 = when 20 < x < 35  15 0 when x ≥ 35  0    x−20

15 , A2 = (60 − x)/15    1

A3 =

  0

x−45 15



1.2

1

when x ≤ 20 or ≥ 60 when 20 < x < 35 when 45 < x < 60 when 35 ≤ x ≤ 45

when x ≤ 45 when 45 < x < 60 when x ≥ 60.

OPERATION ON FUZZY SETS

The three basic operations on classical /crisp sets are complementation, intersection, and union can be generalized to fuzzy sets in more than one way. A particular generalization is known as standard fuzzy set operations. Standard complementation: The standard complementation A¯ of a fuzzy set A with ¯ = 1 − A(x). respect to the universal set X is defined for all x ∈ X by the equation A(x) ¯ are called equilibrium points of X. Elements of X for which A(x) = A(x)

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

For the standard complement, clearly, membership grades of equilibrium point is 0.5. Standard intersection and standard union: Given two fuzzy sets A and B, their standard intersection, (A ∩ B)(x) and standard union, (A ∪ B)(x) are defined for all x ∈ X by the equations; (A ∩ B)(x) = min{A(x), B(x)} (A ∪ B)(x) = max{A(x), B(x)}. Due to the associativity of min and max, these deformations can be extended to any finite number of fuzzy sets. 1.2.1

FUZZY COMPLEMENTS

The general concept of fuzzy complement is a function. c : [0, 1] → [0, 1] that possess certain properties to justify the meaning of fuzzy complements: Axiom c1 : c(0) = 1 and c(1) = 0 (boundary condition). Axiom c2 : for all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b) (monotonocity). According to Axiom c1 , function c is required to produce correct complements for crisp sets. According to Axiom c2 , it is required to be monotonic decreasing: when a membership grade in A increases (by changing x), the corresponding membership grade in cA must not increase as well; it may decrease or, at least, remain the same. Many functions satisfy both Axioms c1 and c2 . For any particular fuzzy set A, different fuzzy sets cA can be said to constitute its complements, each being produced by a distinct function c. All functions that satisfy the axioms form the most general class of fuzzy complements. It is rather obvious that the exclusion or weakening of either of these axioms would add to this class some functions totally unacceptable as complements. Indeed, a violation of Axiom c1 would include functions that do not conform to the ordinary complement for crisp sets. Axiom c2 is essential since we intuitively expect that an increase in the degree of membership in a fuzzy set must result in either a decrease or, in the extreme case, no change in the degree of membership in its complement. These axioms are called axiomatic skeleton for fuzzy complements. The function c may possess some other properties, which could be useful in various situations: Axiom c3 : c is a continuous function; Axiom c4 : c is involutive i.e. c(c(a)) = a for each a ∈ [0, 1].  1 : for a ≤ t Example 1.1. : c(a) = where a ∈ [0, 1] and t ∈ [0, 1]. 0 : for a > t Example 1.2. : (a) = 21 (1 + cos πa) (non-involutive fuzzy complement)

5

Fuzzy Set and Basic Operation

Example 1.3. : cλ (a) = (−1, ∞).

1−a 1+λ a

(Sugeno class of fuzzy complement) where λ ∈

Example 1.4. : c(a) = 1 − a where a ∈ [0, 1] 1.2.2

FUZZY INTERSECTION: T-NORM

The intersection of two fuzzy sets A and B is specified by a binary operation on the unit interval; that is, a function of the form ∗ : [0, 1] × [0, 1] → [0, 1] having certain properties to give meaningful fuzzy intersection: Axiom ∗1 : a ∗ 1 = a (boundary condition) Axiom ∗2 : b ≤ d implies a ∗ b ≤ a ∗ d (monotonicity) Axiom ∗3 : a ∗ b = b ∗ a (commutativity) Axiom ∗4 : a ∗ (b ∗ a) = (a ∗ b) ∗ d (associativity) The function ∗ may possess some other properties like, Axiom ∗5 : ∗ is a continuous function Axiom ∗6 : a ∗ a ≤ a for all a ∈ (0, 1) (sub idempotency). It is easy to see that the first three axioms ensure that the fuzzy intersection defined in the formula becomes the classical set intersection when sets A and B are crisp: 0 ∗ 1 = 0 and 1 ∗ 1 = 1 follow directly from the boundary condition; 1 ∗ 0 = 0 follows then from commutativity, whereas 0 ∗ 0 = 0 follows from monotonicity. When one argument of ∗ is 1 (expressing a full membership), the boundary condition and commutativity also ensure, as our intuitive conception of fuzzy intersection requires, that the membership grade in the intersection is equal to the other argument. Monotonicity and commutativity express the natural requirement that a decrease in the degree of membership in set A or B cannot produce an increase in the degree of membership in the intersection. Commutativity ensures that the fuzzy intersection is symmetric, that is, indifferent to the order in which the sets to be combined are considered. The last axiom, associativity, ensures that we can take the intersection of any number of sets in any order of pairwise grouping desired; this axiom allows us to extend the operation of fuzzy intersection to more than two sets. It is often desirable to restrict the class of fuzzy intersection (t - norms) by considering various additional requirements. Three of the most important requirements are expressed as Axioms ∗5 , ∗6 , and ∗7 . The axiom of continuity prevents a situation in which a very small change in the membership grade of either set A or B would produce a large change in the membership grade is (A ∩ B)(x). Example 1.5. a ∗ b = min{a, b} for all a, b ∈ [0, 1]. (Standard intersection). a ∗ b = a · b (algebraic product).

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

a ∗ b = max{0, a + b − 1} (bounded difference).   a : when b = a b : when a = 1 (Drastic intersection) a∗b =  0 : otherwise. The power of this class of t-norms is expressed in Theorem 1: Theorem 1.1. For all a, b ∈ [0, 1], a ∗min b ≤ a ∗ b ≤ min(a, b) where ∗min denotes the drastic intersection. In context to crisp intersection, the fuzzy intersection, i.e. t-norm, can be generated in many ways. The method of generating class of t-norm is known as characterization theorem of t-norm Before starting this theorem, we define the following: Definition 1.1. Decreasing generator: A decreasing generator is continuous, strictly decreasing function f : [0, 1] → R such that f (1) = 0. The pseudo-inverse of f (denoted by f −1 ) is a function f −1 = R → [0, 1] given by  for a ∈ (−∞, 0)  1 f −1 (a) for a ∈ [0, f (0)] f −1 (a) =  0 for a ∈ ( f (0), ∞) where f −1 is the ordinary inverse of f . Example 1.6. Yager class of decreasing generator: fw (a) = (1 − a)w (w > 0). Many other classes of decreasing generator can be found in Ref. [2]. Theorem 1.2. Characterization theorem of t-norms: Let ∗ be a binary operation on the unit interval. Then, ∗ is an Archimedean t-norm iff there exists a decreasing generator f such that a ∗ b = f −1 ( f (a) + f (b)) for all a, b ∈ [0, 1]. Example 1.7. Given a Yager class of decreasing generator: fw (a) = (1 − a)w (w > 0). The corresponding t-norm a ∗w b is given by 1

a ∗w b = 1 − min(1, [(1 − a)w + (1 − b)w ] w ) The power of this class of t-norm is expressed in the following theorem Theorem 1.3. Let ∗w denote the class of Yager t-norm, then a ∗min b ≤ ∗w ≤ min(a, b).

Fuzzy Set and Basic Operation

1.2.3

7

FUZZY UNION: T-CONORMS

Fuzzy unions are close parallels of fuzzy intersection as they are almost opposite in nature. Definition 1.2. A fuzzy union t-conorm is a binary operation on the unit interval in the form ♢ : [0, 1] × [0, 1] → [0, 1] that satisfies at least the following axioms, for all a, b, d ∈ [0, 1] Axiom ♢1 : ♢(a, 0) = a (boundary condition) Axiom ♢2 : b ≤ d implies ♢(a, b) ≤ ♢(a, d) (monotonicity) Axiom ♢3 : ♢(a, b) = ♢(b, a) (commutativity) Axiom ♢4 : ♢(a, ♢(b, d)) = ♢(♢(a, b), d) (associativity) The aforementioned unions are called as axiomatic skeleton. Axiom ♢5 : ♢ is continuous. Axiom ♢6 : ♢(a, a) > a for all A ∈ (0, 1) (super idempotency). Example 1.8. ♢(a, b) = max(a, b) (standard union) ♢(a, b) = a + b − ab (algebraic sum) ♢(a, b) = min(1 − a + b)(bounded union) when b = 0  a b when a = 0 ♢(a, b) or ♢max (a, b) =  1 when otherwise. Similar to fuzzy intersection, the following theorems express the power of tconorm Theorem 1.4. For all a, b ∈ [0, 1], we have max(a, b) ≤ ♢(a, b) ≤ ♢max (a, b). To give characterization theorem of t-conorm. We need the following: Definition 1.3. Increasing generator: An increasing generator is a continuous function g : [0, 1] → R such that g(0) = 0, g is strictly increasing. The pseudo inverse of −1 g is denoted  by g and defined by for a ∈ (−∞, 0)  0 g−1 (a) for a ∈ [0, g(1)] g−1 (a) =  1 for a ∈ (g(1), ∞). where g−1 is the ordinary inverse of g.

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

1.2.4

CHARACTERIZATION THEOREM OF T-CONORM

Theorem 1.5. Let ♢ be a binary operation on the unit interval. Then ♢ is t-conorm iff there exists an increasing generator g such that ♢(a, b) = g−1 (g(a) + g(b)) for all a, b ∈ [0, 1]. Yager class of increasing generator and corresponding t-conorm are as follows: 1 gw (a) = aw (w > 0) and ♢w (a, b) = min(1, (aw + bw ) w ) Theorem 1.6. Let ♢w denote the class of Yager t-conorms, then max(a, b) ≤ ♢w (a, b) ≤ ♢max (a, b) for all a, b ∈ [0, 1]. More on this topic, we refer the readers to the excellent book by Klir and Yuan [2].

1.3

CONCLUSION

The basic concepts of fuzzy sets and corresponding operations are fundamental source on which most of the fuzzy mathematics rely. This chapter is written with the aforementioned objective.

REFERENCES 1. Zadeh L.A., Fuzzy sets. Information and Control, 1965;8:338–353. 2. Klir G. J., Yuan B., Fuzzy Sets and Fuzzy Logic, Theory and Applications, PHI Publication, Upper Saddle River, NJ, 2012.

and Motivation 2 Origin of Fuzzy Metric 2.1

INTRODUCTION

The concept of fuzziness found its place in probabilistic metric spaces due to Menger [11]. The main reason behind this was that, in some cases, uncertainty in the distance between two points was due to fuzziness rather than randomness. Due to the naturality of the concept of distance, it has been studied and formulated with respect to fuzzy framework by various mathematicians e. g. ([14] and others). In their papers, the constructions of fuzzy metric involved normality and convexity of fuzzy sets including its α-cuts. In this regard, the fuzzy metric introduced by Kaleva et al. [13] received much attention. It is a well-known fact that in practice when measuring a distance, we are not able, in general, to measure it precisely. This can be explicitly seen from the fact that when measuring the same distance several times, the results may differ. Usually, the average value is taken as an appropriate approximation in such a case. There are at least two approaches that enable us to describe and somehow handle this situation. The first, probabilistic and statistical approach has already been developed for many years by Menger [11]. A brief survey and the latest development on this topic can be found in Refs. [6,9,15]. The probabilistic approach is based on the idea that uses distribution function instead of non-negative real numbers as values of the distances. The fuzzy approach to the notion of distance was first observed and formulated by Kramosil and Michalek [10] in 1975. These concepts still remain interesting for several mathematicians.

2.2

KRAMOSIL AND MICHALEK FUZZY METRIC SPACES

Definition 2.1. Kramosil and Michalek [10] The triple (X, M, ∗) is a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × [0, ∞) satisfying the following axioms: (KM1) (KM2) (KM3) (KM4) (KM5)

M(x, y, 0) = 0; M(x, y,t) = 1, for all t > 0 if and only if x = y; M(x, y,t) = M(y, x,t); M(x, y,t) ∗ M(y, z, s) ≤ M(x, z,t + s); The function M(x, y, ·) : [0, ∞) → [0, 1] is left continuous, for all x, y, z ∈ X and t, s > 0.

In what follows, fuzzy metric spaces in the sense of Kramosil and Michalek [10] will be referred as KM-fuzzy metric space. DOI: 10.1201/9781003427797-2

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

Example 2.1. [16] Let X = R, the set of all real numbers. Define a ∗ b = ab for all a, b ∈ [0, 1]. For all x, y ∈ X, t ≥ 0, define ( t , if x, y ∈ X,t > 0, M(x, y,t) = t+d(x,y) 0, if x, y ∈ X,t = 0. Then M is a KM-fuzzy metric on R Example 2.2. [12] Let X be a set with at least two elements. If we define the fuzzy set M by M(x, x,t) = 1 for all x ∈ X, t > 0, and ( 0, if t ≤ 1, M(x, y,t) = 1, if t > 1. for all x ∈ X, x ̸= y, then (X, M, ∗) is KM-fuzzy metric space under any continuous t-norm ∗. Remark 2.1. Any KM-fuzzy metric defined on X is equivalent to a statistical metric space or a generalized Menger space in the sense that for all x, y ∈ X and for all t ∈ (−∞, ∞), M(x, y,t) = Fxy (t). The following example of KM-fuzzy metric space can be found in Ref. [17] Example 2.3. (1) Let X = [k, +∞) with k > 0. Define a map M : X × X × [0, ∞) → √ min{x,y}+ √t [0, 1] by M(x, y,t) = max{x,y}+ t for all t > 0 and M(x, y,t) = 0 for t = 0. Then (X, M, ∗) is a KM-fuzzy metric space where t-norm ∗ is the product t-norm. However, (X, M, ∗) is not a KM-fuzzy metric space in case of min t-norm. (2) Let (X, d) be a metric space. Define a map M : X × X × [0, ∞) → [0, 1] by   h −d(x, y) i M(x, y,t) = exp g(t) for all t > 0 and M(x, y,t) = 0 for t = 0, where g(t) : [0, ∞) → [0, ∞) is a left continuous and increasing function and satisfies lim g(t)) = ∞.

t→∞

Then (X, M, ∗) is a KM-fuzzy metric space. Kramosil and Michalek expressed their views on the possibilities of introducing fuzziness into the topological spaces theory, especially the notion of convergence. Perhaps these aspects might become an inspiration for M. Grabiec [3] who initiated the study of convergent sequence in KM-fuzzy metric space and established fuzzy Banach contraction theorem and Fuzzy Edelstein contraction theorem.

Origin and Motivation of Fuzzy Metric

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On the other hand, in Refs. [4,16], it was observed that the notion of completeness (defined in Chapter 3) of fuzzy metric given by Grabiec has a disadvantage; in fact, if d is the Euclidean metric in R, then the induced fuzzy metric (Md , ∗) of Example 2.1 given in Ref. [16] is not G-complete. Therefore, to retrieve more faithfully, the classical notion of metric to the fuzzy context, George and Veeramani modified in Refs. [4,5] some axioms of the ones established by Kramosil and Michalek and introduced a new concept of fuzzy metric (now known as GV-fuzzy metric). Indeed, there are still various aspects (in terms of properties) of KM-fuzzy metric like- T- completeness in the context of domain theory and a new type of Banach contraction theorem in the realm of this metric. We will discuss these results in detail in coming chapters.

2.3

GEORGE AND VEERAMANI FUZZY METRIC SPACES

Definition 2.2. George and Veeramani [4] The triple (X, M, ∗) is called a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following axioms: (GV1) (GV2) (GV3) (GV4) (GV5)

M(x, y,t) > 0; M(x, y,t) = 1 if and only if x = y; M(x, y,t) = M(y, x,t); M(x, y,t) ∗ M(y, z, s) ≤ M(x, z,t + s); M(x, y, .) : (0, ∞) → [0, 1] is continuous for all x, y, z ∈ X and t, s > 0.

The axiom (GV1) is justified by the authors because in the same way that a classical metric space does not take the value ∞, then M cannot take the value 0. The axiom (GV2) is equivalent to the following: M(x, x,t) = 1 for all x ∈ X, t > 0 and M(x, y,t) < 1 for all x ̸= y, t > 0 The axiom (GV2) gives the idea that only when x = y the degree of nearness of x and y is perfect, or simply 1, and then M(x, x,t) = 1 for each x ∈ X and for each t > 0. (we observe that the M in Example 2.2 (above) does not satisfy axiom (GV2)). In this manner, the value 0 and ∞ in the classical theory of metric space are identified with 1 and 0, respectively, in this fuzzy theory. Finally, in (GV5), the authors only assume that the variable t behaves nicely, i.e., they assume that for fixed x and y, the function t → M(x, y,t) is continuous without any imposition for M as t → ∞. In what follows, fuzzy metric spaces in the sense of George and Veeramani [4] will be referred as GV-fuzzy metric space. Example 2.4. Let X = R. Define a ∗ b = ab for all a, b ∈ [0, 1] and   h |x − y| i−1 M(x, y,t) = exp t for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗) is a GV-fuzzy metric space.

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The next example shows that every metric space induces a fuzzy metric space. Example 2.5. Let (X, d) be a metric space. Define a ∗ b = ab for all a, b ∈ [0, 1] and M(x, y,t) =

kt n kt n + md(x, y)

for all k, m, n ∈ N. Then (X, M, ∗) is a GV-fuzzy metric space. In particular, taking k = m = n = 1, we get t M(x, y,t) = , t + d(x, y) that is called a standard fuzzy metric. Note that the aforementioned example holds even with the t-norm a ∗ b = min{a, b}. Example 2.6. Let X = N, a ∗ b = ab and ( x/y, if x ≤ y, M(x, y,t) = y/x, if y ≤ x. It is well known that (X, M, ∗) is a fuzzy metric space. Remark 2.2. It is interesting to note that there exists no metric on X satisfying n t , if x, y ∈ X,t > 0, M(x, y,t) = t+d(x,y) where M(x, y,t) is as defined in the aforementioned example. Also note that the aforementioned function M is not a fuzzy metric with the t-norm a ∗ b = min{a, b}. The following examples of GV-fuzzy metric can be found in Ref. [8]. Here (∧) stand for min. 2.3.1

GENERALIZING WELL-KNOWN FUZZY METRICS

Example 2.7. Let f : X → R+ be one-to-one onto function and let g : R+ → [0, +∞] be an increasing continuous function. Fixed α, β > 0, define M by α

(min{ f (x), f (y)}) +g(t) β M(x, y,t) = ( (max{ f (x), f (y)})α +g(t) )

Then (M, ·) is Fuzzy metric on X . Proof. We only show the triangle inequality: Suppose f (x) ≤ f (z). We can distinguish three cases: f (x) ≤ f (y) ≤ f (z), f (y) ≤ f (x) ≤ f (z) or f (x) ≤ f (z) ≤ f (y)

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Now, if we put α

α

+g(t+s) β f (y) +g(t+s) β M(x, z,t + s) = ( ff (x) (y)α +g(t+s) ) · ( f (z)α +g(t+s) )

it is easy to verify the inequality in the three cases. M(x, z,t + s) ≥ M(x, y,t) · M(y, z, s) holds, since g is increasing. Proof in the case f (x) > f (z) is similar. Now, if we take f as the corresponding identity function and α = β = 1, then we obtain the following three examples as particular cases. (A) Let X = R+ ,and let g be the identity function.Then above M becomes min{x,y}+t max{x,y}+t .

M(x, y,t) =

(B) Let X = N, and take g as the zero function.Then above M becomes M(x, y) =

min{x,y} max{x,y} .

(C) Let X = [−k, +∞](k > 0), and take g as the constant function g(t) = k. Then above M becomes M(x, y) =

min{x,y}+k max{x,y}+k

and this is a stationary fuzzy metric. It is easy to verify that (M, ∧) is not a fuzzy metric in the last three examples. 2.3.2

FUZZY METRICS DEFINED BY MEANS OF A METRIC

In the next two examples g : R+ → R+ is an increasing continuous function, and d is metric on X. Example 2.8. Let m > 0. Define the function M M(x, y,t) =

g(t) g(t)+m·d(x,y) .

Then (M, ·) is a fuzzy metric on X. As a particular case, if we take g(t) = t n , where n ∈ N and m = 1, then M(x, y,t) =

tn t n +d(x,y) .

So (M, ∧) is a fuzzy metric space. In particular n = 1 the well known standard fuzzy metric. On the other hand, if we take g as a constant function, g(t) = k > 0 and m = 1, we obtain

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M(x, y,t) =

k k+d(x,y)

and so M(, ·) is stationary fuzzy metric space on X but, in general (M, ∧) is not. Example 2.9. Define the function M by M(x, y,t) = e(−d(x,y)/g(t)) . Then (M, ·) is a fuzzy metric on X. As a particular case, if we take g as the identity function, then M(x, y,t) becomes M(x, y,t) = e(−d(x,y)/t) . In this case (M, ∧) is fuzzy metric on X. On the other hand, if we take g as a constant function g(t) = K > 0, then M(x, y,t) becomes M(x, y,t) = e(−d(x,y)/k) . and so M(, ·) is fuzzy metric space on X but, in general (M, ∧) is not. Example 2.10. Let (X, d) be a bounded metric space and suppose d(x, y) < k for all x, y ∈ X. Let g : R+ → [k, +∞] be an increasing continuous function. Define the function M by M(x, y,t) = 1 − d(x,y) g(t) . Then (M, £) is fuzzy metric on X but, (M, ·) is not. If we take g as a constant function g(t) = K > k, then M(x, y,t) becomes M(x, y) = 1 − d(x,y) K and so (M, £) is stationary fuzzy metric on X, but (M, ·) is not. 2.3.3

DISCRETE FUZZY METRIC

Example 2.11. Let φ ; R+ → [0, 1] be an increasing continuous function. Define the function M by ( 1 if x=y M(x, y,t) = . φ (t) if x ̸= y Then (M, ∧) is fuzzy metric on X. In particular if φ is a constant function φ (t) = k ∈ [0, 1] Then M(x, y,t) becomes ( 1 if x = y M(x, y) = k if x ̸= y that will call discrete fuzzy discrete metric due to analogy with the classical discrete metric.

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Origin and Motivation of Fuzzy Metric

2.3.4

FUZZY METRICS DEDUCED FROM METRICS

Example 2.12. If (M, ∗) and (N, ∗) are fuzzy metrics on X, then it is known that M ∧ N and M ∗ N (supposed M ∗ N(x, y,t) > 0 for each x, y ∈ X) are fuzzy metric on X. The following examples are other ways to construct fuzzy metric deduced from other fuzzy metrics. Example 2.13. Let (N, ·) be a fuzzy metric on X and let φ : R+ → R+ an increasing continuous function. Define the function M by M(x, y,t) =

φ (t)+N(x,y,t) . φ (t)+1

Then (M, ·) is a fuzzy metric on X. Proof. We only show the triangular inequality. Let a, b ∈ [0, 1]. Since φ is increasing, it is easy to show that φ (t+s)+a φ (t+s)+1

· φφ (t+s)+b (t+s)+1 ≤

φ (t+s)+ab φ (t+s)+1 .

φ (t)+a φ (t)+1

· φφ (s)+b (s)+1 ≤

Then, for all x, y, z ∈ X,t > 0, we have

φ (t) + N(x, y,t) φ (s) + N(x, y, s) · φ (t) + 1 φ (s) + 1 φ (t + s) + N(x, y,t) · N(y, z, s) ≤ φ (t + s) + 1 φ (t + s) + N(x, z,t + s) ≤ = M(x, z,t + s). □ φ (t + s) + 1

M(xy,t) · M(y, z, s) =

Example 2.14. Let (N, ∗) be a fuzzy metric on X and let φ : R+ →]0, 1] be an increasing continuous function. Suppose N(x, y,t) ∗ φ (t) ̸= 0 for all x, y ∈ X,t > 0. Define the function M by  1 :x=y M(x, y,t) = N(x, y,t) ∗ φ (t) : x ̸= y Then (M, ∗) is a fuzzy metric on X. As a particular case, take X = {1} ∪ {1  − 1/(n + 1) : n ∈ N} and ∗ the usual prodt :t 0 where φ : R+ →]0, 1] is an increasing continuous function. Define the function M by  N(x, y,t) : x, y ∈ A or x, y ∈ B M(x, y,t) = N(x, y,t) ∗ φ (t) : elsewhere Then (M, ∗) is a fuzzy metric on X. The following are three particular cases in which φ is given by  t :0 0, Md (x, y,t) = t+d(x,y) Example 2.17. Let X = (0, +∞), a ∗ b = ab for all a, b ∈ [0, 1] and M(x, y,t) = min{x,y} max{x,y} for all x, y ∈ X and t > 0. Clearly, (X, M, ∗) is a non-Archimedean fuzzy metric space. Definition 2.4. [7] A fuzzy metric space (X, M, ∗) is said to be stationary if M does not depend on t, i.e., if for each x, y ∈ X, the function Mxy is constant. Obviously, stationary fuzzy metrics are strong. The standard fuzzy metric Md is a fuzzy ultrametric if and only if d is an ultrametric. Consequently, if d is a metric which is not ultrametric, then (Md , min) is a non-strong fuzzy metric on X. In Ref. [7], it was asked to find a non-strong fuzzy metric (X, M, ∗) where ∗ ̸=min. In 2011, Gutierrez and Romaguera [2] gave a positive answer to the aforementioned question by providing the following examples: Example 2.18. Let X = {x, y, z}, ∗ = · and M : X × X × (0, ∞) → [0, 1] defined for each t > 0 as: M(x, x,t) = M(y, y,t) = M(z, z,t) = 1, M(x, z,t) = M(z, x,t) = t M(y, z,t) = M(z, y,t) = t+1 , M(x, y,t) = M(z, x,t) = M(y, x,t) = (X, M, .) is a non-strong fuzzy metric on X.

{t 2 } , {(t+2)2 }

then

Example 2.19. Let X = {x, y, z}, ∗ = TL (the Lukasiewicz t-norm) and M : X × X × (0, ∞) → [0, 1] defined for each t > 0 as: M(x, x,t) = M(y, y,t) = M(z, z,t) = 2t+1 1, M(x, z,t) = M(z, x,t) = M(y, z,t) = M(z, y,t) = 2t+2 , M(x, y,t) = M(z, x,t) = t M(y, x,t) = t+2 , then (X, M, .) is a non-strong fuzzy metric on X. Further, Gutierrez and Romaguera [2] asked to find a non-strong fuzzy metric (X, M, ∗) for continuous t- norm that are greater than the product but different from minimum.

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In 2014 [1], Castro et al. gave a positive answer to the aforementioned question by providing the following example: {2ab} Example 2.20. Let X = {x, y, z}, a ∗ 1 b = {1+a+b−ab} for all a, b ∈ [0, 1] and M : 2 X × X × (0, ∞) → [0, 1] defined for each t > 0 as: M(x, x,t) = M(y, y,t) = M(z, z,t) = 1, {t} , M(x, z,t) = M(z, x,t) = M(y, z,t) = M(z, y,t) = {t + 3}

M(x, y,t) = M(z, x,t) = M(y, x,t) = metric on X.

{t 2 } , {(t+3)2 }

then (X, M, .) is a non-strong fuzzy

For more examples on related topic, we refer the readers to Ref. [1]. Remark 2.3. There are some other especial types of fuzzy metric, namely, principle fuzzy metric and s-fuzzy metric. These fuzzy metrics are based on the notions of convergent sequence. Therefore, we postponed the discussion on this topic for a while. Moreover, there are theoretical properties of fuzzy metrics which have applications in a variety of practice problems such as filter color image and to measure the degree of consistency of elements in a dataset. We shall discuss these aspects of fuzzy metrics in Chapter 7.

2.4

CONCLUSIONS

The notion of fuzzy metric spaces is introduced for the first time by I. Kramosil and J. Michalek in 1975; thus, releasing axioms to the fuzzy metric spaces requires a function of the distance with supremum 1, in relation to the axiomatic probability of metric spaces. The modified definition of the fuzzy metric spaces is introduced by A. George and P. Veeramani in 1994, which relieves axiomatic of the fuzzy metric spaces, and it was desired that the infimum of the function of the distance is 0, in relation to the probability approximation space. Today, they are studying the fuzzy metric spaces in terms of both definitions. These types of fuzzy metrics are interesting for engineering problems mainly due to the following two advantages with respect to classical metric. First, the values given by fuzzy metrics are in the interval (0, 1] regardless of the nature of the distance concept being measured. This implies that it is easy to combine different distance criteria that may originally be in quite different ranges, but fuzzy metrics take to a common range. Second, fuzzy metrics match perfectly with the employment of other fuzzy techniques. The value given by a fuzzy metric can be directly employed or interpreted as a fuzzy certainty degree. For instance, in Ref. [8], fuzzy metrics are applied successfully to color image filtering, improving some filters when replacing classical metrics and allowing the design of new filter methods. This allows us to straightforwardly include fuzzy metrics as part of the complex systems.

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REFERENCES 1. Castro-Compan F., Tirado P., On Yager and Hamacher t-norms and fuzzy metric spaces. International Journal of Intelligent Systems, 2014;29:1173–1180. 2. Gutierrez G. J., Romaguera S., Examples of non-strong fuzzy metric spaces. Fuzzy Sets and Systems, 2011;162:91–93. 3. Grabiec M., Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 1988;27:385– 389. 4. George A., Veeramani P., On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 1994;64:395–399. 5. George A., Veeramani P., On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 1997;90(3):365–368. 6. Gopal D., Kumam P., Agarwal P., Metric Structures and Fixed Point Theory. CRC Press, Boca Raton, FL. 2021. 7. Gregori V., Morillas S., Sapena A., On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 2010;161:2193–2205. 8. Gregori V., Morillas S., Sapena A., Examples of fuzzy metrics and applications. Fuzzy Sets and Systems, 2011;170:95–111. 9. Hadızi´c O., Pap E., Fixed Point Theory in PM-Spaces. Kluwer Academic Publishers, Dordrecht, 2001. 10. Kramosil I., Michalek J., Fuzzy metric and statistical metric spaces. Kybernetica, 1975;11:336–344. 11. Menger K., Statistical metric. Proceedings of the National Academy of Sciences of the United States of America, 1942;28:535–537. 12. Mihet D., Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 2008;159:739–744. 13. Kaleva O., Seikkala S., On fuzzy metric spaces. Fuzzy Sets and Systems, 1984;12(3):215– 229. 14. Diamond P., Kloeden P., Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 1990;35,2:241–249. 15. Schweizer B., Sklar A., Statistical metric spaces. Pacific Journal of Mathematics, 1960;10;313–334. 16. Vasuki R., Veeramani P., Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 2003;135:415–417. 17. Wang K., T-complete KM-fuzzy metric spaces via domain theory. Fuzzy Sets and Systems, 2022;437:69–80. 18. Mihet D., On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 2007;158:915–921. 19. Gregori V., L´opez-Crevill´en A., Morillas S., Sapena A., On convergence in fuzzy metric spaces. Topology and Its Applications, 2009;156:3002–3006

in Fuzzy 3 Convergence Metric Spaces 3.1

INTRODUCTION

The notion of convergence and completeness in fuzzy metric space in the sense of Kramosil and Michalek [14] (in short, KM-fuzzy metric space)was first introduced by M. Grabiec [1] in order to prove fuzzy version of Banach contraction theorem and Edelstein contraction theorem. Definition 3.1. [1] Let (X, M, ∗) be a fuzzy metric space. Then i. a sequence {xn }n∈N in X is said to be convergent to x ∈ X, if lim M(xn , x,t) = 1 n→∞ for all t > 0. ii. a sequence {xn }n∈N in X is said to be Cauchy (or G-Cauchy) if lim M(xn , xn+p ,t) = 1 for each p ∈ N and t > 0. n→∞

A fuzzy metric space in which every Cauchy (or G-Cauchy) sequence is convergent is called complete (or G-complete). Definition 3.2. A fuzzy metric space (X, M, ∗) is called compact if every sequence contains a convergent subsequence. In Refs. [2,17], it has been observed that the notion of G-completeness has disadvantage, and so its applicability was drastically reduced because the concept of Gcompleteness is so weak that even compact spaces are not necessarily G-complete. The following example illustrates the aforementioned fact. Example 3.1. Let X = [−1, 1]. Obviously (X, d) is a compact metric space, where d is the Euclidean metric. Therefore, (X, Md , .) is a compact √ fuzzy metric space. Let {xn }n∈N be a sequence in (X, Md , .) given by {xn } = sin n, for n ∈ N. It is easy to show that lim M(xn , xn+p ,t) = 1 for each t > 0. So {xn } is a non-convergent n→∞

G-Cauchy sequence, therefore (X, Md , .) is a compact fuzzy metric space that is not G-complete. Remark 3.1. In (Tirado) [16], it is proved that each complete non-Archimedean or strong fuzzy metric space is G-complete. So the notion of G-completeness seems very interesting in the context of non-Archimedean fuzzy metric space.

DOI: 10.1201/9781003427797-3

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To overcome the inconveniences on the notion of G-completeness and to make compatible with the notion of Cauchy sequence in classical metric, George and Veeramani [2,3] presented the following: Definition 3.3. George and Veeramani [2] Let (X, M, ∗) be fuzzy metric space. Then a sequence {xn }n∈N in X is said to be a Cauchy sequence (or M-Cauchy sequence) if, for each ε ∈ (0, 1) and t > 0, there is n0 ∈ N such that M(xn , xm ,t) > 1 − ε for all n, m ≥ n0 . A fuzzy metric space in which every Cauchy sequence (M-Cauchy sequence) is convergent is called complete (M-complete). It is called compact if every sequence contains a convergent subsequence. Remark 3.2. George and Veeramani [2] the metric space (X, d) is complete if and only if the standard fuzzy metric space (X, Md , ∗) is complete. The fact that the significant difference between fuzzy metric and classical metric is that the first one includes in its definition a parameter t. Due to this, several well-motivated notions of convergent and Cauchy sequences in fuzzy metric are introduced. In fact there are well-developed theories on convergent sequences and so the completions of fuzzy metric spaces. Now we discuss all these aspects one by one. To do so, we first recall some topological notions such as open ball, closed ball, etc. with respect to GV-fuzzy metric space. Definition 3.4. [2,3] Let (X, M, ∗) be fuzzy metric space. We define open ball B(x, r,t) with centre x ∈ X and radius r, 0 < r < 1, t > 0 as B(x, r,t) = {y ∈ X : M(x, y,t) > 1 − r.} The following results can be found in Refs. [2,3] Result 1: Every open ball is an open set. Result 2: Let (X, M, ∗) be fuzzy metric space. Define τ = {{A ⊂ X : x ∈ X} if and only if there exists t > 0 and r, 0 < r < 1 such that B(x, r,t) ⊂ A}. Then τ is a topology on X. Remark 3.3. Since B(x, 1n , 1n ) where {n = 1.2, ....} is a local base at x, the aforementioned topology is first countable, Result 3: Every GV-fuzzy metric space is Housdorff. Result 4: Let (X, d) be a metric space. Let M(x, y,t) =

t t + d(x, y)

be the fuzzy metric defined on X. Then the topology τ0 induced by the metric d and the topology τ induced by the fuzzy metric M are the same. Now, we are ready to discuss the notion of pointwise convergence given by Mihet [15].

Convergence in Fuzzy Metric Spaces

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3.2 p-CONVERGENT SEQUENCES Definition 3.5. Let (X, M, ∗) be a fuzzy metric space. A sequence {xn } in X is said to be pointwise convergent to x ∈ X (we write xn → p x) if there exists t > 0 such that lim M(xn , x,t) = 1.

n→∞

It is easy to see that, endowed with the point convergence, a GV-fuzzy metric space (X, M, ∗) is a space with the convergence in the sense of Fr´echet, that is, one of the following holds: a. Every sequence in X has at most one limit point. b. Every constant sequence, xn = x, ∀n ∈ N, is convergent and lim xn = x. n→∞ c. Every subsequence of a convergent sequence is also convergent and has the same limit as the whole sequence. Remark 3.4. It is worth noting that if the point convergence in a fuzzy metric space (X, M, ∗) is Fr´echet, then (GV2) holds (so the uniqueness of the limit in the point convergence characterizes, in a sense, a fuzzy metric space in the sense of George and Veeramani). Indeed, let x, y ∈ X with x ̸= y. If M(x, y,t) = 1 for some t > 0, then the sequence {xn }n∈N ⊂ X defined as x, y, x, y, . . . has two distinct limits, for the equality M(x, x,t) = M(y, x,t) = 1 implies xn → p x, while M(x, y,t) = M(y, y,t) = 1 implies xn → p y. In the following example, we will see that there exist p-convergent but not convergent sequences. Example 3.2. Let {xn }n∈N ⊂ (0, ∞) with xn → 1 and X = {xn } ∪ {1}. Define M(xn , xn ,t) = 1 for all n ∈ N and t > 0, M(1, 1,t) = 1 for all t > 0, M(xn , xm ,t) = min{xn , xm } for all n, m ∈ N and t > 0 and  min{xn ,t}, if 0 < t < 1, M(xn , 1,t) = xn , if t > 1, for all n ∈ N. Then (X, M, TM ), where TM (a, b) = min{a, b}, is a fuzzy metric space (see [6], Example 3.2]). Since lim M(xn , 1, 12 ) = 12 , {xn } is not convergent. Nevern→∞

theless, it is p-convergent to 1 for lim M(xn , 1, 2) = 1. n→∞

Indeed, in Chapter 4, we shall discuss the role of p-convergent in the context of existence of fixed point for fuzzy contractive type mappings. Since this new notion of p-convergence gives rise of some especial class of fuzzy metric theory, therefore, we now look into the theory of completion of GV -fuzzy metric spaces. Note that a topological space (X, τ) admits a compatible fuzzy metric if there is a fuzzy metric M on X such that τ = τM . Thus, from Result 3, every metrizable topological space admits a compatible fuzzy metric. Conversely, we shall prove that every fuzzy metric space is metrizable.

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To prove the promised result, we need the following: Lemma 3.1. [5] A T1 topological space (X, τ) is metrizable if and only if it admits a compatible uniformity with a countable base. Lemma 3.2. [3] Let (X, M, ∗) be a fuzzy metric space. Then τM is a Housdorff topology and for each x ∈ X, B(x, 1n , 1n :){n = 1.2, ....} is a neighbourhood base at x for the topology τM . Theorem 3.1. [5] Let (X, M, ∗) be a fuzzy metric space. Then, (X, τM ) is a metrizable topological space. Proof: For each n ∈ N define 1 1 Un = {(x, y) ∈ X × X : M(x, y, ) > 1 − .}. n n We shall prove that Un : n ∈ N is a base for a uniformity U on X whose induced topology coincides with τM : We first note that for each n ∈ N, {(x, x) ∈ X ⊆ Un , Un+1 ⊆ Un and Un = U − 1n . On the other hand, for each n ∈ N, there is, by the continuity of ∗, an m ∈ N such that m > 2n and (1 −

1 1 1 ) ∗ (1 − ) > (1 − ). m m n

Then, Um ⊙Um ⊆ Un : Indeed, let (x, y) ∈ Um and (y, z) ∈ Um . Since M(x, y, .) is nondecreasing M(x, z, n1 ) ≥ M(x, z, m2 ). So 1 1 1 M(x, z, ) ≥ M(x, y, ) ∗ M(y, z, ) n m m 1 1 1 ≥ (1 − ) ∗ (1 − ) > (1 − ). m m n Therefore (x, z) ∈ Un . Thus Un : n ∈ N is a base for a uniformity U on X. Since for each x ∈ X and each n ∈ N, 1 1 1 1 Un (x) = {y ∈ X : M(x, y, ) > 1 − = B(x, , )}, n n n n we deduce, from Lemma 3.2, that the topology induced by U coincides with τM . By Lemma 3.1,(X, τM ) is a metrizable topological space. Corollary 3.1. A topological space is metrizable if and only if it admits a compatible fuzzy metric. Let (X, M, ∗) be a fuzzy metric space. Then, (X, τM ) is a metrizable topological space. The proofs of the following important results can be found in Ref. [5] Theorem 3.2. [5] Let (X, M, ∗) be a complete fuzzy metric space. Then, (X, τM ) is completely metrizable.

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Corollary 3.2. A topological space is completely metrizable if and only if it admits a compatible complete fuzzy metric. Definition 3.6. Let (X, M, ∗) and (Y, N, ⋄) be two fuzzy metric spaces. Then (a) A mapping f from X to Y is called an isometry if for each x, y ∈ X and each t > 0, M(x, y,t) = N( f x, f y,t). (b) (X, M, ∗) and (Y, N, ⋄) are called isometric if there is an isometry from X to Y . Definition 3.7. Let (X, M, ∗) be a fuzzy metric spaces. A fuzzy metric completion of (X, M, ∗) is a complete fuzzy metric (Y, N, ⋄) such that (X, M, ∗) is isometric to a dense subspace of Y . Definition 3.8. A fuzzy metric space (X, M, ∗) is called completable if it admits a fuzzy metric completion. Proposition 3.1. If a fuzzy metric space has a fuzzy metric completion, then it is unique up to isometry. Proposition 3.2. Let (X, d) be metric space. Then, the standard fuzzy metric space (X, Md , .) is completable, and its fuzzy metric completion is the standard fuzzy metric space of the metric completion of (X, d). Lemma 3.3. Let {an } be a Cauchy sequence in a fuzzy metric space (X, M, ∗) and let t > 0. If for each k ∈ N there exists lim M(ak , an ,t), then limk limn (M(ak , an ,t) = 1. n

Lemma 3.4. Let (X, M, ∗) be a fuzzy metric space. Then, for each metric d on X compatible with M, the following hold: (i) A sequence in X is Cauchy in (X, M, ∗) if and only if it is Cauchy in (X, d), (ii) (X, M, ∗) is complete if and only if (X, d) is complete. Lemma 3.5. Let A be a dense subset of a fuzzy metric space (X, M, ∗). If every Cauchy sequence of points of A converges in X, then (X, M, ∗) is complete. Definition 3.9. Let (X, M, ∗) be a fuzzy metric space. Then a pair {an }, {bn } of Cauchy sequences in X, is called a. point-equivalent if there is s > 0 such that lim M(an , bn , s) = 1. n

b. equivalent, denoted by {an } ∼ {bn }, if lim M(an , bn ,t) = 1 for all t > 0. n

Theorem 3.3. [5] A fuzzy metric space (X, M, ∗) is completable if and only if it satisfies the following conditions: (c1) Given Cauchy sequences {an }, {bn } in X, then t → lim M(an , bn , s) is a continn

uous function on (0, +∞) with values in (0, 1]. (c2) Each pair of point-equivalent Cauchy sequences is equivalent.

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The following examples deal with some natural conjuncture that one may consider in the light of Theorem 3.1. Example 3.3. Consider a ∗ b = max{a + b − 1, 0}. Now let {xn }, {yn } (where n ≥ 3) be two sequences of distinct points such that A ∩ B = φ where A = {xn } : n ≥ 3 and B = {yn } : n ≥ 3. Put A ∪ B = X and define a fuzzy set M on X 2 × [0, ∞) by M(xn , xm ,t) = M(yn , ym ,t) = 1 − [

1 1 − ], (n ∧ m) (n ∨ m)

M(xn , ym ,t) = M(yn , xn ,t) =

1 1 + n m

for all n, m ≥ 3, t > 0. It was shown in Example 3.2 of Ref. [5] that (X, M, ∗) is a fuzzy metric space for which both {xn }, {yn } are Cauchy sequences in X. It is easy to check that (X, M, ∗) satisfies condition (c2) of Theorem 3.3. However, since lim M(xn , yn ,t) = 0 for any n

t > 0, it follows that of (X, M, ∗) does not satisfy condition (c1) of Theorem 3.3 and thus it is not completable. (A direct proof of the fact that (X, M, ∗) is not completable is given in Ref. [16]. Definition 3.10. A fuzzy metric space (X, M, ∗) is said to stationary if M does not depend on t, i.e if for each x, y ∈ X, the function M(x, y, ) is constant. The fuzzy metric space of Example 3.3 (above) is stationary. Obviously, each stationary fuzzy metric space (X, M, ∗) satisfies condition (c2) of Theorem 3.3. On the other hand, if {an }, {bn } are (not necessarily Cauchy) sequences in X, it is clear that there exits subsequence {an }k , {bn }k of them such that the sequence M({an }k , {bn }k )k converges to some α ∈ [0, 1] with respect to Euclidean metric. If, in addition, the sequences {an }, {bn } are Cauchy, then it is easy to check, and by using triangle inequality of fuzzy metric, the sequence M({an }, {bn })n also converges to α with respect to Euclidean metric. From these facts and Theorem 3.3, we deduce the following result. Proposition 3.3. A standard fuzzy metric space (X, M, ∗) is completable if and only if lim M({an }, {bn })n > 0 for each pair of Cauchy sequences {an }n , {bn })n in X. n

Definition 3.11. A fuzzy metric space (X, M, ∗) is called principle (or simply, M principle) if {BM (x, r,t) : r ∈ [0, 1]} is a local base at x ∈ X, for each x ∈ X and each t > 0. Let (X, M, ∗) be a non-stationary fuzzy metric. Define the family of functions {Mt : t > 0} where, each t > 0, Mt : X × X × [0, 1] → (0, 1] is given by Mt (x, y) = M(x, y,t). Then (X, M, ∗) is strong iff (X, Mt , ∗) is a stationary fuzzy metric for each t > 0.

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Also, the sequence {xn } in X is M-Cauchy iff {xn } is Mt -Cauchy for each t > 0. Here, we denote ([0, ∞], M0 , .) as stationary fuzzy metric space where M0 is defined as min{x, y} M0 (x, y) = max{x, y} It is easy to verify that τM0 is the usual topology of R restricted to [0, ∞]. Also, ([0, ∞], M ∗ , .) will be the fuzzy metric space where M ∗ is defined by M ∗ (x, y,t) =

(min{x, y} + t) (max{x, y} + t)

Its subspace is ([0, ∞], M ∗ , .). The proofs of the following results can be found in Ref. [9]. Proposition 3.4. Consider the fuzzy metric M ∗ on [0, ∞] (respectively on [0, ∞]) ∗ is the usual topology of R restricted to [0, ∞], (respectively on [0, ∞]); i. τM ii. M ∗ is principal; iii. M ∗ is strong.

Proposition 3.5. Consider M ∗ on [0, ∞]. Then i. t>0 Mt∗ is not a fuzzy metricVon [0, ∞]. ii. Consider M ∗ on [0, ∞]. Then t>0 Mt∗ is a fuzzy metric on M0 . V

Theorem 3.4. ([0, ∞], M0 , .) is complete. Corollary 3.3. ([0, ∞], M0 , .) is not precompact. Proposition 3.6. ([0, ∞], Mt∗ , .) is not complete for each t > 0. Corollary 3.4. ([0, ∞], M ∗ , .) is not complete. Lemma 3.6. Take t > 0 and consider the fuzzy metric space ([0, ∞], Mt∗ , .). Let {xn } be a sequence in [0, ∞]. Then {xn } is Mt∗ - Cauchy iff {xn } converges in [0, ∞]. Corollary 3.5. Consider the fuzzy metric space ([0, ∞], M ∗ , .). Then a sequence {xn } in [0, ∞] is Mt∗ - Cauchy iff {xn } converges in [0, ∞]. Theorem 3.5. ([0, ∞], M ∗ , .) is completable. Corollary 3.6. ([0, ∞], Mt∗ , .) is the completion of ([0, ∞], Mt∗ , .) for each t > 0. Remark 3.5. Using similar arguments to the aforementioned theorem, one can show that ([0, ∞], Mt∗ , .) is complete. Now the mapping i : ([0, ∞], M ∗ , .) → ([0, ∞], M ∗ , .) given by i(x) = x for each x ∈ [0, ∞], is an isometry and by (i) of Proposition 3.4 ∗ ), and since the completion of a fuzzy metric space is [0, ∞] is dense in ([0, ∞], τM

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unique up to isometry, then ([0, ∞], M ∗ ,′ ) is the completion of ([0, ∞], M ∗ , .). The process of obtaining the completion of ([0, ∞], M ∗ , .) naturally suggests the following question: Open Problem. To find a fuzzy metric space (X, M, ∗) where for two M- Cauchy sequences {an }n , {bn }) in X the assignment f (t) = lim{an }n , {bn },t) for all t > 0, n does not define a continuous function on t? Remark 3.6. It is known that the completion of a strong fuzzy metric is strong. On the other hand, we have just obtained above that the completion of the principal fuzzy metric space ([0, ∞], M ∗ , .) is ([0, ∞], M ∗ , .), which is also principal. Now the next is an open question. Open Problem. If the principal fuzzy metric space (X, M, ∗) admits completion (X, M, ∗), is it also principal? Let us again come back to point-wise convergence and principal fuzzy metric space. First, we give various examples of principal fuzzy metric space. a. Stationary fuzzy metrics are obviously principal. b. The well-known standard fuzzy metric is principal. c.   h |x − y| i M(x, y,t) = exp t d.

is principal. M(x, y,t) =

(min{x, y} + t) (max{x, y} + t)

is principal. Theorem 3.6. The fuzzy metric space (X, M) is principal if and only if all p- convergent sequences are convergent. Proof. Suppose that M is principal and that (xn ) is a sequence which is p-convergent to x0 , for t0 > 0. 
Let ε ∈ [0, 1] and t > 0. Since M is principal, then B x0 , n1 ,t0 : n ∈ N is a local
base at x0 . Hence, we can find m ∈ N such that B x0 , m1 ,t0 ⊂ B (x0 , ε,t). Since limn M (xn , x0 ,t0 ) = 1, we can find δ ∈ [0, 1], with δ < m1 , and n1 ∈ N such that xn ∈ B (x0 , δ ,t0 ) for all n ⩾ n1 , and thus xn ∈ B (x0 , ε,t) for all n ⩾ n1 . Hence, M (xn , x0 ,t) > 1 − ε for all n ⩾ n1 , and so limn M (xn , x0 ,t) = 1. The aforementioned argument is valid for all t > 0, then (xn ) converges to x0 . For the converse, assume that M is not principal. We will construct a p-convergent sequence which is not convergent.

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 If M is not principal we can find x0 ∈ X and t > 0 such that B x0 , 1n ,t0 : n ∈ N
is not a local base at x0 . Then, we can find t > 0 and r ∈ [0, 1] such that B x0 , n1 ,t0 ⊈ B (x0 , r,t) for all n ∈ N. Now, by induction, we form the sequence (xn ) as follows. For each n ∈ N, we take
xn ∈ B x0 , n1 ,t0 \B (x0 , r,t). Now, given ε ∈ [0, 1] we choose n1 ∈ N with n11 < ε. Hence, for m ⩾ n1 , we have M (xm , x0 ,t0 ) > 1 −

1 1 > 1− > 1−ε m n1

and, since ε is arbitrary, limn M (xn , x0 ,t0 ) = 1, and so (xn ) is p-convergent to x0 . On the other hand, by construction xn ∈ X\B (x0 , r,t) for all n ∈ N, and so (xn ) does not converge to x0 and, by Corollary 3.6, (xn ) is not convergent. The following examples illustrate the last theorem. Example 3.4. Let ϕ : R+ → [0, 1] be an increasing continuous function. Define the function M on X 2 × R+ by ( 1, x=y M(x, y,t) = ϕ(t), x ̸= y It is easy to verify that (M, ·) is a fuzzy metric on X. Now, for each x ∈ X and t > 0 we have B(x, 1 − ϕ(t),t) = {x} and so M is principal. Further, τM is the discrete topology, and then only the constant sequences are convergent, so they are the only p-convergent sequences in X. Next we give an example of a complete fuzzy metric space, which is not principal. Example 3.5. Let X = R+ and let ϕ : R+ → [0, 1] be a function given by ϕ(t) = t if t ⩽ 1 and ϕ(t) = 1 elsewhere. Define the function M on X 2 × R+ by ( 1, x = y, M(x, y,t) = min{x,y} max{x,y} · ϕ(t), x ̸= y It is easy to verify that (M, ·) is a fuzzy metric on X and, since M(x, y,t) < t, whenever t ∈ [0, 1] and x ̸= y, it is obvious that the only Cauchy sequences in X are the constant sequences and so, X is complete.
This fuzzy metric is not principal. In fact, notice that B x, 12 , 21 = {x} for each x ∈ X and so τM is the discrete topology. Now, if we set x = 1 and t = 1, we have 1 B(1, r, 1) = [1 − r, 1−r ] for all r ∈ [0, 1] and so {B(1, r, 1) : r ∈ [0, 1]} is not a local base at x = 1, since {1} is open. Now, consider the sequence (xn ) in X given by xn = 1 − n1 , n ∈ N. We have M (xn , 1, 1) = 1 − 1n for all n ∈ N, so limn M (xn , 1, 1) = 1 and (xn ) is p-convergent to 1, but (xn ) is not convergent.

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In the following example, (X, M) is a fuzzy metric space which is not principal and non-completable. Example 3.6. Let X =]0, 1], A = X ∩ Q, B = X\A. Define the function M on X 2 × R+ by  min{x,y}  x, y ∈ A or x, y ∈ B,t > 0,  max{x,y} ,  min{x,y} M(x, y,t) = max{x,y} ,    min{x,y} · t, elsewhere. max{x,y}

It is easy to verify that (M, ·) is a fuzzy metric on X. Now, we show that M is not principal. For it, we will see that {B(1, r, 1) : r ∈ [0, 1]} is not a local base at x = 1.  In fact, B(1, 1r,
1) = [11− r, 1] for all 1r ∈1
[0, 1].1 Now, since 1 ∈ A we have y ∈ B : M 1, y, 2 > 1 − 2 = 0, / so B 1, 2 , 2 = [ 2 , 1] ∩Q and clearly B(1, r, 1) ⊈
B 1, 21 , 12 for all r ∈ [0, 1], and then M is not principal. Now, it is easy to verify that for n ⩾ 2,  (  nx [x − nx , n−1 ] ∩ A, x ∈ A, 1 1 = B x, , x nx n n [x − n , n−1 ] ∩ B, x ∈ B, and clearly τM is not the discrete topology, although it is finer than the usual topology of R relative to X. Finally, let (xn ) be an increasing sequence contained in B which converges to 1 for the usual topology of R relative to X. Thus, for each r ∈ [0, 1], there exists n1 ∈ N such that xn ∈ [1 − r, 1] for all n ⩾ n1 , and by the definition of M we have M (xn , 1, 1) > 1 − r, for all n ⩾ n1 , and so limn M (xn , 1, 1) = 1, since r is arbitrary. Then, (xn ) is p-convergent to 1 . Further, (xn ) is a Cauchy sequence. Indeed, since min{xn ,xm } M (xn , xm ,t) = max{x for all t > 0, we have that M (xn , xm ,t) ⩾ xn1 > 1 − r, for n ,xm } all t > 0,
and so (xn ) is a Cauchy sequence. Now, (xn ) is not convergent since xn ∈ / B 1, 12 , 12 , for all n ∈ N and then by Corollary 3.4,(X, M) is non-completable. Continuing the aforementioned study, we give the next definition. Definition 3.12. Let (X, M) be a fuzzy metric space. A sequence (xn ) in X is said to be p-Cauchy if for each ε ∈ [0, 1] there are n0 ∈ N and t0 > 0 such that M (xn , xm ,t0 ) > 1 − ε for all n, m ⩾ n0 , i.e. limm,n M (xn , xm ,t0 ) = 1 for some t0 > 0. In such a case, we say that (xn ) is p-Cauchy for t0 > 0, or, simply, (xn ) is p-Cauchy. Clearly, (xn ) is a Cauchy sequence if and only if (xn ) is p-Cauchy for all t > 0 and, obviously, p-convergent sequences are p-Cauchy. Definition 3.13. The fuzzy metric space (X, M) is called p-complete if every p-Cauchy sequence in X is p-convergent to some point of X. In such a case, M is called p-complete.

Convergence in Fuzzy Metric Spaces

31

Obviously, p-completeness and completeness are equivalent concepts in stationary fuzzy metrics, and it is easy to verify that the standard fuzzy metric Md is p-complete if and only if Md is complete. Proposition 3.7. Let (X, M) be a principal fuzzy metric space. If X is p-complete then X is complete. Proof. Let (xn ) be a Cauchy sequence in X. Then (xn ) is p-Cauchy and so (xn ) p-converges to some point x0 ∈ X, and, since X is principal, (xn ) converges to x0 . The assumption that X is principal cannot be removed in the last proposition as shown in the following example. Example 3.7. Consider the fuzzy metric space (X, M, ∗) of Example 3.7. The sequence (xn ) satisfies limm,n M (xn , xm ,t) = 1 for all t > 0, so (xn ) is a Cauchy sequence and in consequence X is not complete, since (xn ) is not convergent. Next we show that X is p-complete. Let (xn ) be a p-Cauchy sequence in X. Then, with an easy argument one can verify that (xn ) must be a convergent sequence to 1 with respect to the usual topology of R relative to X. Now, limn M (xn , 1, 1) = limn xn = 1 and hence (xn ) is p-convergent to 1. One could expect p-Cauchy sequences to be Cauchy sequences in principal fuzzy metric spaces. In fact, this property is satisfied by all examples (a,b,c) above. Nevertheless, as shown in the following example, it is not true, in general, for any principal fuzzy metric and, in consequence, the converse of the aforementioned proposition is not true. Example 3.8. Let X = [0, 1] and define the function M on X 2 × R+ by   x = y, 1, M(x, y,t) = xyt, x ̸= y,t ⩽ 1,   xy, x ̸= y,t > 1 It is easy to verify that (M, ·) is a fuzzy metric on X. Now, let x ∈ X and t > 0. If we take r ∈ [0, 1] such that 1 − r > x then B(x, r,t) = {x} and so M is principal. Further, τM is the discrete topology and then the only convergent sequences or (by Theorem 3.6) p-convergent sequences, are the constant sequences.
Now, X does not have Cauchy sequences, since M x, y, 12 ⩽ 12 for each x, y ∈ X, and so X is complete. Let (xn ) be a strictly increasing sequence convergent to 1 in the usual topology of R, relative to X. We have limm,n M (xn , xm , 1) = 1 and then (xn ) is p-Cauchy. Nevertheless (xn ) is not p-convergent since (xn ) is not constant, and thus X is not pcomplete. Remark 3.7. It is an open problem to characterize those fuzzy metric spaces where the family of p-Cauchy sequences and Cauchy sequences agree, or further, when it is satisfied that completeness is equivalent to p-completeness.

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Most recently, [13] extended the study of characterization of fuzzy metric spaces by introducing the notions of p-completeness and w-completeness. First we recall the followings: Definition 3.14. Let (X, M, ∗) be a fuzzy metric space. The fuzzy diameter of a (non-empty) set A of X, with respect to t, is the function φA : [0, +∞] → [0, 1] given by φA (t) = inf{M(x, y,t) : x, y ∈ A}, for each t > 0. Proposition 3.8. The function φA is well-defined and, in addition, it satisfies the following: i. If s < t then φA (s) ≤ φA (t) ii. If A ⊂ B then φA (t) ≥ φB (t) iii. φA (t) = 1 for some t if and only if A is a singleton set.

3.3

p-FUZZY DIAMETER

We start this section by recalling a definition introduced by George and Veeramani [3]. Definition 3.15. Let (X, M, ∗) be a fuzzy metric space. A collection of non-empty sets {Ai }i∈I in X is said to have fuzzy diameter zero if for each r ∈ [0, 1] and t > 0 we can find ir,t ∈ I (depending on r and t) such that M(x, y,t) > 1 − r for all x, y ∈ Ai . According to the previous concept, we introduce the following weaker definition. Definition 3.16. Let (X, M, ∗) be a fuzzy metric space. A collection of non-empty sets {Ai }i∈I of X has p-fuzzy diameter zero if there exists t0 > 0 such that for each r ∈ [0, 1] we can find ir ∈ I (depending on r) such that M (x, y,t0 ) > 1 − r for each x, y ∈ Ai . We also say that {Ai } has p-fuzzy diameter zero for t0 . In the following, by a nested sequence of sets {An }, we mean a sequence of nonempty sets {An } of X satisfying An+1 ⊂ An for all n ∈ N. Then, {An } has fuzzy diameter zero if and only if given r ∈ [0, 1] and t > 0, there exists nr,t ∈ N such that M(x, y,t) > 1 − r for all x, y ∈ An with n ≥ nr,t , or equivalently, limn φAn (t) = 1 for all t > 0 (see Ref. [13], Proposition 3.2). We will omit the proofs of the following propositions because they are immediate or can be obtained by mimicking the corresponding ones in Ref. [13]. Proposition 3.9. Let {An } be a nested sequence of sets of a fuzzy metric space X. Then {An } has p-fuzzy diameter zero if and only if there exists t0 > 0 such that for each r ∈ [0, 1] there exists nr ∈ N such that M (x, y,t0 ) > 1 − r for all x, y ∈ An , n ≥ nr . Remark 3.8. If {An } has p-fuzzy diameter zero for t0 > 0 then, obviously, it has p-fuzzy diameter zero for each t ≥ t0

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33

Proposition 3.10. Let {An } be a nested sequence of sets of a fuzzy metric space X T which has p-fuzzy diameter zero, with non-empty intersection. Then, An = {x}, for some x ∈ X. Proposition 3.11. Let {An } be a (nested) eventually constant sequence of sets of a fuzzy metric space X, i.e., there exists n0 ∈ N such that An = A for all n ≥ n0 . Then {An } has p-fuzzy diameter zero if and only if A is a singleton set. In general, a nested sequence of sets {An } has p-fuzzy diameter zero if the sequence contains small sets whose fuzzy diameter, for some t0 > 0, tends to 1. We formalize this in the following proposition. Proposition 3.12. Let {An } be a nested sequence of sets of a fuzzy metric space X. Then the following conditions are equivalent: i. {An } has p-fuzzy diameter zero for some t0 > 0. ii. limn φAn (t0 ) = 1 for some t0 > 0. It is easy to conclude that every family of sets of X which has fuzzy diameter zero has p-fuzzy diameter zero. The converse is false as we show in the following example given by Mihet [15]. Example 3.9. Let {xn } be a strictly increasing sequence of positive real numbers that converges to 1, in the usual topology of R. Consider the fuzzy metric space (X, M, ∧) where X = {x1 , x2 , . . . , } ∪ {1}, ∧, is the minimum t-norm and M is defined as follows: M(x, x,t) = 1 for all x ∈ X,t > 0 M (xn , xm ,t) = M (xm , xn ,t) = xn ∧ xm for all t > 0, if n ̸= m M (xn , 1,t) = M (1, xn ,t) = xn ∧ t for all t > 0 Let An = {xn , xn+1 , . . .} ∪ {1} for each n ∈ N. Clearly, {A
n } is a nested sequence, and it is easy to verify that limn φAn (1) = 1 and limn φAn 12 = 12 . Then, by Proposition 3.12, {An } has p-fuzzy diameter zero, and from Proposition 3.2 in Ref. [8], it has not fuzzy diameter zero.

3.4 p-ACCUMULATION We start this section with a natural definition. Definition 3.17. Let A be a non-empty set of a fuzzy metric space X. A point x ∈ X is called a p-accumulation point (briefly, p-acc point) of A if there exists t0 > 0 such that for each r ∈ [0, 1] we have that (B (x, r,t0 ) − {x}) ∩ A ̸= 0. / In such a case, if necessary, we will say that x is a p-acc point for t0 . Proposition 3.13. Let (X, M, ∗) be a fuzzy metric space and let A ⊆ X. A point x ∈ X is a p-acc point of A, for t0 , if and only if there exists a sequence {an } in A − {x} such that limn M (x, an ,t0 ) = 1, i.e., {an } is p-convergent to x for t0 .

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Proof. Suppose

x is a p-acc point of A. For n = 2, 3, . . . we have that B x, n1 ,t0 − {x} ∩ A ̸= 0. / Then, we can construct a sequence {an }, taking an ∈ A with an ̸= x, such that M (x, an ,t0 ) > 1 − n1 for each n ≥ 2. Then, limn M (x, an ,t0 ) = 1 Conversely, suppose that {an } is a sequence in A−{x} such that limn M (x, an ,t0 ) = 1. Then, for ε ∈ [0, 1] we can find nε such that M (x, an ,t0 ) > 1 − ε for all n ≥ nε , i.e., an ∈ B (x, ε,t0 ) with an ̸= x. Then (B (x, ε,t0 ) − {x}) ∩ A ̸= 0. / ∼t0

Definition 3.18. The p-closure of a set A of X for t0 > 0, denoted by A , is the set A ∪ {x ∈ X : x isSa p-accumulation point of A for t0 }. The p-closure of A, denoted by ∼t ˜ will be A˜ = t>0 A, A. Under this notation, the following proposition is immediate. Proposition 3.14. Let (X, M, ∗) be a fuzzy metric space and let A ⊆ X. Then, ∼t0

i. x ∈ A if and only if for each ε ∈ [0, 1] we have that B (x, ε,t0 ) ∩ A ̸= 0. / ii. x ∈ A if and only if there exists a sequence {an } in A such that limn M (x, an ,t0 ) = 1, i.e., {an } is p-convergent to x, for t0 . ∼ iii. A¯ ⊂A, for all t > 0, where A¯ denotes the closure of A in τM . ∼t1

iv. If t1 ≥ t0 , then A ⊃ At0 . ∼ ¯ for all t > 0, and then A˜ = A. ¯ v. If X is principal then A= A, Definition 3.19. Let {xn } be a sequence in a fuzzy metric space X. A point x of X is called a p-cluster point of {xn } for t0 > 0 if {xn } is frequently in B (x, r,t0 ) for each r ∈ [0, 1], i.e., for each r ∈ [0, 1] we have that given n ∈ N we can find m ≥ n such that xm ∈ B (x, r,t0 ). Remark 3.9. If x is a p-cluster point of {xn } for t0 , then, obviously, x is a p-cluster point of {xn } for each t ≥ t0 . Also, if {xn } is eventually constant, i.e., there exists n0 ∈ N such that xn = x for all n ≥ n0 , then x is the unique cluster point of {xn } (and so a p-cluster point for each t > 0). Proposition 3.15. Let {xn } be a sequence in a fuzzy metric space X. A point  x ∈ X is a p-cluster point of {xn } for t0 if and only if there exists a subsequence xnk of {xn } which is p-convergent to x for t0 . Proof. Let x ∈ X be a p-cluster point of {xn } for t0 . Since {xn } is frequently in
B (x, r,t0 ) for each r ∈ [0, 1], then for m = 2 we can take xn2 ∈ B x, 12 ,t0 . By induction on m, we can
construct the subsequence {xnm } of {xn } where nm > nm−1 and xnm ∈ B x, m1 ,t0 . Then M (x, xnm ,t0 ) > 1 − m1 for each m ≥ 2 and thus limn M (x, xnm ,t0 ) = 1. Conversely, suppose x is not a p-cluster point of {xn } for t0 > 0. Then, we can find r0 ∈ [0, 1] such that {xn } is not frequently in B (x, r0 ,t0 ). Therefore {xn } is eventually in X − B (x, r0 ,t0 ). Thus, every subsequence of {xn } is eventually in X − B (x, r0 ,t0 ), and so it cannot be p-convergent to x for t0 .

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Next we show a characterization of p-cluster points by means of p-closure. T Theorem 3.7. Let {xn } be a sequence in a fuzzy metric space X. Then A˜ n is the set of p-cluster points of {xn } for t0 , where An = {xm : m ≥ n} for every n ∈ N.

Proof. Suppose x is a p-cluster point of {xn } for t0 . Then {xn } is frequently in B (x, ε,t0 ) for all ε ∈ [0, 1] and thus, for each r ∈ [0, 1] we have that An ∩ B (x, r,t0 ) ̸= 0/ T for all n ∈ N. Then, x ∈ An for all n ∈ N i.e., x ∈ t˜n . Conversely, if x is not a p-cluster point of {xn } for t0 , then there exists r0 ∈ [0, 1] such that {xn } is not frequently in B (x, r0 ,t0 ), i.e., for some n0 ∈ N we have that xn ∈ / B (x, r0 ,t0 ) for all n ≥ n0 . Then B (x, r0 ,t0 ) ∩ An = 0/ for n ≥ n0 , and therefore x is not in An . Example 3.10. Consider the fuzzy metric space (X, M, ∧) of Example 3.8. a. Let {yn } be a non-eventually constant sequence in X. We claim that {yn } is p-convergent if and only if {yn } is convergent to 1, in the usual topology of R. Further, in that case {yn } is p-convergent to 1, only for t ≥ 1. Indeed, suppose that {yn } is p-convergent to x < 1. Then M (x, yn ,t) ≤ max{x,t} < 1 whenever yn ̸= x and t < 1, and M (x, yn ,t) ≤ x whenever yn ̸= x and t ≥ 1. Hence {yn } is not p-convergent to x, for any t > 0. Now, suppose {yn } is p-convergent to 1 for some t > 0. We claim that t ≥ 1. Indeed, in another case, if t < 1, M (1, yn ,t) ≤ t < 1 whenever yn ̸= 1 and hence {yn } is not p-convergent for t < 1. Finally, {yn } is p-convergent to 1 for t ≥ 1 if and only if limn M (1, yn ,t) = limn yn = 1, i.e., if and only if {yn } is a convergent sequence to 1, in the usual topology of R. b. Let A ⊂ X. If x is a p-accumulation point of A then necessarily x = 1. Indeed, by Proposition 3.13, we can find a sequence {yn } in A − {x} which is p-convergent to x. Now, {yn } is not eventually constant, since in that case {yn } converges in A − {x}. So, by (a), {yn } is p-convergent to 1 for t ≥ 1, and further {yn } is convergent to 1, in the usual topology of R. c. Let A be a nonempty subset of X, and suppose 1 ∈ / A. Since there are not non∼ eventually p-convergent sequences for t < 1, then by (a), we have that A= A ∼t

for each 0 < t < 1, and A = A ∪ {1} if and only if A contains a sequence {yn } that converges to 1, in the usual topology of R, and t ≥ 1. We continue approaching the following question: given a nested sequence of sets {An }of a fuzzy metric space X with p-fuzzy diameter zero, can we find t > 0 such that A˜ tn has p-fuzzy diameter zero? In the next result, we answer affirmatively to such a question. First, we prove the following useful lemma. Lemma 3.7. Let A be a subset of the fuzzy metric space (X, M, ∗). Then φ∼t (3t) ≥ φA (t) for all t > 0.

A

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Proof. Fix t0 > 0. Let x, y ∈ A0 . By (ii) of Proposition 3.14 we can find two sequences {xn } and {yn } in A, which are p-convergent for t0 , to x and y, respectively. Then M (x, y, 3t0 ) ≥ M (x, xn ,t0 ) ∗ M (xn , yn ,t0 ) ∗ M (yn , y,t0 ) ≥ ≥ M (x, xn ,t0 ) ∗ φA (t0 ) ∗ M (yn , y,t0 ) and when n tends to ∞ we have that M (x, y, 3t0 ) ≥ φA (t0 ) and hence φA ∼0 (3t0 ) ≥ φA (t0 ). The announced answer to the aforesaid question is provided below. Proposition 3.16. Let {An } be a nested sequence of setsof a fuzzy metric space X. If {An } has p-fuzzy diameter zero for some t0 > 0, then Aˆ n has p-fuzzy diameter zero, for some t1 ≥ t0 . Proof. Suppose {An } has p-fuzzy diameter zero for t0 > 0. By the previous lemma  we have that limn φ∼τ0 (3t0 ) ≥ limn φAn (t0 ) = 1 and hence, by Proposition 3.12, A˜ n has p-fuzzy diameter zero for t1 = 3t0 ≥ t0 . The converse of the preceding proposition is not true, in general. Indeed, if for each n ∈ N we consider An = {xm : m ≥ n} in the fuzzy metric space of Example 3.10, then by (c) in Example 3.11 we know that A˜ tn = An for each n ∈ N, when we consider 0 < t0 < 1. Moreover, ∼t0 has p-fuzzy diameter zero for t1 = 1 ≥ t0 because of lim φAn (t0 ) = 1 = lim φAn (1) = 1. n

n

However, we can prove the following version related to the reciprocal of Proposition 3.16. Proposition 3.17. Let {An } be a nested sequence of sets of the fuzzy metric space X. If there exists t0 > 0 such that {An } has p-fuzzy diameter zero for some t1 > 0, then {A t0 > 0 such that  n } has p-fuzzy diameter zero for t1 . Proof. Suppose there exist Aˆ n has p-fuzzy diameter zero for some t1 > 0. Since An ⊂ A˜ tn (for all t > 0) then A limn φAn (t1 ) ≥ limn φAn0 (t1 ) = 1, and hence {An } has p-fuzzy diameter zero for t1 . On account of Propositions 3.16 and 3.17, we obtain the following two corollaries. Corollary 3.7. Let {An } be a nested sequence of sets of a fuzzy metric space X. The following conditions are equivalent: i. {An } has p-fuzzy diameter zero.  ii. There exists t0 > 0 such that Aˆ n has p-fuzzy diameter zero. Corollary 3.8. Let {An } be a nested sequence of sets of the fuzzy metric space X. If there exists t0 > 0 such that {An } has  p-fuzzy diameter zero, then we can find t1 > 0 such that for each t > t1 we have that A˜ n has p-fuzzy diameter zero.

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3.5 p-COMPLETENESS This section is devoted to characterize w-p-completeness by means of nested sequences. With this aim, we start recalling the following notion weaker than Cauchy sequence introduced by Gregori et al. [4]. Definition 3.20. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called pCauchy for t0 > 0 if given ε ∈ [0, 1] we can find nε ∈ N such that M (xm , xn ,t0 ) > 1−ε for all m, n ≥ nε , or equivalently limm,n M (xm , xn ,t0 ) = 1. Obviously, {xn } is Cauchy if and only if it is p-Cauchy for all t > 0. Under this notation, we have the following proposition. Proposition 3.18. Every p-convergent sequence is p-Cauchy. Proof. Suppose {xn } is a p-convergent sequence to x for t0 . Let ε ∈ [0, 1]. We can choose δ ∈ [0, 1] such that (1 − δ )∗ (1 − δ ) > 1 − ε. Then, there exists nε ∈ N such that M (xn , xm , 2t0 ) ≥ M (xn , x,t0 ) ∗ M (x, xm ,t0 ) > (1 − δ ) ∗ (1 − δ ) > 1 − ε, for all m, n ≥ nε , and hence {xn } is p-Cauchy for 2t0 . Observe that in the previous demonstration that it is actually shown that a pconvergent sequence for t0 > 0 is p-Cauchy for 2t0 . So, it arises the following open question. Question 5.1 Is every p-convergent sequence for t0 > 0 a p-Cauchy sequence for t0 ? Obviously, the converse of Proposition 3.18 is not true, in general. Indeed, if we consider the fuzzy metric space (X, M, ∧) of Example 3.10 and take Y = X − {1}. Then, {xn } is a p-Cauchy sequence in Y which is not p-convergence. Nevertheless, such a reciprocal becomes true when a p-Cauchy sequence in addition has a cluster point, as shows the following result. Proposition 3.19. Every p-Cauchy sequence with a p-cluster point is p-convergent. Proof. Let {xn } be a p-Cauchy sequence for t1 > 0 and suppose that x is a p-cluster point of {xn } for t2 > 0. Let ε ∈ [0, 1] and consider δ ∈ [0, 1] such that (1 − δ ) ∗ (1 − δ ) > 1 − ε. Then, for such a δ ∈ [0, 1] we can find nδ ∈ N such that it satisfies (simultaneously) M (xm , xn ,t1 ) > 1 − δ and M (x, xn ,t2 ) > 1 − δ for all n ≥ nδ . Then M (x, xn ,t1 + t2 ) ≥ M (x, xnε ,t2 ) ∗ M (xnε , xn ,t1 ) ≥ (1 − δ ) ∗ (1 − δ ) > 1 − ε for all n ≥ nδ and so limn M (x, xn ,t1 + t2 ) = 1, and hence {xn } is p-convergent to x (for t1 + t2 > 0 ). An immediate corollary of the previous result is the following one. Corollary 3.9. If {xn } is a p-Cauchy sequence in a fuzzy metric space X, then it can have at most one p-cluster point.

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Definition 3.21. A fuzzy metric space (X, M, ∗) is called w - p-complete (respectively p-complete) if every p-Cauchy sequence is p-convergent (respectively, convergent). (Compare with definition of p-complete in Ref. [4].) It is also said that M or X is complete. The relationship among completeness, p-completeness, and w-p-completeness is shown in the following diagram of implications.

In Example 3.19 of Ref. [4], there is a complete principal fuzzy metric space which is not w-p-complete. If X is principal then, obviously, every w-p-complete space is p-complete. (Indeed, assume X is a principal w - p-complete fuzzy metric space. Let {xn } be a p-Cauchy sequence in X. Then {xn } is p-convergent and, by Proposition 3.19, {xn } is convergent.) From the aforementioned definitions, we obtain the following corollary. Corollary 3.10. X is p-complete if and only if X is principal and w-p-complete. Proof. Suppose X is p-complete, then, obviously, X is w - p-complete. Now, let {xn } be a p-convergent sequence in X. Then, by Proposition 3.18, {xn } is p-Cauchy and so {xn } is convergent. Hence, by Proposition 3.7, X is principal. The converse has just be seen in the last paragraph. Next, we characterize w − p-complete fuzzy metric spaces by means of a nested sequence of sets of X. Theorem 3.8. Let (X, M, ∗) be a fuzzy metric space. Then X is w-p-complete if and only if for every nested sequence {An } which has p-fuzzy diameter zero there exists T t > 0 such that A˜ tn = {x}, for some x ∈ X. Proof. Suppose X is w − p-complete. Let {An } be a nested sequence which has pfuzzy diameter zero for t0 > 0. We construct a sequence {an } taking an ∈ An for each n ∈ N. Since {An } has p-fuzzy diameter zero, given r ∈ [0, 1] there exists nr ∈ N such that M (x, y,t0 ) > 1−r for all x, y ∈ An with n ≥ nr . In particular, M (am , an ,t0 ) > 1−r for all m, n ≥ nr , i.e., {an } is p-Cauchy, and therefore, by hypothesis {an } is pconvergent to (some) x ∈ X, for (some) t ≥ t0 . In addition, {An } has also p-fuzzy diameter zero for that t > 0 attending to Remark 3.8. ˜t Now, am ∈ An for all m ≥ n and  then, by (ii) of Proposition 3.14 x ∈ An for all t ˜ n ∈ N. Now, by Proposition 3.16, An has p-fuzzy diameter zero for some t1 ≥ t. T Therefore, by Proposition 3.10, A˜ tn = {x}.

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Conversely, let {xn } be a p-Cauchy sequence in X for t0 > 0. Define An = {xn , xn+1 , . . .} for all n ∈ N. For a given r ∈ [0, 1] we can find nr ∈ N such that M (xm , xn ,t0 ) > 1 − r for all m, n ≥ nr . Then {An } is a nested sequence that has pT t ˜ fuzzy diameter zero for t0 > 0. By hypothesis, there exists t > 0 such  that An = {x}. ˜ Now, by Corollary 3.8, there exists t1 > max {t0 ,t} such that An has p-fuzzy diameter zero. Moreover, by (iv) of Proposition 3.14, x ∈ An for all n ∈ N, and by Proposition 3.10, ∼tt1n = {x}. Then, for ε ∈ [0, 1] we can find nε ∈ N such that M (y, z,t1 ) > 1 − ε for all y, z ∈ A˜ n and n ≥ nε . In particular, M (x, xn ,t1 ) > 1 − ε for all n ≥ nε , i.e., {xn } is p-convergent to x (for t1 ), and hence X is w − p-complete. Example 3.11. Consider the fuzzy metric space (X, M, ∗) of Examples 3.10 and 3.11, which is not principal (see Example 3.8 in Ref. [4]). We will prove that X satisfies Theorem 3.8 and thus X is w-p-complete. Let {An } be a nested sequence of sets of X which has p-fuzzy diameter zero. If {An } is eventually constant, i.e., there exists n0 ∈ N such that An = A for n ≥ n0 , then by Proposition 3.10, A = {x} for some x ∈ X, and by (ii) of Proposition 3.14, for each n ≥ n0 , A˜ n = {x} for all t > 0. Suppose now that {An } is not eventually constant, and without loss of generality, that it has p-fuzzy diameter zero for some t1 ≥ 1. For each n ∈ N take yn ∈ An and consider the sequence {yn }. Take ε ∈ [0, 1]. There exists nε ∈ N such that, for each n ≥ nε we have that M (x, y,t0 ) = min{x, y} > 1 − ε for all x, y ∈ An with x ̸= y. Obviously, 1 − ε < yn ≤ 1 for all n ≥ nε and then {yn } converges to 1, in the usual topology of R. Then by (a) of Example 3.11, {yn } is p-convergent to 1 for t0 = 1. Now, ym ∈ An for all m ≥ n and then 1 is a p-acc point  T of An for t0 = 1, n ∈ N. So 1 ∈ An . Now, by Proposition 3.16 we have that A˜ n has T T p-fuzzy diameter zero and 1 ∈ A˜ n , and by Proposition 3.10, A˜ n = {1}. Finally, by Corollary 3.10, X is not p-complete since it is not principal.

3.6

ONLY FOR THE STANDARD FUZZY METRIC

Let (X, d) be a metric space and Md the standard fuzzy metric deduced from d. If {xn } is a sequence in X, it is well known [5] that {xn } is d-Cauchy if and only if it is Md -Cauchy, and also {xn } is d-convergent if and only if it is Md -convergent, since τ(d) = τMd . Further, (X, d) is complete if and only if (X, Md ) is complete. Proposition 3.20. Let A be a non-empty subset of (X, Md ). Then φA (t) = for t > 0. Proof. Let t > 0. Then 

 t φA (t) = inf {Md (x, y,t) : x, y ∈ A} = inf : x, y ∈ A = t + d(x, y) t t = = . t + sup{d(x, y) : x, y ∈ A} t + diam(A)

t t+diam(A)

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Proposition 3.21. Let {An } be a nested sequence of sets of X. The following conditions are equivalent: i. {An } has p-fuzzy diameter zero in (X, Md ). ii. limn diam (An ) = 0. iii. {An } has fuzzy diameter zero in (X, Md ). Proof. By the last proposition, limn φAn (t0 ) = 1 for some t0 > 0 is equivalent to limn diam (An ) = 0 and it is equivalent to limn φAn (t) = 1 for all t > 0. In a similar way, the following proposition can be obtained. Proposition 3.22. Let {xn } be a sequence in the standard fuzzy metric space (X, Md ). Then i. {xn } is p-Cauchy if and only if {xn } is Cauchy. ii. {xn } is p-convergent if and only if {xn } is convergent. Further, iii. (X, Md ) is w-p-complete if and only if (X, Md ) is p-complete if and only if (X, Md ) is complete. Now, as a corollary of our Theorem 3.8, we obtain the well-known characterization of the completeness of a metric space by means of a nested sequence of closed sets. Corollary 3.11. Let (X, d) be a metric space. They are equivalent: i. (X, d) is complete. ii. Every nested sequence of closed sets {Fn } with limn diam (Fn ) = 0 has a singleton intersection. Proof. Suppose (X, d) is complete. Then (X, Md ) is complete, and consequently, it is w-p-complete. Let {Fn } be a sequence of closed sets with lim diam (Fn ) = 0. Then, by Proposition 3.21, {Fn } has p-fuzzy diameter zero in (X, Md ), and hence, by T Theorem 3.8, there exists t0 > 0 such that ∼ F¯n = {x}. Now, F˜n t = Fn (= Fn ) for T all t > 0 since Md is principal and then Fn = {x}. Conversely, let {An } be a nested sequence of sets of X which has p-fuzzy diameter zero in (X, Md ). Then, by Proposition 3.21, {An } has fuzzy diameter zero, and following  the arguments in the proof of by Lemma 3.1 of Ref. [13], we conclude that A¯
n has fuzzy diameter zero in (X, Md ). Now, by Proposition 3.21, T limn diam A¯ n = 0. Then, by hypothesis, A¯ n = {x} and by (v) of Proposition 3.14, T t A˜ n = {x}, for all t > 0, since Md is principal. Hence, by Theorem 3.8, we have that (X, Md ) is w-p-complete, and by (iii) of Proposition 3.22, (X, Md ) is complete. Consequently, (X, d) is complete. In Ref. [9], the authors strengthen the condition of convergence on t by introducing the concept of s- convergent. Such type of convergence (including p- convergence) provide a way to classify the behavior of different types of fuzzy metrics. In this section, we discuss these ideas and the corresponding theory.

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Definition 3.22. Let (X, M, ∗) be a fuzzy metric space.
We will say that a sequence {xn } in X is s-convergent to x0 ∈ X if limn M xn , x0 , 1n = 1 or, equivalently, if for
each r ∈ [0, 1], there exists n0 ∈ N such that limn M xn , x0 , 1n > 1 − r, for all n ≥ n0 . Under this terminology, the following consequences are immediate: Consequences i. If M is stationary then convergent sequences are s- convergent. ii. Constant sequences are s- convergent. iii. If τM is the discrete topology then convergent sequences are s- convergent. Proposition 3.23. Let (X, M, ∗) be a fuzzy metric space. Each s- convergent sequences in X is convergent The following example illustrates that the converse of the aforementioned proposition may not be true. Example 3.12. On [0, ∞] consider the principal fuzzy metric (M, ∗), where M is defined by (min{x, y} + t) M ∗ (x, y,t) = (max{x, y} + t) for all x, y ∈ ([0, ∞]), t > 0. Then the sequence ( n1 ) converges to 0 but it is not s convergent to 0. Proposition 3.24. Let (X, M, ∗) be a fuzzy metric space. i. Each subsequence of an s- convergent sequence in X is s-convergent ii. Each convergent sequence in X admits an s- convergent subsequence. Definition 3.23. We will say that (X, M, ∗) is an s- fuzzy metric space or simply M-fuzzy metric if every convergent sequence is s- convergent. From the Consequence (i–iii) and the aforementioned definition, we have: Corollary 3.12. Let (X, M, ∗) be a fuzzy metric space. i. If τM is the discrete topology then M is an s- fuzzy metric. ii. If M is stationary then M s an s- fuzzy metric. Now, considering Theorem 3.6, we have the following result: Corollary 3.13. Each p-convergent sequence {xn } in X is s-convergent if and only if X is a principal s- fuzzy metric space. If (X, M, ∗) be a fuzzy metric space, we define the mapping NM on X 2 given by t>0 M(x, y,t) for all x, y ∈ X. Now, we are interested to discuss those non-stationary fuzzy metric space (X, M, ∗) such that (NM , ∗) is a stationary fuzzy metric on X and V

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we establish a relationship between those fuzzy metrics and s-fuzzy metrics. Notice that if X is a set with at least two elements and d is a metric on X it is obvious that V t>0 Md (x, y,t) = 0 for x ̸= y, and so NM (d) is not a fuzzy metric on X. Lemma 3.8. Let (X, M, ∗) be a fuzzy metric space on X. then i. (NM , ∗) is a stationary fuzzy metric on X if and only if NM (x, y) > 0 for all x, y ∈ X. In such a case: ii. τNM ≻ τM . Theorem 3.9. Let (X, M, ∗) be a fuzzy metric space on X such that NM (x, y) > 0 for each x, y ∈ X. Then τNM = τM if and only if M is an s- fuzzy metric. Example 3.13. (a) Let X =]0, 1] be endowed with the usual metric d of R. Define ( 1 − 21 d(x, y)t , 0 < t ≤ 1 M(x, y,t) = 1 − 21 d(x, y), t > 1 Then (X, M, L) is a fuzzy metric space and τM is τ(d). V On the other hand, NM (x, y) = t>0 M(x, y,t) > 0 for all x, y ∈ (0, 1), since ( 1, x = y NM (x, y) = M(x, y,t) = 1 ̸ y 2, x = t>0 ^

By Lemma 3.8, we have (NM , L) be a fuzzy metric space on X, and it is obvious that τNM is the discrete topology. Therefore τNM ̸= τM. (b) the fuzzy metric given in Example 3.14 is an s- fuzzy metric on X, but (τNM , .) is not a fuzzy metric on X.

3.7

A CLASSIFICATION OF FUZZY METRIC SPACES

Attending to above consequences, we have the following implications: s − convergence ⇒ convergence ⇒ p − convergence In fact, we can conclude the diagram of inclusion in Figure 3.1. Next we give examples that show that all inclusions (non-trivial) in the diagram are strict. Example 3.14. (A non-stationary principal s-fuzzy metric space). Let ([0, ∞], M, .) be the fuzzy metric space, where M is the fuzzy metric of Example 3.13. It is known min{x, y} + t min{x, y} that M is principal [12]. Now, NM (x, y) = Λt>0 = > 0 for max{x, y} + t max{x, y} each x, y ∈ [0, ∞]. Then by Theorem 3.9 we have that M is an s-fuzzy metric, since T NM = T M,.

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Remark 5.2. Since the completion of the fuzzy metric space of Example 3.15 is the fuzzy metric space of Example 3.13 [7], the completion of an s-fuzzy metric space is not necessarily an s-fuzzy metric space. Example 3.15. (A non-stationary non-principle s-fuzzy metric space) Let X = ]0, 1], A = X ∩ Q, B = X\A. Define the function M on X 2 × R+ by  min{x,y}  x, y ∈ A or x, y ∈ B,t > 0,   max{x,y} , min{x,y} M(x, y,t) = max{x,y} ,    min{x,y} · t, elsewhere. max{x,y}

It is easy to verify that (M, ·) is a fuzzy metric on X. Now, we will show that M is not principal. For it, we will see that {B(1, r, 1) : r ∈ [0, 1] is not a local base at x = 1.  In fact, B(1, 1r,
1) = [11− r, 1] for all r1 ∈1
[0, 1].1 Now, since 1 ∈ A we have y ∈ B : M 1, y, 2 > 1 − 2 = 0, / so B 1, 2 , 2 = [ 2 , 1] ∩ Q and clearly B(1, r, 1) ⊈
B 1, 12 , 12 for all r ∈ [0, 1], and then M is not principal. Now, we will see that M is an s-fuzzy metric on X. For this we will prove that ∩t>0 B(x, r,t) is a neighbourhood of x, for each x ∈ X and each r ∈ [0, 1]. Fix x ∈ X and r ∈ [0, 1]. It is(easy to verify that for t ∈ [0, 1 − r]: x [x(1 − r), 1−r ] ∩ A, x ∈ A, B (x, r,t) = ∩t>0 B(x, r,t) = x [x(1 − r), 1−r ] ∩ B, x ∈ B, On the other hand, if n ≥ 2,    [x(1 − 1 ), x ] ∩ A, x ∈ A, 1 1 n 1− 1 n B x, , = [x(1 − 1n ), x 1 ] ∩ B, x ∈ B, n n 1− n

Therefore, if we take n ∈ N such that 0
0 B(x, r,t) and so ∩t>0 B(x, r,t) is a neighbourhood of x and so M is an s-fuzzy metric. Example 3.16. (A non-principal non-s-fuzzy metric space). Let A = R ∩ Q, B = R − A. Let d be the usual metric on R. Define the function M on R2 × R+ by  t · Md (x, y,t), (x ∈ A, y ∈ B) or(y ∈ A, x ∈ B), t ∈ [0, 1], M(x, y,t) = Md (x, y,t), elsewhere. We will show that (R, M, ·) is a fuzzy metric space. Obviously, M satisfies a(GV1),(GV3) and (GV5). Suppose that M(x, y,t) = 1 for x ∈ A,y ∈ B and t ∈ [0, 1]. Then t · Md (x, y,t) = 1 but since t ∈ [0, 1] we have that Md (x, y,t) > 1, a contradiction. Therefore, M(x, y,t) = Md (x, y,t) = 1 and so x = y. The converse is immediate. Now, we will that M satisfies (GV4). Suppose that x, y ∈ A, z ∈ B and let t, s > 0 such that t + s ∈ [0, 1]. Then M(x, z,t + s) = (t + s) · Md (x, y,t + s) > s · Md (x, y,t) · Md (y, z, s) = M(x, y,t) · M(y, z, s). The other cases are proved in a similar way. We will see that M is neither principal nor an s-fuzzy metric. For this, we will give a p-convergent sequence which is not convergent and a convergent sequence which is not s-convergent. Consider the sequence { πn }. Then limn M( πn , 0, 1) = limn 1+1 π = 1 and so { πn } is p-convergent, but

( 1 )2 limn M( πn , 0, 12 ) = limn 1 2 π 2+ n

n

=

1 2

and so

Now, consider the sequence { n1 }. But limn M( 1n , 0, 1n ) = s-convergent.

{ πn } 1 n

1+1 n n

is not convergent.

=

1 2

and so { 1n } is not

Example 3.17. (A non-stationary non-principle s-fuzzy metric which generates the discrete topology) Let X = [0, ∞] and let ϕ : R+ → [0, 1] be a function given by ( t, t ∈ [0, 1] ϕ(t) = 1, otherwise Define the function M on X 2 × R+ by ( 1, x=y M(x, y,t) = ϕ(t), x = ̸ y Then (M, ·) is a non- principal fuzzy metric on X and that τM is the discrete topology, so M is an s- fuzzy metric. Clearly M is non-stationary. In 2014, Ricarte and Romaguera [12] established a nice correspondence between fuzzy metric and domain theory. In this connection, they introduced the concept of standard Cauchy ( briefly std− Cauchy) which is stronger than Cauchy sequence.

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Definition 3.24. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called std− Cauchy if for each ε ∈ [0, 1] there exists nε ∈ N, such that M (xn , xm ,t) >

t t +ε

for all n, m ≥ nε and t > 0. X is called std− Complete if every std− Cauchy sequence in X is convergent. Definition 3.25. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called stdt convergent to x0 ∈ X if given ε ∈ [0, 1] there exists nε ∈ N such that M (xn , x0 ,t) > t+ε for all n ≥ nε and t > 0. The following example illustrates the fact that the exists std−convergent sequence which is not std− Cauchy. Example 3.18. Let d be the usual metric on R restricted to [0, ∞] and consider the standard fuzzy metric Md induced by d. Let X = [0, ∞]. We define on X × X × [0, ∞] the function ( 1, x=y M(x, y,t) = Md (x, 0,t) · Md (0, y,t) x ̸= y Then (X, M, .) is a fuzzy metric space. If we consider a sequence xn = { 1n }, then {xn } is std−convergent to 0. However, it is not std− Cauchy. The aforementioned example asserts that the concept of std−convergent is not compatible with std− Cauchy. So it is natural to think for compatibility aspects among various convergences like- s− convergence, s− Cauchy etc. Such consideration has been well studied by Gregori and Mi˜nana [8]. Now, let’s come to our promise about various aspects on G− convergence and M− convergence. In the following, we shall discuss some relationships among convergence, completeness, and compactness. Definition 3.26. A fuzzy metric space (X, M, ∗) is called compact if every sequence has a convergent subsequence. Note that every compact GV -fuzzy metric space is complete in George and Veeramani’s sense [3]. However, the same may not be G-complete. Example 3.1 illustrate this fact. On the other hand, in the case of non-Archimedean fuzzy metric space, each compact non-Archimedean fuzzy metric space is G-complete. Now to frame a diagram for convergences, we first mention the following: Definition 3.27. We will say that a sequence {xn } in a fuzzy metric space (X, M, ∗) is G convergent to x0 if {xn } has a subsequence converging to x0 (i.e., x0 is a cluster point of {xn } ) and limn M (xn , xn+1 ,t) = 1 for all t > 0.

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Obviously, the next diagram of implications if fulfilled. Next we give appropriate examples which prove that there are not any other implications in the last diagram, and then we conclude that G-Cauchyness and Gconvergence constitute a compatible pair. In these examples, Md is the standard fuzzy metric deduced from the usual metric of R, restricted to the corresponding sets of R. Example 3.19. (A G-Cauchy non-G-convergent non-Cauchy sequence) Consider the sequence {xn } given by xn = ∑ni=1 1i defined in the fuzzy metric space (R, Md , ·). It is well-known that {xn } is a G-Cauchy non-Cauchy sequence. Clearly, {xn } is not G-convergent since it has no cluster points. Example 3.20. √ (A G-convergent non-Cauchy sequence) Consider the sequence {xn } where xn = sin n defined in the fuzzy metric space ([−1, 1], Md , ·). It is easy to verify that {xn } is G− Cauchy. We will see that each point of [−1, 1] is a cluster point of {xn } and hence {xn } is G convergent. Indeed, let y0 ∈ [−1, 1] and take t0 ∈ [0, 2π]. such that sint0 = y0 . Take ε > 0 with [y0 − ε, y0 + ε] ⊂ [−1, 1] and let N ∈ N. By continuity of sint there exists δ > 0 such that |t − t0 | < δ implies |sint − y0 | < ε and by the periodicity of sint we have that |t − t0 + 2kπ| δ implies |sint − y0 | < ε for all k ∈ Z+ . √ N such that √ √ | n − n − 1| < δ for all n ≥ n0  √ Let k0 = min k ∈ Z+ : n0 < t0 + 2kπ . The sequence {n0 + i}∞ i=0 tends to +∞ and two consecutive terms of this sequence satisfy (1). Therefore some i0 ∈ N satisfies √ √ that n0 + i0 ∈ [t0 + 2kπ − δ ,t0 + 2kπ + δ ] and then sin n0 + i0 ∈ [y0 − ε, y0 + ε] with n0 + i0 > N. Therefore the sequence {xn } is frequently in [y0 − ε, y0 + ε] and hence y0 is a cluster point of {xn }. With slight modifications, it can be proved that −1 and 1 are also cluster points of {xn }. Clearly, {xn } is not Cauchy (and consequently {xn } is not convergent). Example 3.21. (A Cauchy non-G-convergent sequence) Consider the sequence {xn } in the fuzzy metric space ([0, +∞]Md , ·) where xn = n1 . Hence {xn } is Cauchy but it is not G-convergent since it has not cluster points. 3.7.1

p-CONVERGENCE AND p-CAUCHYNESS

To obtain a fixed point theorem in a fuzzy metric space, D. Mihet [15] gave the following concept weaker than convergence. Definition 3.28. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called p convergent to x0 if limn M (xn , x0 ,t0 ) = 1 for some t0 > 0.

Convergence in Fuzzy Metric Spaces

47

This concept has been used by the authors in Ref. [8] to characterize t-continuous mappings. In Ref. [7], the authors continue the study started by Mihet and first they gave the following concept. Definition 3.29. A fuzzy metric space (X, M, ∗) is said to be principal (or simply, M is principal) if {B(x, r,t) : r ∈ [0, 1]} is a local base at x ∈ X, for each x ∈ X and each t > 0. It is worth to notice that many fuzzy metric spaces are principal [7,11]. Then the authors obtained the following characterization. Proposition 3.25. A fuzzy metric space (X, M, ∗) is principal if and only if every p-convergent sequence in X is convergent in (X, τM ). Further, the authors of Ref. [7] gave the following concept of Cauchyness deduced in a natural way from the p-convergence concept. Definition 3.30. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called pCauchy if there exists t0 > 0 such that for each ε ∈ [0, 1] there exists n0 ∈ N such that M (xn , xm ,t0 ) > 1 − ε for all n, m ≥ n0 , or equivalently, limn,m M (xn , xm ,t0 ) = 1 for some t0 > 0. The authors proved that the next diagram of implications is fulfilled.

Now, it is left to the reader to verify that there is not any other implication in such diagram, and so p-convergence and p-Cauchyness is a compatible pair. 3.7.2 s-CONVERGENCE AND s-CAUCHYNESS The following concept stronger than convergence, called s-convergence, tries to extend the classical metric formulation of convergence using a simple limit. This concept leads to define a class of fuzzy metric spaces called s-fuzzy metric. In a sfuzzy metric space each point has a local base which is not defined by balls, but by neighbourhoods which, as in the classical case, only depend on (the radius) r ∈ [0, 1] V (Proposition 3.26 ). Now, if N(x, y) = t>0 M(x, y,t) > 0 for all x, y ∈ X then (N, ∗) is a stationary fuzzy metric on X and the authors (Ref. [11], Theorem 4.2) proved that (X, M, ∗) is an s-fuzzy metric space if and only if τN = τM Definition 3.31. A sequence
{xn } in a fuzzy metric space (X, M, ∗) is s-convergent to x0 ∈ X if limn M xn , x0 , 1n = 1 A fuzzy metric space in which every convergent sequence is s-convergent is called an s-fuzzy metric space. We also say that M is an s-fuzzy metric. s-fuzzy metric spaces are characterized in Ref. [11] as follows.

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Proposition 3.26. M is an s-fuzzy metric if and only if t>0TB(x, r,t) is a neighbourhood of x, for all x ∈ X and for all r ∈ [0, 1], or equivalently, { t>0 B(x, r,t) : r ∈ [0, 1]} is a local base at x, for each x ∈ X. T

The corresponding concept of Cauchyness deduced in a natural way from the s convergence is the following. Definition 3.32. A
sequence {xn } in a fuzzy metric space (X, M, ∗) is s∗ -Cauchy if limn,m M xn , xm , 1n = 1 Proposition 3.27. Every s∗ -Cauchy sequence is Cauchy. Proof. Suppose that {xn } is s∗ -Cauchy. Let t > 0 and take n0 ∈ N such that n10 < t.  
Then we have that M (xn , xm ,t) ≥ M xn , xm , n10 ≥ M xn , xm , 1n for all n ≥ n0 and all m ∈ N. Then limn M (xn , xm ,t) = 1. Unfortunately, an s-convergent sequence is not necessarily s∗ -Cauchy, as shown in the next example. Example 3.22. Let R be endowed with its usual metric d. Consider the standard fuzzy metric space (R, Md , ·). We will see that the sequence {xn } given by xn = n12 for all n ∈ N, is s-convergent to 0 but it is not s∗ -Cauchy. We have that   1 1 1 = lim 1 n 1 = lim lim Md xn , 0, =1 n n n n 1 + 1n n + n2 and thus {xn } is s-convergent to 0. Now, we will see that {xn } is not s∗ -Cauchy. Suppose that {xn } is s∗ -Cauchy, that is   1 1 n =1 = lim lim Md xn , xm , n,m 1 n,m n + 1 − 1 n

n2

m2

Now, for large values of n and m, if we take m ∈ N with m = lim n

√ n we have that

1 1 1 1 n = lim 1 1n 1 = lim = , 1 1 n n 1 1 2 + − 2 − √ n n n n2 n + n2 − ( n)2

a contradiction. Consequently, s∗ -Cauchy is not compatible with s-convergence. To overcome this inconvenience, we introduce here the following definition. Definition 3.33.
A sequence in a fuzzy metric space (X, M, ∗) is s-Cauchy if limm,n M xn , xm , m+n mn = 1.

Convergence in Fuzzy Metric Spaces

49

It is easy to verify that an s-Cauchy sequence is Cauchy. Next we will see that s convergence and s-Cauchyness is a compatible pair. Proposition 3.28. Every s-convergent sequence in a fuzzy metric space (X, M, ∗) is s-Cauchy. Proof. Let ε ∈ [0, 1]. Take µ ∈ [0, 1] such that (1 − µ) ∗ (1 − µ) ≥ 1 − ε. Let {xn }
be an s-convergent sequence to x0 . Then there exists n0 ∈ N such that M xn , x0 , 1n ≥ 1 − µ for all n ≥ n0 . Now we have


1 1 M xn , xm , m+n mn ≥ M xn , x0 , n ∗ M xm , x0 , m ≥ (1 − µ) ∗ (1 − µ) ≥ 1 − ε for all m, n ≥ n0 and hence {xn } is an s-Cauchy sequence. So, the following diagram of implications is fulfilled.

To prove that s-convergence and s-Cauchyness is a compatible pair we will see, with appropriate examples, that no other implication is fulfilled in the previous diagram. Example 3.23. (An s-Cauchy non-convergent sequence) Let X = [1, +∞] and consider the fuzzy metric space (X, M, ·) where M is the stationary fuzzy metric defined min{x,y} by M(x, y) = max{x,y} It is easy to verify that the sequence xn = 1 + n1 is an s-Cauchy sequence in X which is not convergent.

Example 3.24. (A convergent non- s-Cauchy sequence) Let (X, Md , ·) be the standard fuzzy metric space induced by (X, d) where X = R and d is the usual metric on R. Consider the sequence {xn } defined by xn = n1 for all n ∈ N. Clearly {xn } is convergent and so it is Cauchy. Suppose now that {xn } is an s-Cauchy sequence. Then we have that for all ε ∈
[0, 1] there exists nε such that M xn , xm , m+n all n, m ≥ nε or equivalently ≥ 1−ε for mn m+n m−n 1 ε mn m+n + 1 − 1 = 1+ m−n ≥ 1 − ε for all m, n ≥ nε . In consequence m+n ≤ 1−ε for all | m+n | |n m| mn m, n ≥ nε , which is a contradiction. 3.7.3

STRONG CONVERGENCE AND STRONG CAUCHYNESS

In the fuzzy context, there is another way of regarding the classical formulation of the concept of convergence using the role ε and n0 and strengthening the imposition on t. In fact, in Ref. [10], the authors have given the following condition stronger than s-convergence called strong convergence.

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

Definition 3.34. A sequence {xn } in a fuzzy metric space (X, M, ∗) is called strong convergent to x0 ∈ X if given ε ∈ [0, 1] there exists nε , depending on ε, such that M (xn , x0 ,t) > 1 − ε, for all n ≥ nε and for all t > 0. The first goal of this concept is that s-fuzzy metric spaces are characterized because every convergent sequence is strong convergent. In the mentioned paper, the authors have introduced the following concept deduced in a natural way from the concept of strong convergence. Definition 3.35. A sequence {xn } in a fuzzy metric space (X, M, ∗) is strong Cauchy if given ε ∈ [0, 1] there exists nε , depending on ε, such that M (xn , xm ,t) > 1 − ε, for all n, m ≥ nε and for all t > 0. The second goal of Ref. [10] is that the authors have proved that strong convergence and strong Cauchy is a compatible pair.

3.8 3.8.1

RELATING THE CONCEPTS RELATING WEAK CONCEPTS

The only relations among the concepts that appeared previously are illustrated in the next diagram of implications.

The following examples prove that there is not any other relation between these concepts. Example 3.25. A p-Cauchy Sequence Which Is Not G-Cauchy Let {xn } be a strictly increasing sequence of positive real numbers converging to 1, with respect to the usual topology of R. Put X = {xn } ∪ {1}. Define the fuzzy set M on X 2 × R+ given by M(x, x,t) = 1 for each x ∈ X and t > 0 and M (xn , xm ,t) = min {xn , xm ,t} for all m, n ∈ N,t > 0. Then (M, ∧) is a fuzzy metric on X. Consider the sequence {an } where an = 1 if n is even and an = xn if n is odd. It is left to the reader to verify that {an } is p-Cauchy but it is not G-Cauchy. Example 3.26. A G-Cauchy Sequence Which Is Not p-Cauchy) Consider the standard fuzzy metric space (R, Md , ·) where d is the usual metric on R. Let {xn } be the sequence given by xn = ∑ni=1 1i . It is well-known that {xn } is G-Cauchy. It is left to the reader to show that {xn } is not p-Cauchy.

Convergence in Fuzzy Metric Spaces

51

Example 3.27. (A p-Convergent (Non- G-Convergent) Non- G-Cauchy Sequence) Let X = R+ and let ϕ : R+ → [0, 1] be a function given by ϕ(t) = t if t ≤ 1 and ϕ(t) = 1 elsewhere. Define the function M on X 2 × R+ by ( 1 x=y M(x, y,t) = min{x,y} max{x,y} · ϕ(t) x ̸= y It is easy to verify that (M, ·) is a fuzzy metric on X and it is obvious that the only (G) Cauchy sequences in X are the constant sequences, since M(x, y,t) < t whenever t ∈ [0, 1] and x ̸= y. Now, consider the sequence {xn } in X given by xn = 1 − 1n , n ∈ N. We have that M (xn , 1, 1) = 1 − n1 for all n ∈ N, so limn M (xn , 1, 1) = 1 and {xn } is p-convergent to 1, but {xn } is not G-Cauchy. Example 3.28. (A G-convergent (non- p-convergent) non- p-Cauchy sequence) Consider the standard fuzzy metric space ([−1, 1], Md , ·) where √ d is the usual metric on R restricted to [−1, 1]. Consider the sequence {xn } = sin n. In Example 3.21 we have seen that {xn } is G-convergent. It is left to the reader to verify that {xn } is not p-Cauchy. 3.8.2

RELATING STRONG CONCEPTS

The following diagram of implications shows all the relations (except convergence implies Cauchy), among all the concepts summarized in Section 3.7.

The following examples (jointly with the results of Section 3.7) shows that there is not any other implication among these concepts. Example 3.29. (A strong convergent non-standard Cauchy (non-standard convergent) sequence) Consider √the fuzzy metric space (X, M, ∗) where X = min{x,y}+ √t and ∗ is the usual product on [0, 1] [20]. [1, +∞], M(x, y,t) = max{x,y}+ t Consider the sequence {xn } where xn = 1 + n1 , n ∈ N. It is left to the reader to verify that {xn } is strong convergent (to 1), and it is not standard Cauchy.

Example 3.30. (A standard convergent (non-s-convergent) non- s-Cauchy sequence) Consider the standard fuzzy metric space (R, Md , ·) where d is the usual metric on R. The sequence {xn } where xn = n1 , n ∈ N is standard convergent to 0. It is left to the reader to show that it is not s-Cauchy.

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

COMPACTNESS AND COMPLETENESS

In this section, we study the relationship between compactness and all the concepts of completeness based on the aforementioned Cauchy concepts. Definition 3.36. We will say that a fuzzy metric space (X, M, ∗), or simply the fuzzy metric (M, ∗), is p-complete if every p-Cauchy sequence in X converges in (X, τM ). This definition is extended to the other concepts of Cauchyness appeared before. We have seen that compactness does not imply G-completeness. Now we will prove that every compact space is p-complete. We start with the following lemma. Lemma 3.9. (Gregori et al. [7]) Let {xn } be a p-convergent sequence to x0 . Then:  (i) Each subsequence xnk of {xn } is p-convergent to x0 . (ii) If {xn } is convergent, then it converges to x0 . Proposition 3.29. Compact fuzzy metric spaces are principal. Proof. let (X, M, ∗) be a compact fuzzy metric space. Suppose that X is not principal. Then
there exist x0 ∈ X and t0 > 0 such that the family of open balls  B x0 , 1n ,t0 : n ∈ N is not a local base at x0 , that is, there exist r ∈ [0, 1] and
t > 0 such that for all n ∈ N it is satisfied B x0 , 1n ,t0 ⊈ B (x0 , r,t). By induction
we construct a sequence {xn } where xn ∈ B x0 , 1n ,t0 \B (x0 , r,t) for all n > 1. By construction {xn } is not convergent and also limn M (x0 , xn ,t0 ) = 1 and hence {xn } is p-convergent to x0 .  Since X is compact we can find a subsequence xnk of {xn } which  converges in X. By (i) of Lemma 3.9 xnk is p-convergent to x0 and by (ii) xnk converges to x0 , a contradiction since xn ∈ / B (x0 , r,t) for all n ∈ N. So, X is principal. Theorem 3.10. Compact Fuzzy Metric Spaces Are p-Complete Proof. let (X, M, ∗) be a compact fuzzy metric space and let {xn } be a p-Cauchy sequence in X. Since X is compact we can find a subsequence xnk of {xn } which converges to x0 ∈ X. Suppose that limn M (xn , xm ,t0 ) = 1 for some t0 > 0 since {xn } is p-Cauchy. Then we have

M (xn , x0 , 2t0 ) ≥ M xn , xnk ,t0 ∗ M xnk , x0 ,t0 and taking limit as n and k tend to +∞ we obtain limn M (xn , x0 , 2t0 ) = 1 and so {xn } is p-convergent to x0 . Now, by Proposition 3.29 {xn } is convergent and hence X is p complete. Remark 3.10. The converse of Theorem 3.10 is not true. Indeed, it is easy to verify that (R, Md , ·), where d is the usual metric on R, is p-complete and it is not compact. As a consequence of the last paragraphs and due to the relationship among the concepts of Cauchyness given in Sections 3.5 and 3.6, we have the following diagram of implications.

Convergence in Fuzzy Metric Spaces

53

To continue this work, we propose to study the next question. Question 1: For each one of the aforementioned convergence concepts, we consider the class P of all convergent sequences. In which cases (X, M, ∗) is a space with convergence in the sense of Fr´echet? In such a case, does there exist a topology in which convergent sequences are precisely the sequences in the class P ?

3.10

CONCLUSION

We survey some concepts of convergence, Cauchyness, completeness, and compactness that appeared separately in the context of fuzzy metric spaces in the sense of George and Veeramani. For each one of the concepts mentioned above, we discuss its appearance, interest, advantages, and inconveniences. We also focus our attention on completion and on fixed point theory, since it is a high activity area of research. Future research direction on the topic is also suggested.

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11. Gregori V., Mi˜nana J.-J., Morillas S., A note on convergence in fuzzy metric spaces. Iranian Journal of Fuzzy Systems, 2014;11(4):75–85. 12. Ricarte L. A., Romaguera S., A domain-theoretic approach to fuzzy metric spaces. Topology and Its Applications, 2014;163:149–159. 13. Gregori V., Mi˜nana J. J., Roig B., Sapena A., A characterization of strong completeness in fuzzy metric spaces. Mathematics, 2020;8(6):861. https://doi.org/10.3390/math8060861. 14. Kramosil I., Michalek J., Fuzzy metric and statistical metric spaces. Kybernetica, 1975;11:336–344. 15. Mihet D., On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 2007;158:915–921. 16. Tirado P., On compactness and G-completeness in fuzzy metric spaces. Iranian Journal of Fuzzy Systems, 2012;9(4):151–158. 17. Vasuki R., Veeramani P., Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 2003;135(3):415–417.

4 4.1

Theory of Fuzzy Contractive Mappings and Fixed Points

INTRODUCTION

Fuzzy fixed point theory is a fuzzy extension of classical fixed point theory. The fixed point theory in fuzzy metric spaces (in the sense of Kramosil and Mich´alek [25]) was first introduced by Grabiec in Ref. [14] where a fuzzy metric version of Banach and Edelstein fixed point theorems was proved. But his method was not appropriate to retrieve metric Banach contraction. This is due to the complexity exhibited in the nature of fuzzy metric spaces and corresponding theory. Some of these issues are discussed in Refs. [20,23,26,28,35,44]. But at the same time considering the applicabilities of fuzzy concept, various researchers made attempt to translate some of the well-known classical fixed point theorems in fuzzy settings. Some instances of these works can be found in Refs. [16,17,31,36,41,46] and references cited therein. The work in this direction is still going on.

4.2

FUZZY CONTRACTIVE MAPPINGS

To obtain fuzzy version of classical Banach contraction theorem, Gregori and Sapena [20] introduced the following concepts: Definition 4.1. Let (X, M, ∗) be a fuzzy metric space. A mapping f : X → X is said to be fuzzy contractive if there exists k ∈ (0, 1) such that   1 1 −1 ≤ k −1 M( f (x), f (y),t) M(x, y,t) for each x, y ∈ X and t > 0. Definition 4.2. Let (X, M, ∗) be a fuzzy metric space. A sequence {xn } in X is said to be fuzzy contractive if there exists k ∈ (0, 1) such that   1 1 −1 ≤ k −1 M(xn+1 , xn+2 ,t) M(xn , xn+1 ,t) for all t > 0, n ∈ N. Recall that a sequence {xn } in a metric space (X, d) is said to be contractive if there exists k ∈ (0, 1) such that d(xn+1 , xn+2 ) ≤ kd(xn , xn+1 ) for all n ∈ N. DOI: 10.1201/9781003427797-4

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Proposition 4.1. Let (X, Md , ∗) be the standard fuzzy metric space induced by the metric d on X. The sequence {xn } in X is contractive in (X, d) iff {xn } is fuzzy contactive in (X, Md , ∗). Theorem 4.1. [15] A sequence {xn } in a fuzzy metric space (X, M, ∗) converges to x if and only if M(xn , x,t) → 1 as n → ∞. Theorem 4.2. [Fuzzy Banach contraction theorem] Let (X, M, ∗) be a complete fuzzy metric space (in the sense of George and Veeramani) in which fuzzy contractive sequences are Cauchy sequences. Let f : X → X be a contractive mapping being k the contractive constant. Then f has a unique fixed point. Proof. Fix x ∈ X. Let xn = f n (x) for each n ∈ N. Then it follows that, for all t > 0,   1 1 − 1 ≤ k − 1 M( f (x), f 2 (x),t) M(x, x1 ,t) and, by induction,   1 1 −1 ≤ k −1 M(xn+1 , xn+2 ,t) M(xn , xn+1 ,t) for all n ∈ N. Then {xn } is a fuzzy contractive sequence. So it is a Cauchy sequence, and hence xn converges to y for some y ∈ X. Now, we see that y is a fixed point for f . In fact, by Theorem 4.1, we have   1 1 −1 ≤ k −1 → 0 M( f (y), f (xn ),t) M(y, xn ,t) as n → ∞. Then lim M( f (y), f (xn ),t) = 1 for each t > 0 and so lim f (xn ) = f (y), n→∞

i.e., lim xn+1 = f (y) and then f (y) = y.

n→∞

n→∞

To show uniqueness, assume f (z) = z for some z ∈ X. Then, for all t > 0, we have 1 1 −1 = −1 M(y, z,t) M( f (y), f (z),t)   1 ≤k −1 M(y, z,t)   1 n ≤ ... ≤ k −1 → 0 M(y, z,t) as n → ∞. Hence M(y, z,t) = 1 and then y = z. This completes the proof. Now, suppose (X, Md , ∗) is a complete standard fuzzy metric space induced by the metric d on X. From result 4 (Chapter 3), (X, d) is complete and so, if {xn } is a fuzzy contractive sequence, by Proposition 4.1, it is contractive in (X, d) and hence convergent. So, from Theorem 4.2, we have the following corollary, which can be considered as the fuzzy version of the classic Banach contraction theorem on complete metric space.

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Corollary 4.1. Let (X, Md , ∗) be a complete standard fuzzy metric space and f : X → X be a fuzzy contractive mapping. Then f has a unique fixed point. Theorem 4.3. [Fuzzy Banach contraction theorem] Let (X, M, ∗) be a G-complete fuzzy metric space (in the sense of Kramosil and Michalek) and f : X → X be a fuzzy contractive mapping. Then f has a unique fixed point. Proof. Let k ∈ (0, 1) and since f is fuzzy contractive, so f satisfies   1 1 −1 ≤ k −1 . M( f (x), f (y),t) M(x, y,t) Fix x ∈ X. Let xn = f n (x), n ∈ N. We have seen in the proof of Theorem 4.2 that {xn } is a fuzzy contractive sequence satisfying   1 1 −1 ≤ k −1 M(xn+1 , xn+2 ,t) M(xn , xn+1 ,t) for each n ∈ N. Thus we have 1 − 1 ≤ k2 M(xn+1 , xn+2 ,t)



1

 −1

M(xn−1 , xn ,t)   1 ≤ . . . ≤ kn −1 M(x1 , x2 ,t) → 0 as n → ∞

and so lim M (xn , xn+1 ,t) = 1 for all t > 0. Then, for a fixed p ∈ N, we have n→∞

    t t M(xn , xn+p ,t) ≥ M xn , xn+1 , ∗ . . . ∗ M xn+p−1 , xn+p , p p p

z }| { → 1∗...∗1 = 1 and so {xn } is a G-Cauchy sequence. Therefore, {xn } converges to y for some y ∈ X. Now, imitating the proof of Theorem 4.2, one can prove that y is the unique fixed point for f . This completes the proof. Remark 4.1. In Theorem 4.3, it has been proved that each fuzzy contractive sequence is G-Cauchy sequence, whereas, in Theorem 4.2, it was assumed that fuzzy contractive sequences are M-Cauchy sequence. This arises the following question: Question (Gregori and Sapena [20]). Is a fuzzy contractive sequence a Cauchy sequence in George and Veeramani’s sense? The aforementioned problem generated much interest to fuzzy fixed point theorist to work on various aspects of fuzzy contractive mapping and associated fixed point. In this direction, Tirado [39,40] introduced the following:

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Definition 4.3. We say that the mapping T is Tirado’s contraction [39] (see also [28]) if the following condition is satisfied: there exists k ∈ (0, 1) such that 1 − M(T x, Ty,t) ≤ k (1 − M(x, y,t)) for all x, y ∈ X and t > 0. The constant k is called the contractive constant of T. Tirado [39] proved the following theorem as a consequence of his study. Theorem 4.4. Let (X, M, ∗L ) be a complete fuzzy metric space. If T is a Tirado’s contraction on X, then T has a unique fixed point. On the other hand, Mihet [27] established the following theorem utilizing the concept of point convergent and improving the result of Gregori and Sapena [20]. Theorem 4.5. Let (X, M, ∗) be a GV-fuzzy metric space and f : X → X be a fuzzy contractive mapping. Suppose that, for some x ∈ X, the sequence {xn }n∈N defined by xn = f n (x) of its iterates has a p-convergent subsequence. Then f has a unique fixed point. It should be noted that a similar theorem does not hold in KM-fuzzy metric spaces. This is illustrated in the following: Example 4.1. Let X be the set N = {1, 2, . . .}. We define (for p ̸= q) the fuzzy mapping M by  0 if t = 0,  M(p, q,t) = 1 − 2− min{p,q} , if 0 < t ≤ 1,  1, if t > 1. As 1 − 1/2− min(p,r) ≥ min{1 − 1/2− min(r,q) and 1 − 1/2− min(p,q) } for all p, q, r ∈ N, (X, M, TM ) is a KM-fuzzy metric space satisfying M(x, y,t) ̸= 0 for all x, y ∈ X and t > 0. The mapping f : N → N defined by f (x) = x + 1 is fuzzy contractive. Indeed, if t > 1, then we have   1 1 1 −1 = 0 ≤ −1 M( f (p), f (q),t) 2 M(p, q,t) for all p, q ∈ N, while, if 0 < t ≤ 1 and p < q, then we have 1 1 − 1 = p+1 M( f (p), f (q),t) 2 −1   1 1 1 ≤ p+1 = −1 . 2 − 2 2 M(p, q,t) As lim M( f n (x), 1, s) = 1 for all x ∈ X and s > 1, it follows that xn → p 1. Neverthen→∞ less, 1 is not a fixed point of f . On the other hand, Yun et al. [45] introduced the notion of minimal slop of a map between fuzzy metric spaces and studied various properties of fuzzy contractive mapping, which complement the aforementioned question proposed by Gregori and Sapena [20].

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4.3

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FUZZY Ψ-CONTRACTIVE MAPPINGS

In 2008, Mihet [28] provided a partial answer to the aforementioned question proposed by Gregori and Sapena in affirmative by introducing the notion of fuzzy Ψcontractive mapping as follows: Definition 4.4. [28] Let Ψ be the class of all mapping ψ : [0, 1] → [0, 1] such that ψ is continuous, nondecreasing and ψ(t) > t for all t ∈ (0, 1). Let (X, M, ∗) be a fuzzy metric space and ψ ∈ Ψ. 1. A mapping f : X → X is called a fuzzy ψ-contractive mapping if the following implication takes place: M(x, y,t) > 0 =⇒ M( f (x), f (y),t) ≥ ψ(M(x, y,t)). 2. A fuzzy ψ-contractive sequence in a fuzzy metric space (X, M, ∗) is any sequence {xn }n∈N in X such that M(xn+2 , xn+1 ,t) ≥ ψ(M(xn+1 , xn ,t)) for all n ∈ N and t > 0. Example 4.2. Let X = [0, ∞), a ∗ b = min{a, b} ∀a, b ∈ [0, 1] and ( 0, if t ≤ |x − y|, M(x, y,t) = 1, if t > |x − y|. It is well known that (X, M, ∗) is KM-fuzzy metric space. Let ψ be a mapping in Ψ. Since ψ(1) = 1 and M(x, y,t) > 0 =⇒ M(x, y,t) = 1 =⇒ ψ(M(x, y,t)) = 1. It follows that any fuzzy contractive mapping on (X, M, ∗) satisfying |x − y| < t =⇒ | f (x) − f (y)| < t, that is, | f (x) − f (y)| ≤ |x − y|, ∀x, y ∈ X. Conversely, if f : X → X is such that | f (x) − f (y)| ≤ |x − y| for all x, y ∈ X, then f is a fuzzy ψ contractive mapping for all ψ ∈ Ψ such that ψ(0) = 0. Thus the mapping f : X → X, f (x) = x + 1, g(x) = x are fuzzy ψk -contractive on (X, M, ∗). Remark 4.2. [Mihet [26], Example 4.4] The sequence {xn }n∈N defined by xn = n+1 in the fuzzy metric space considered in the aforementioned Example 4.2, although fuzzy ψk -contractive, is not an M-Cauchy sequence.

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We note that, for every k ∈ (0, 1), the mapping ψk : [0, 1] → [0, 1] defined by t is in Ψ and a ψk -fuzzy contractive mapping is a fuzzy contracψk (t) = t + k(1 − t) tive mapping in the sense of Geogori and Sepena [20]. Theorem 4.6. Let (X, M, ∗) be an M-complete non-Archimedean fuzzy metric space and f : X → X be a a fuzzy ψ-contractive mapping. If there exists x ∈ X such that M(x, f (x),t) > 0 for all t > 0, then f has a fixed point. Proof. Let x ∈ X be such that M(x, f (x),t) > 0 for all t > 0 and xn = f n (x) for each n ∈ N. Then we have M(x1 , x2 ,t) ≥ ψ(M(x0 , x1 ,t)) ≥ M(x0 , x1 ,t) > 0, ∀t > 0. Hence we have M(x2 , x3 ,t) ≥ ψ(M(x1 , x2 ,t)) ≥ M(x1 , x2 ,t) > 0, ∀t > 0. By induction, M(xn+1 , xn+2 ,t) ≥ M(xn , xn+1 ,t) > 0 for all t > 0. Therefore, for every t > 0, M(xn , xn+1 ,t)n∈N is a nondecreasing sequence of numbers in (0, 1]. Fix a t > 0 and denote lim M(xn , xn+1 ,t) by l. We have l ∈ (0, 1] (for M(x0 , x1 ,t) > 0) and since n→∞

M(xn , xn+1 ,t) ≥ ψ(M(xn−1 , xn ,t)) and ψ is continuous, l ≥ ψ(l). This implies l = 1 and so lim M(xn , xn+1 ,t) = 1, ∀t > 0. n→∞

If {xn } is not an M-Cauchy sequence, then there are ε ∈ (0, 1) and t > 0 such that, for each k ∈ N, there exist m(k), n(k) ∈ N with m(k) > n(k) ≥ k and M(xm(k) , xn(k) ,t) ≤ 1 − ε. Let for each k ≥ 1, m(k) be the least integer exceeding n(k) satisfying the aforementioned property, that is, M(xm(k)−1 , xn(k)−1 ,t) > 1 − ε and M(xm(k) , xn(k) ,t) ≤ 1 − ε. Then, for each positive integer k ≥ 1, 1 − ε ≥ M(xm(k) , xn(k) ,t) ≥ ∗(M(xm(k)−1 , xn(k) ,t), M(xm(k)−1 , xm(k) ,t)) ≥ ∗(1 − ε, M(xm(k)−1 , xm(k) ,t)). Since lim ∗(1 − ε, M(xm(k)−1 , xm(k) ,t)) = ∗(1 − ε, 1) = 1 − ε, it follows that k→∞

lim M(xm(k) , xn(k) ,t) = 1 − ε.

k→∞

Let us denote M(xn(k) , xn(k)+1 ,t) by zn . Then we have M(xm(k) , xn(k) ,t) ≥ ∗2 (zn , M(xm(k)+1 , xn(k)+1 ,t), zm ) ≥ ∗2 (zn , M(xm(k) , xn(k) ,t), zm ).

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Letting k → ∞, we obtain 1 − ε ≥ ∗2 (1, ψ(1 − ε), 1) = ψ(1 − ε) > 1 − ε, which is a contradiction. Thus, {xn } is a Cauchy sequence. If lim xn = y, then, from M( f (y), f (xn ),t) ≥ ψ(M(y, xn ,t)), it follows that xn+1 → n→∞

f (y). From here, we deduce that M(y, f (y),t) ≥ ∗2 (M(y, xn ,t), M(xn , xn+1 ,t), M(xn+1 , f (y),t)) −−−→ 1 n→∞

for all t > 0 and hence f (y) = y. This completes the proof. Theorem 4.7. Let (X, M, ∗) be an M-complete non-Archimedean fuzzy metric space satisfying the condition M(x, y,t) > 0 for all t > 0 and f : X → X be a fuzzy ψcontractive mapping. Then f has a unique fixed point. Example 4.3. Let X = (0, ∞), a ∗ b = ab for all a, b ∈ [0, 1] and M(x, y,t) =

min(x, y) max(x, y)

for all t ∈ (0, ∞) and x, y √ > 0. Then (X, M, ∗) is an M-complete non-Archimedean fuzzy metric space. Since t > t for all t ∈ (0, 1), the mapping f : X → X defined by √ √ f (x) = x is a fuzzy ψ-contractive mapping with ψ(t) = t. Thus all the conditions of Theorem 4.7 are satisfied and so the fixed point of f is x = 1. Some other generalizations of results of Geogori and Sepena [20] and Mihet [29] can be found in Refs. [2,16,39,42,43].

4.4 α-φ -FUZZY CONTRACTIVE MAPPINGS We start this section by introducing the notions of α-φ -fuzzy contractive and αadmissible mappings in fuzzy metric spaces. Denote by Φ the family of all right continuous functions φ : [0, +∞) → [0, +∞) with φ (r) < r for all r > 0. Remark 4.3. Note that, for every function φ ∈ Φ, limn→+∞ φ n (r) = 0 for each r > 0, where φ n (r) denotes the n-th iterate of φ . Definition 4.5. [17] Let (X, M, ∗) be a fuzzy metric space in the sense of George and Veeramani. We say that f : X → X is an α-φ -fuzzy contractive mapping if there exist two functions α : X × X × (0, +∞) → [0, +∞) and φ ∈ Φ such that     1 1 −1 ≤ φ −1 (4.1) α(x, y,t) M( f x, f y,t) M(x, y,t) for all x, y ∈ X and t > 0.

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Remark 4.4. If α(x, y,t) = 1 for all x, y ∈ X and t > 0 and φ (r) = kr for all r > 0 and for some k ∈ (0, 1), then Definition 4.5 reduces to the definition of the fuzzy contractive mapping given by Gregori and Sapena [20]. It follows that a fuzzy contractive mapping is an α-φ -fuzzy contractive mapping, but the converse is not necessarily true (see Example 4.4 given below). Definition 4.6. Let (X, M, ∗) be a fuzzy metric space in the sense of George and Veeramani. We say that f : X → X is α-admissible if there exists a function α : X × X × (0, +∞) → [0, +∞) such that, for all t > 0, x, y ∈ X, α(x, y,t) ≥ 1 =⇒ α( f x, f y,t) ≥ 1. Definition 4.7. [Di Bari and Vetro [9]] Let (X.M, ∗) be a fuzzy metric space in the sense of George and Veeramani. The fuzzy metric M is said to be triangular if the following condition holds:       1 1 1 −1 ≤ −1 + −1 (4.2) M(x, y,t) M(x, z,t) M(y, z,t) for all x, y, z ∈ X and t > 0. Now, we are ready to state and prove our first result of this section. Theorem 4.8. [17] Let (X, M, ∗) be a G-complete fuzzy metric space in the sense of George and Veeramani. Let f : X → X be an α-φ -fuzzy contractive mapping satisfying the following conditions: a. f is α-admissible; b. there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0; c. f is continuous. Then f has a fixed point, that is, there exists x∗ ∈ X such that f x∗ = x∗ . Proof. Let x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0. Define the sequence {xn } in X by xn+1 = f xn , for all n ∈ N. If xn = xn+1 for some n ∈ N, then x∗ = xn is a fixed point of f . Assume that xn ̸= xn+1 for all n ∈ N. Since f is α-admissible, we have α(x0 , x1 ,t) = α(x0 , f x0 ,t) ≥ 1 =⇒ α( f x0 , f x1 ,t) = α(x1 , x2 ,t) ≥ 1. By induction, we get

α(xn , xn+1 ,t) ≥ 1

for all n ∈ N and t > 0. By Equation (4.3), we have     1 1 −1 = −1 M(xn , xn+1 ,t) M( f xn−1 , f xn ,t)   1 ≤ α(xn−1 , xn ,t) −1 . M( f xn−1 , f xn ,t)

(4.3)

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Using (4.1) with x = xn−1 and y = xn from the aforementioned inequality, by the property of φ (φ (r) < r for all r > 0), we obtain 

   1 1 −1 ≤ φ −1 M(xn , xn+1 ,t) M(xn−1 , xn ,t)   1 < −1 . M(xn−1 , xn ,t)

Consequently, M(xn , xn+1 ,t) > M(xn−1 , xn ,t) for all n ∈ N and thus M(xn−1 , xn ,t) is an increasing sequence of positive real numbers in [0,1]. Let S(t) = lim M(xn−1 , xn ,t). Now, we show that S(t) = 1 for all t > 0. We n→+∞

suppose that there is t0 > 0 such that S(t0 ) < 1. Then, from 

   1 1 −1 ≤ φ −1 M(xn , xn+1 ,t0 ) M(xn−1 , xn ,t0 )

as n → +∞, using the right continuity of the function φ , we deduce that 1 −1 ≤ φ S(t0 )



 1 1 −1 < − 1, S(t0 ) S(t0 )

which is a contradiction and so we get lim M(xn−1 , xn ,t) = 1 for all t > 0. Then, for a fixed p ∈ N, we have

n→+∞

    t t M(xn , xn+p ,t) ≥ M xn , xn+1 , ∗ M xn+1 , xn+2 , p p p   z }| { t ∗ · · · · · · ∗ M xn+p−1 , xn+p , → 1∗······∗1 = 1 p as n → +∞ and thus {xn } is a G-Cauchy sequence. Therefore, {xn } converges to x∗ for some x∗ ∈ X. Now, the continuity of f implies that f xn → f x∗ and so lim M( f xn , f x∗ ,t) = 1 for all t > 0. It follows that n→+∞

lim M(xn+1 , f x∗ ,t) = lim M( f xn , f x∗ ,t) = 1

n→+∞

n→+∞

for all t > 0, that is, xn → f x∗ . By the uniqueness of the limit, we get x∗ = f x∗ , that is, x∗ is a fixed point of f . This completes the proof. In the next theorem, we omit the continuity hypothesis of f : Theorem 4.9. [17] Let (X, M, ∗) be a G-complete fuzzy metric space in the sense of George and Veeramani. Let M be triangular and f : X → X be an α-φ -fuzzy contractive mapping satisfying the following conditions:

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a. f is α-admissible; b. there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0; c. if {xn } is a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for all n ∈ N and xn → x as n → +∞, then α(xn , x,t) ≥ 1 for all n ∈ N. Then f has a fixed point. Proof. Following the proof of Theorem 4.8, we get that {xn } is a G-Cauchy sequence in the G-complete fuzzy metric space (X, M, ∗). Then, there exists x∗ ∈ X such that xn → x∗ as n → +∞. On the other hand, from Equation (4.3) and the hypothesis (c), we have α(xn , x∗ ,t) ≥ 1 (4.4) for all n ∈ N and t > 0. Now, using, successively, Equations (4.2), (4.4), and (4.1), also in view of (GV-3), we obtain       1 1 1 − 1 ≤ − 1 + − 1 M( f x∗ , x∗ ,t) M( f x∗ , f xn ,t) M(xn+1 , x∗ ,t)     1 1 ≤ α(xn , x∗ ,t) − 1 + − 1 M( f xn , f x∗ ,t) M(xn+1 , x∗ ,t)     1 1 ≤φ − 1 + − 1 . M(xn , x∗ ,t) M(xn+1 , x∗ ,t) Letting n → +∞, since φ is continuous at r = 0, we obtain   1 − 1 = 0, M( f x∗ , x∗ ,t) that is, f x∗ = x∗ . This completes the proof. The following example shows that the generalization given by Definition 4.5 offers many possibilities to study the existence of a fixed point for a mapping:  Example 4.4. Let X = 1n : n ∈ N ∪ {0, 2}, a ∗ b = ab for all a, b ∈ [0, 1] and t M(x, y,t) = t+|x−y| for all x, y ∈ X and t > 0. Clearly, (X, M, ∗) is a G-complete fuzzy metric space. Define the mapping f : X → X by  2 x   , 4 fx =   2,

if x ∈ X \ {2}, if x = 2,

and the function α : X × X × (0, +∞) → [0, +∞) by   1. if x, y ∈ X \ {2}, α(x, y,t) =  0, otherwise,

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for all t > 0. Clearly, f is an α-φ -contractive mapping with φ (r) = r/2 for all r ≥ 0. In fact, if at least one between x and y is equal to 2, then α(x, y,t) = 0 and so (4.1) holds trivially. Otherwise, if both x and y are in X \ {2}, then α(x, y,t) = 1 and so (4.1) becomes     1 1 1 −1 ≤ −1 , M( f x, f y,t) 2 M(x, y,t) which is always true since x + y ≤ 2. Now, let x, y ∈ X such that α(x, y,t) ≥ 1 for all t > 0, this implies that x, y ∈ X \{2} and, by the definitions of f and α, we have fx =

x2 y2 ∈ X \ {2}, f y = ∈ X \ {2}, α( f x, f y,t) = 1, ∀t > 0, 4 4

that is, f is α-admissible. Further, there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0. Indeed, for x0 = 1, we have α(1, f (1),t) = 1. Finally, let {xn }n∈N be a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for all n ∈ N and xn → x ∈ X as n → +∞. By the definition of the function α, it follows that xn ∈ X \ {2} for all n ∈ N and hence x ∈ X \ {2}. Therefore α(xn , x,t) = 1 for all n ∈ N. Thus all the hypotheses of Theorem 4.8 are satisfied. Here 0 and 2 are two fixed points of f . However, f is not a fuzzy contractive mapping [20]. To see this, consider x = 2 and y = 1, then we have 

   1 7 k 1 −1 = ≰ =k −1 M( f x, f y,t) 4t t M(x, y,t)

since k ∈ (0, 1). Remark 4.5. Let (X, M, ∗) be a fuzzy metric space in the sense of George and Veeramani. A sequence {xn }n∈N is said to be fuzzy contractive if there exists k ∈ (0, 1) such that     1 1 −1 ≤ k −1 M(xn+1 , xn+2 ,t) M(xn , xn+1 ,t) for all n ∈ N and for all t > 0. In the conclusions of their paper, Yun et al. [45] observed that every fuzzy contractive sequence is Cauchy in both George and Veeramani sense and Grabiec sense. Here, in proving Theorems 4.8 and 4.9, we used the G-completeness of the fuzzy metric space (X, M, ∗). Thus, it will be interesting to see whether these results will remain true in a M-complete fuzzy metric space. Now, we give a sufficient condition to obtain the uniqueness of the fixed point in the previous theorems. Precisely, we consider the following hypothesis: (H) for all x, y ∈ X and t > 0, there exists z ∈ X such that α(x, z,t) ≥ 1 and α(y, z,t) ≥ 1.

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Theorem 4.10. [17] Adding the condition (H) to the hypotheses of Theorem 4.8 (resp. Theorem 4.9), we obtain the uniqueness of the fixed point of f . Proof. Suppose that x∗ and y∗ are two fixed points of f . If α(x∗ , y∗ ,t) ≥ 1, then, by Equation (4.1), we conclude easily that x∗ = y∗ . Assume that α(x∗ , y∗ ,t) < 1, it follows from (H) that there exists z ∈ X such that α(x∗ , z,t) ≥ 1 and α(y∗ , z,t) ≥ 1.

(4.5)

Since f is α-admissible, from Equation (4.5), we get α(x∗ , f n z,t) ≥ 1 and α(y∗ , f n z,t) ≥ 1

(4.6)

for all n ∈ N and t > 0. Using (4.1) and (4.6), we have     1 1 −1 = −1 M(x∗ , f n z,t) M( f x∗ , f ( f n−1 z),t)   1 ∗ n−1 ≤ α(x , f z,t) −1 M( f x∗ , f ( f n−1 z),t)   1 ≤φ −1 . ∗ M(x , f n−1 z,t) This implies that 

   1 1 n −1 ≤ φ − 1 , ∀n ∈ N. M(x∗ , f n z,t) M(x∗ , z,t)

Then, letting n → +∞, we have

f n z → x∗ .

(4.7)

f n z → y∗ .

(4.8)

Similarly, for n → +∞, we get also Using (4.7) and (4.8), the uniqueness of the limit gives us x∗ = y∗ . This completes the proof. In view of Remark 4.4 and to show the usefulness of our theorems, we prove the following classical theorem of Gregori and Sapena [20]: Theorem 4.11. Let (X, M, ∗) be a G-complete fuzzy metric space in the sense of George and Veeramani. Let f : X → X be a fuzzy contractive mapping. Then f has a unique fixed point. Proof. Let α : X × X × (0, +∞) → [0, +∞) be the function defined by α(x, y,t) = 1 for all x, y ∈ X and t > 0. Define also φ : [0, +∞) → [0, +∞) by φ (r) = kr for all r > 0. Then f is an α-φ -contractive mapping. It is easy to show that all the hypotheses of Theorems 4.8 and 4.10 are satisfied. Consequently, f has a unique fixed point. This completes the proof. Following [6,8,32], we show that the obtained theorems are also useful to deduce easily some fixed point results in ordered fuzzy metric spaces. We begin by giving the following two definitions:

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Definition 4.8. Let ⪯ be an order relation on X. We say that f : X → X is a nondecreasing mapping with respect to ⪯ if x ⪯ y implies f x ⪯ f y. Definition 4.9. Let (X, ⪯) be a partially ordered set and (X, M, ∗) be a fuzzy metric space in the sense of George and Veeramani. We say that f : X → X is a fuzzy order φ contractive mapping if there exists φ ∈ Φ such that the following implication holds:     1 1 −1 ≤ φ − 1 , ∀t > 0. x, y ∈ X, x ⪯ y =⇒ M( f x, f y,t) M(x, y,t) Theorem 4.12. Let (X, ⪯) be a partially ordered set and (X, M, ∗) be a G-complete fuzzy metric space in the sense of George and Veeramani. Let φ ∈ Φ be such that f : X → X is a fuzzy order φ -contractive mapping and suppose that the following conditions hold: a. f is a non-decreasing mapping with respect to ⪯; b. there exists x0 ∈ X such that x0 ⪯ f x0 , M(x0 , f x0 ,t) > 0 for all t > 0; c. if {xn } is a nondecreasing sequence in X such that xn → x ∈ X as n → +∞, then xn ⪯ x for all n ∈ N. Then f has a fixed point. Proof. Define the function α : X × X × (0, +∞) → [0, +∞) by  1, if x ⪯ y, α(x, y,t) = 0, otherwise, for all t > 0. The reader can show easily that f is α-φ -contractive and α-admissible. Now, let {xn } be a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for all n ∈ N and xn → x ∈ X as n → +∞. By the definition of α, we have xn ⪯ xn+1 for all n ∈ N. From (c), this implies that xn ⪯ x for all n ∈ N, which gives us that α(xn , x,t) = 1 for all n ∈ N and t > 0. Thus, all the hypotheses of Theorem 4.9 are satisfied and f has a fixed point. This completes the proof.

4.5

β -ψ-FUZZY CONTRACTIVE MAPPINGS

In this section, we present the notions of β -ψ-fuzzy contractive and β -admissible mappings in fuzzy metric spaces due to Gopal et al. [17]. Let Ψ be the class of all functions ψ : [0, 1] → [0, 1] such that a. ψ is non-decreasing and left continuous; b. ψ(r) > r for all r ∈ (0, 1). It can easily be shown (see, e.g, [42]) that, if ψ ∈ Ψ, then ψ(1) = 1 and lim ψ n (r) = 1 for all r ∈ (0, 1).

n→+∞

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Definition 4.10. Let (X, M, ∗) be a fuzzy metric space. We say that f : X → X is a β -ψ-fuzzy contractive mapping if there exist two functions β : X × X × (0, +∞) → (0, +∞) and ψ ∈ Ψ such that M(x, y,t) > 0 =⇒ β (x, y,t)M( f x, f y,t) ≥ ψ (M(x, y,t))

(4.9)

for all t > 0 and x, y ∈ X with x ̸= y. Remark 4.6. If β (x, y,t) = 1 for all x, y ∈ X and t > 0, then Definition 4.10 reduces to the definition of the fuzzy ψ-contractive mapping given by Mihet [27]. It follows that a fuzzy ψ-contractive mapping is a β -ψ-fuzzy contractive mapping, but the converse is not true always (see Example 4.22 given below). Definition 4.11. Let (X, M, ∗) be a fuzzy metric space. We say that f : X → X is β -admissible if there exists a function β : X × X × (0, +∞) → (0, +∞) such that, for all t > 0, x, y ∈ X, β (x, y,t) ≤ 1 =⇒ β ( f x, f y,t) ≤ 1. Theorem 4.13. Let (X, M, ∗) be a M-complete non-Archimedean fuzzy metric space and f : X → X be a β -ψ-fuzzy contractive mapping satisfying the following conditions: a. f is β -admissible; b. there exists x0 ∈ X such that β (x0 , f x0 ,t) ≤ 1 for all t > 0; c. for each sequence {xn } in X such that β (xn , xn+1 ,t) ≤ 1 for all n ∈ N and t > 0, there exists k0 ∈ N such that β (xm+1 , xn+1 ,t) ≤ 1 for all m, n ∈ N with m > n ≥ k0 and t > 0; d. if {xn } is a sequence in X such that β (xn , xn+1 ,t) ≤ 1 for all n ∈ N and t > 0 and xn → x as n → +∞, then β (xn , x,t) ≤ 1 for all n ∈ N and t > 0. Then f has a fixed point. Proof. Let x0 ∈ X such that β (x0 , f x0 ,t) ≤ 1 for all t > 0. Define the sequence {xn } in X by xn+1 = f xn for all n ∈ N. If xn+1 = xn for some n ∈ N, then x∗ = xn is a fixed point of f . Assume xn ̸= xn+1 for all n ∈ N. Since f is β -admissible, we have β (x0 , f x0 ,t) = β (x0 , x1 ,t) ≤ 1 =⇒ β ( f x0 , f x1 ,t) = β (x1 , x2 ,t) ≤ 1. By induction, we get

β (xn , xn+1 ,t) ≤ 1

(4.10)

for all n ∈ N and t > 0. Now, applying (4.9) with x = xn−1 and y = xn and using (4.10), we obtain M(xn , xn+1 ,t) = M( f xn−1 , f xn ,t) ≥ β (xn−1 , xn ,t)M( f xn−1 , f xn ,t) ≥ ψ(M(xn−1 , xn ,t)).

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By induction, we get M(xn , xn+1 ,t) ≥ ψ n (M(x0 , x1 ,t)), ∀ n ∈ N. Since lim ψ n (r) = 1 for all r ∈ (0, 1), we deduce that n→+∞

lim M(xn , xn+1 ,t) = 1, ∀t > 0.

n→+∞

Now, if the sequence {xn } is not an M-Cauchy sequence, then there are ε ∈ (0, 1), t > 0 and k0 ∈ N (by (c)) such that, for each k ∈ N with k ≥ k0 , there exist m(k), n(k) ∈ N with m(k) > n(k) ≥ k and M(xm(k) , xn(k) ,t) ≤ 1 − ε and β (xm(k) , xn(k) ,t) ≤ 1. Let, for each k ≥ 1, m(k) be the least positive integer exceeding n(k) satisfying the aforementioned property, that is, M(xm(k)−1 , xn(k) ,t) > 1 − ε and M(xm(k) , xn(k) ,t) ≤ 1 − ε. Then, for each positive integer k ≥ k0 , we have 1 − ε ≥ M(xm(k) , xn(k) ,t) ≥ M(xm(k)−1 , xn(k) ,t) ∗ M(xm(k)−1 , xm(k) ,t) (by (NA)) ≥ (1 − ε) ∗ M(xm(k)−1 , xm(k) ,t). Since lim (1 − ε) ∗ M(xm(k)−1 , xm(k) ,t) = (1 − ε) ∗ 1 = 1 − ε, it follows that n→+∞

lim M(xm(k) , xn(k) ,t) = 1 − ε.

n→+∞

Now, by (NA) and the condition (c), we get M(xm(k) , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ M(xm(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ M(xm(k)+1 , xn(k)+1 ,t) ∗ M(xn(k)+1 , xn(k) ,t) = M(xm(k) , xm(k)+1 ,t) ∗ M( f xm(k) , f xn(k) ,t) ∗ M(xn(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ β ( f xm(k) , f xn(k) ,t) ∗ M( f xm(k) , f xn(k) ,t) ∗ M(xn(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ ψ(M(xm(k) , xn(k) ,t)) ∗ M(xn(k) , xn(k)+1 ,t). Letting k → +∞, we obtain 1 − ε ≥ 1 ∗ ψ(1 − ε) ∗ 1 = ψ(1 − ε) > 1 − ε, which is a contradiction and so {xn } is a Cauchy sequence. Since X is M-complete, there exists x∗ ∈ X such that limn→+∞ xn = x∗ .

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On the other hand, from Equation (4.10) and the hypothesis (d), we have β (xn , x∗ ,t) ≤ 1, ∀t > 0. Now, by (NA) and (4.9), we get M( f x∗ , x∗ ,t) ≥ M( f x∗ , f xn ,t) ∗ M(xn+1 , x∗ ,t) ≥ β (xn , x∗ ,t)M( f xn , f x∗ ,t) ∗ M(xn+1 , x∗ ,t) ≥ ψ(M(xn , x∗ ,t)) ∗ M(xn+1 , x∗ ,t). Letting n → +∞ and since ψ(1) = 1, we conclude that f x∗ = x∗ . This completes the proof. The following example shows the usefulness of Definition 4.10: min{x,y} Example 4.5. Let X = (0, +∞), a∗b = ab for all a, b ∈ [0, 1] and M(x, y,t) = max{x,y} for all x, y ∈ X and t > 0. Clearly, (X, M, ∗) is a M-complete non-Archimedean fuzzy metric space. Define the mapping f : X → X by  √  x, if x ∈ (0, 1], fx =  2, otherwise,

and the function β : X × X × (0, +∞) → (0, +∞) by   1, if x, y ∈ (0, 1], β (x, y,t) =  2, otherwise, √ for all t > 0. It is easy to show that f is a β -ψ-contractive mapping with ψ(r) = r for all r ∈ [0, 1]. Clearly, f is β -admissible. Further, there exists x0 ∈ X such that β (x0 , f x0 ,t) ≤ 1 for all t > 0. Indeed, for x0 = 1, we have β (1, f (1),t) = 1. Finally, let {xn }n∈N be a sequence in X such that β (xn , xn+1 ,t) ≤ 1 for all n ∈ N, xn → x ∈ X as n → +∞ and let k0 = 1 be such that, for all m, n ∈ N, m > n ≥ k0 . By the definition of the function β , it follows that xn ∈ (0, 1] for all n ∈ N. Now, if x > 1, we get xn 1 min{xn , x} = ≤ < 1, M(xn , x,t) = max{xn , x} x x which contradicts (1) of Definition 4.3 since limn→+∞ M(xn , x,t) = 1 for all t > 0. Consequently, we obtain that x ∈ (0, 1]. Therefore, β (xn , x,t) = 1 and β (xm+1 , xn+1 ,t) = 1 for all m, n ∈ N. Thus all the hypotheses of Theorem 4.13 are satisfied. Here 1 and 2 are two fixed points of f . However, f is not a fuzzy ψ-contractive mapping [28]. To see this, consider x = 12 and y = 3. Then we have p M( f x, f y,t) =

1/2 ≱ 2

r

1/2 p = M(x, y,t) = ψ(M(x, y,t)). 3

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To ensure the uniqueness of the fixed point, we will consider the following hypothesis: (J) For all x, y ∈ X and t > 0, there exists z ∈ X such that β (x, z,t) ≤ 1 and β (y, z,t) ≤ 1. Theorem 4.14. Adding the condition (J) to the hypotheses of Theorem 4.13, we obtain the uniqueness of the fixed point of f . Proof. The proof can be completed using a similar technique as given in the proof of Theorem 4.10. Therefore, to avoid repetitions, we omit the details. Remark 4.7. Motivated by Samet et al. [32], we proposed the concept of α-φ -fuzzy contractive mapping, which is weaker than the corresponding concept of fuzzy contractive mapping [20] and the concept of β -ψ-fuzzy contractive mapping, which is weaker than the corresponding concept of fuzzy-ψ-contractive mapping [27]. Moreover, we proved two theorems which ensure the existence and uniqueness of fixed points of these new types of contractive mappings. The new concepts lead to further investigations and applications. For example, using the recent ideas in the literature [12], it is possible to extend our results to the case of coupled fixed points in fuzzy metric spaces.

4.6

FUZZY H -CONTRACTIVE MAPPINGS AND α TYPE FUZZY H -CONTRACTIVE MAPPINGS

Recently, Wardowski [44] introduced the concept of fuzzy H -contractive mappings, as a generalization of that of fuzzy contractive mappings, and established the conditions guaranteeing the existence and uniqueness of fixed point for this type of contraction in M-complete fuzzy metric spaces in the sense of George and Veeramani. Definition 4.12. Let H be the family of the mappings η : (0, 1] → [0, ∞) satisfying the following conditions: (H1) η transforms (0, 1] onto [0, ∞); (H2) η is strictly decreasing. Then the mapping f : X → X is called a fuzzy H -contractive mapping (see Wardowski [44]) with respect to η ∈ H if there exists k ∈ (0, 1) satisfying the following condition: η(M( f x, f y,t)) ≤ kη(M(x, y,t)) for all x, y ∈ X and t > 0. Proposition 4.2. Let (X, M, ∗) be a fuzzy metric space and let η ∈ H . A sequence (xn )n∈N ⊂ X is an M-Cauchy sequence if and only if, for every ε > 0 and t > 0, there exist n0 ∈ N such that η (M(xm , xn ,t)) < ε, ∀m, n ≥ n0 .

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Proposition 4.3. Let (X, M, ∗) be a fuzzy metric space and let η ∈ H . A sequence (xn )n∈N ⊂ X is convergent to x ∈ X if and only if lim η(M(xn , x,t)) = 0, ∀t > 0. n→∞

Theorem 4.15. [44] Let (X, M, ∗) be an M-complete fuzzy metric space and f : X → X be a fuzzy H -contractive mapping with respect to η ∈ H such that a. ∏ki=1 M(x, f x,ti ) ̸= 0 for all x ∈ X and k ∈ N and a sequence (ti )i∈N ⊂ (0, ∞) with ti → 0; b. if r ∗ s > 0, then η(r ∗ s) ≤ η(r) + η(s) for all r, s ∈ {M(x, f x,t) : x ∈ X,t > 0}; c. {η(M(x, f x,ti )) : i ∈ N} is bounded for all x ∈ X and any sequence (ti )i∈N ⊂ (0.∞) with ti → 0. Then f has a unique fixed point x∗ ∈ X and, for each x0 ∈ X, the sequence ( f n x0 )n∈N converges to x∗ . In a recent note, Gregori and Minana [22] observed that the main idea of Wardowski [44] is correct and different from the known concepts in the literature but they also showed that the class of fuzzy H -contractive mappings are included in the class of fuzzy Ψ-contractive mappings, as well as they point out some drawbacks of the conditions used in the aforementioned Theorem 4.15. Remark 4.8. [See Gregori and Mi˜nana [22]] If η ∈ H , then the mappings η·k : (0, 1] → [0, ∞) and η −1 : [0, ∞) → (0, 1], defined in its obvious sense, are two bijective continuous mappings which are strictly decreasing. In view of the aforementioned remark, we observe that every fuzzy H contractive mapping is a fuzzy ψ-contraction with ψ(t) = η −1 (kη(t)) for all t ∈ (0, 1] (see [22]). In this direction of research work, a recent paper of Mihet [30] provides a larger prospective and further scope to study fixed points of fuzzy H -contractive mappings. Most recently, Beg et al. [10] introduced a new concept of α-fuzzy H -contractive mapping which is essentially weaker than the class of fuzzy contractive mapping and stronger than the concept of α-φ -fuzzy contractive mapping. For this type of contraction, the existence and uniqueness of fixed point in fuzzy M-complete metric spaces have been established. Definition 4.13. Let (X, M, ∗) be a fuzzy metric space. We say that f : X → X is an α-fuzzy-H -contractive mapping with respect to η ∈ H if there exists a function α : X × X × (0, ∞) → [0, ∞) such that α(x, y,t)η (M( f x, f y,t)) ≤ kη (M(x, y,t))

(4.11)

for all x, y ∈ X and t > 0. Remark 4.9. If α(x, y,t) = 1 for all x, y ∈ X and t > 0, then Definition 4.13 reduces to the Definition 4.18 but converse is not necessarily true (see Example 4.6 given below).

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Definition 4.14. Let (X, M, ∗) be a fuzzy metric space. We say that f : X → X is α-admissible if there exists a function α : X × X × (0, +∞) → [0, +∞) such that α(x, y,t) ≥ 1 =⇒ α( f x, f y,t) ≥ 1 for all x, y ∈ X and t > 0. Now, we are ready to state and prove the following: Theorem 4.16. Let (X, M, ∗) be a M-complete fuzzy metric space, where ∗ is positive. Let f : X → X be an α-fuzzy-H -contractive mapping with respect to η ∈ H satisfying the following conditions: a. b. c. d.

there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1, t > 0; f is α-admissible; η(r ∗ s) ≤ η(r) + η(s), r, s ∈ (0, 1]; if {xn } is a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for each n ∈ N and limn→∞ xn = x, then α(xn , x,t) ≥ 1 for all n ∈ N and t > 0.

Then f has a fixed point x∗ ∈ X. Moreover, the sequence { f n x0 }n∈N converges to x∗ . Proof. Let x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1, t > 0. Define the sequence {xn }n∈N in X by xn+1 = f xn , n ∈ N ∪ {0}. If xn+1 = xn , for some n ∈ N, then x∗ = xn is a fixed point of f . So, assume that xn ̸= xn+1 for each n ∈ N. Since f is α-admissible, we have α(x0 , x1 ,t) = α(x0 , f x0 ,t) ≥ 1 =⇒ α( f x0 , f x1 ,t) = α(x1 , x2 ,t) ≥ 1, ∀t > 0. By induction, we get α(xn , xn+1 ,t) ≥ 1

(4.12)

for all n ∈ N and t > 0. Now, applying (4.11) and using (4.12), we obtain the following: η (M(xn+1 , xn+2 ,t)) = η (M( f xn , fn+1 ,t)) ≤ α(xn , xn+1 ,t)η (M( f xn , fn+1 ,t)) ≤ kη (M(xn , xn+1 ,t)) ≤ kα(xn−1 , xn ,t)η (M( f xn−1 , f xn ,t)) ≤ kkη (M(xn−2 , xn−1 ,t)) ≤ ··· ≤ kn+1 η (M(x0 , x1 ,t)) , ∀t > 0. Since k ∈ (0, 1) and η is strictly decreasing, we have η (M(xn+1 , xn+2 ,t)) < η (M(x0 , x1 ,t)) , ∀t > 0,

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and M(xn+1 , xn+2 ,t) ≥ M(x0 , x1 ,t) > 0, ∀n ∈ N, t > 0.

(4.13)

Now, let us consider any m, n ∈ N with m < n and let {ai }i∈N be a strictly decreasing sequence of positive numbers such that ∑∞ i=1 ai = 1. From (GV-4), (GV-2) and the positivity of ∗, we get ! ! n−1

n−1

M(xm , xn ,t) ≥ M xm , xm ,t − ∑ ait ∗ M xm , xn , ∑ ait i=m

i=m

n−1

!

= M xm , xn , ∑ ait i=m

≥ M (xm , xm+1 , amt) ∗ M (xm+1 , xn+2 , am+1t) ∗ · · · ∗ M (xn−1 , xn , an−1t) for all t > 0. By the condition (c) and (4.13), we get n−1

η (M(xm , xn ,t)) ≤ η

!

∏ M(xi , xi+1 , ait)

i=m n−1

≤

∑ η (M(xi , xi+1 , ait))

i=m n−1

≤

∑ ki η (M(x0 , x1 ,t))

i=m

for all m, n ∈ N with m < n and t > 0. The aforementioned sum is finite, i.e., for any ε > 0, there exist n0 ∈ N such that n−1

η (M(xm , xn ,t)) ≤

∑ ki η (M(x0 , x1 ,t)) < ε

i=m

for all m, n ∈ N with m < n and t > 0. Thus, by Proposition 4.2, it follows that {xn }n∈N is an M-Cauchy sequence in X. By the completeness of X, there exists x∗ ∈ X such that xn → x∗ as n → ∞. Due to Proposition 4.3, we have lim η(M(xn , x∗ ,t)) = 0, ∀t > 0.

n→∞

Now, applying the condition (d) and (4.11), we obtain η (M(xn+1 , f x∗ ,t)) = η (M( f xn , f x∗ ,t)) ≤ α(xn , x∗ ,t)η (M( f xn , f x∗ ,t)) ≤ kη (M(xn , x∗ ,t)) , ∀t > 0, which implies that lim η (M(xn+1 , f x∗ ,t)) = 0, ∀t > 0,

n→∞

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i.e., f x∗ = lim xn+1 = x∗ . n→∞

So, x∗ is a fixed point of f . This completes the proof. The following examples show the usefulness of the aforementioned theorem: t Example 4.6. Let X = R, a∗b = min{a, b} for all a, b ∈ [0, 1] and M(x, y,t) = t+|x−y| for all x, y ∈ X and t > 0. Clearly, (X, M, ∗) is an M-complete fuzzy metric space. Define the mapping f : X → X by

 f (x) =

x2 4,

2,

if x ∈ [0, 1], otherwise.

Also, define η(s) = 1s − 1, s ∈ (0, 1] and α : X × X × (0, ∞) → [0, ∞) by  α(x, y,t) =

1, 0,

if x, y ∈ [0, 1], otherwise.

Clearly, f is an α-fuzzy-H -contractive mapping with k = 12 . Now, let x, y ∈ X such that α(x, y,t) ≥ 1 for all t > 0. This implies that x, y ∈ [0, 1] and, by the definitions of f and α, we have f (x) =

x2 y2 ∈ [0, 1], f (y) = ∈ [0, 1], α( f x, f y,t) = 1, ∀t > 0, 4 4

i.e., f is α-admissible. Further, there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0. Indeed, for any x0 ∈ [0, 1], we have α(x0 , f x0 ,t) = 1 for all t > 0. Finally, let {xn }n∈N be a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for each n ∈ N and limn→∞ xn = x. By the definition of the function α, it follows that xn ∈ [0, 1] for each n ∈ N and hence x ∈ [0, 1]. Therefore, α(xn , x,t) = 1 for each n ∈ N. So, all the hypotheses of Theorem 4.16 are satisfied. Here, 0 and 2 are two fixed points of f . However, f is not a fuzzy H -contractive mapping [44]. To see this, consider x = 2 and y = 1. Then, since k ∈ (0, 1), we have η (M( f x, f y,t)) =

k 7 > = kη(M(x, y,t)), ∀t > 0. 4t t

Now, we give a sufficient condition to obtain the uniqueness of the fixed point in the previous theorem. Precisely, we consider the following hypothesis: (U) For all x, y ∈ X and t > 0, there exists z ∈ X such that α(x, z,t) ≥ 1 and α(y, z,t) ≥ 1.

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Theorem 4.17. Adding the condition (U) to the hypothesis of Theorem 4.16, we obtain the uniqueness of the fixed point of f . Proof. Suppose that x∗ and y∗ are two fixed points of f . If α(x∗ , y∗ ,t) ≥ 1 for some t > 0, then by (4.11), we conclude easily that x∗ = y∗ . Assume α(x∗ , y∗ ,t) < 1 for all t > 0. Then, by (U), there exists z ∈ X such that α(x∗ , z,t) ≥ 1 and α(y∗ , z,t) ≥ 1, ∀t > 0.

(4.14)

Since f is α-admissible, and by Equation (4.14), we get α(x∗ , f n z,t) ≥ 1 and α(y∗ , f n z,t) ≥ 1

(4.15)

for all n ∈ N and t > 0. Now, applying (4.11) and (4.15), we have M(x∗ , f n z,t) = M( f x∗ , f ( f n−1 z),t) and


η (M(x∗ , f n z,t)) = η M( f x∗ , f ( f n−1 z),t)
≤ α(x∗ , f n−1 z,t)η M( f x∗ , f ( f n−1 z),t)
≤ kη M(x∗ , f n−1 z,t) ≤ · · · ≤ kn η (M(x∗ , z,t))

for all n ∈ N and t > 0. By letting n → ∞ in the last relation, we get lim η (M(x∗ , f n z,t)) = 0, ∀t > 0,

n→∞

and lim f n z = x∗ .

n→∞

Similarly, we have lim f n z = y∗ .

n→∞

Finally, the uniqueness of the aforementioned limits gives us x∗ = y∗ . This completes the proof. The assumption that ∗ is positive can be further relaxed in Theorem 4.16. In fact, we can prove the following: Theorem 4.18. Let (X, M, ∗) be a M-complete strong fuzzy metric space for a nilpotent t-norm ∗L , and let f : X → X be an α-fuzzy-H -contractive mapping with respect to η ∈ H satisfying the following conditions: a. there exists x0 ∈ X such that α(x0 , f x0 ,t) ≥ 1 for all t > 0; b. f is α-admissible;

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c. η (r ∗ s) ≤ η(r) + η(s) for all r, s ∈ {M(x, f x,t) : x ∈ X,t > 0}; d. each subsequence {xnk }k∈N of a sequence {xn }n∈N = { f n x0 }n∈N has a following property: α(xnk , xnl ,t) ≥ 1 for all k, l ∈ N with k > l and t > 0; e. if {xn } is a sequence in X such that α(xn , xn+1 ,t) ≥ 1 for all n ∈ N and t > 0 and lim xn = x, then α(xn , x,t) ≥ 1 for all n ∈ N and t > 0. n→∞

Then f has a fixed point x∗ ∈ X. Moreover, the sequence { f n x0 }n∈N converges to x∗ . Proof. Let x0 ∈ X and α(x0 , f x0 ,t) ≥ 1 for all t > 0. Define a sequence {xn }n∈N such that xn = f xn−1 = f n x0 . If xn = xn−1 for some n ∈ N, then x∗ = xn is a fixed point of f . So, assume xn ̸= xn−1 for each n ∈ N. Since f is α-admissible, we have α(x0 , f x0 ,t) = α(x0 , x1 ,t) ≥ 1 =⇒ α( f x0 , f x1 ,t) = α(x1 , x2 ,t) ≥ 1 for all t > 0. By induction, we get α(xn , xn+1 ,t) ≥ 1, n ∈ N, ∀t > 0. By Equation (4.11), we have η (M(x1 , x2 ,t)) = η (M( f x0 , f x1 ,t)) ≤ α(x0 , x1 ,t)η (M( f x0 , f x1 ,t)) ≤ kη (M(x0 , x1 ,t)) , ∀t > 0. Inductively, we have η (M(xn , xn+1 ,t)) ≤ kη (M(xn−1 , xn ,t)) ≤ · · · ≤ kn η (M(x0 , x1 ,t))

(4.16)

for all n ∈ N and t > 0. Since η is strictly decreasing and k ∈ (0, 1), we have M(xn , xn+1 ,t) ≥ M(xn−1 , xn ,t) for all n ∈ N and t > 0. So, for every t > 0, the sequence {M(xn , xn+1 ,t)}n∈N is nondecreasing and bounded, it is convergent, i.e., lim M(xn , xn+1 ,t) = p, ∀t > 0.

n→∞

Let us prove, by the contradiction, that p = 1. Suppose that p < 1. Letting n → ∞ in Equation (4.16), since η is continuous, we have lim η (M(xn , xn+1 ,t)) ≤ k lim η (M(xn−1 , xn ,t)) , ∀t > 0.

n→∞

n→∞

So, we obtain a contradiction η(p) ≤ kη(p) and conclude that p = 1, i.e., lim M(xn , xn+1 ,t) = 1, ∀t > 0.

n→∞

(4.17)

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Let us prove that {xn }n∈N is a Cauchy sequence. Suppose the contrary. Then there exist ε > 0, t0 > 0 and s0 ∈ N such that, for each s ∈ N and s ≥ s0 , there exist m(s), n(s) ∈ N, m(s) > n(s) ≥ s such that
η M(xm(s) , xn(s) ,t0 ) ≥ ε and, by the condition (d), α(xm(s)−1 , xn(s)−1 ,t0 ) ≥ 1. Let, for each s ≥ 1, m(s) be the least positive integer exceeding n(s) satisfy
ing the aforementioned property, i.e., η M(xm(s)−1 , xn(s) ,t0 ) < ε and η M(xm(s) ,
xn(s) ,t0 ) ≥ ε for each s ∈ N. Since η is continuous, there exists 0 < ε1 < 1 such that η(ε1 ) = ε, i.e., M(xm(s)−1 , xn(s) ,t0 ) > ε1 , ∀s ∈ N. (4.18) Then we have
ε ≤ η M(xm(s) , xn(s) ,t0 )
≤ α(xm(s)−1 , xn(s)−1 ,t0 ) η M(xm(s) , xn(s) ,t0 )
≤ k η M(xm(s)−1 , xn(s)−1 ,t0 ) , ∀s ∈ N.

(4.19)

Since fuzzy metric is strong, we obtain  M(xm(s)−1 , xn(s)−1 ,t0 ) ≥ ∗L M(xm(s)−1 , xn(s) ,t0 ), M(xn(s) , xn(s)−1 ,t0 )  = max M(xm(s)−1 , xn(s) ,t0 ) + M(xn(s) , xn(s)−1 ,t0 ) − 1, 0 (4.20) for each s ∈ N. Take ε1 defined in Equation (4.18). Then, by Equation (4.17), there exist s0 ∈ N such that M(xn(s) , xn(s)−1 ,t0 ) > 1 − ε1 , ∀s > s0 .

(4.21)

Now, by Equations (4.18) and (4.21), we get M(xm(s)−1 , xn(s) ,t0 ) + M(xn(s) , xn(s)−1 ,t0 ) > 1, ∀s > s0 .

(4.22)

So, applying (4.19), (4.20), (4.22) and the condition (c), we get ε ≤ η(M(xm(s) , xn(s) ,t0 ))
≤ kη M(xm(s)−1 , xn(s)−1 ,t0 ) 

 ≤ k η M(xm(s)−1 , xn(s) ,t) + η M(xn(s) , xn(s)−1 ,t) for each s > s0 . Letting s → ∞ in the aforementioned expression, we get ε ≤ k ε < ε. So, we get a contradiction. Hence, {xn }n∈N is a Cauchy sequence in X. The rest of the proof follows similar lines to Theorem 4.16. This completes the proof.

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Remark 4.10. In the paper of Wardowski [44], one could find the following open question: “Can the condition (a) in Theorem 4.15 (i.e., Theorem 3.2 in Ref. [44]) be omitted for nilpotent t-norms?” If α(x, y,t) = 1 for all x, y ∈ X and t > 0 in Theorem 4.18, then a partial answer to this question is obtained. Namely, in narrowed space (strong fuzzy metric space), we could expand the class of the t-norms, i.e., in that case Theorem 4.18 holds for the nilpotent t-norm ∗ = ∗L . Open Problem. Can the assumption of strong fuzzy metric in Theorem 4.18 be omitted/further relaxed?

4.7

FUZZY Z -CONTRACTIVE MAPPINGS

Most recently, Shukla et al. [35] unified different classes of fuzzy contractive mappings by introducing a new class of fuzzy contractive mappings called as Fuzzy Z -contractive mappings. First, we define the Z -contraction in GV -fuzzy metric spaces. Denote by Z the family of all functions ζ : (0, 1] × (0, 1] → R satisfying the following condition: ζ (t, s) > s for all t, s ∈ (0, 1). Example 4.7. Consider the following functions ζ from (0, 1] × (0, 1] into R defined by (1) ζ (t, s) = ψ(s), where ψ : (0, 1] → (0, 1] is a function such that s < ψ(s) for all s ∈ (0, 1); 1 + t; (2) ζ (t, s) = s+t s (3) ζ (t, s) = . t Then, in all the cases, ζ ∈ Z . Remark 4.11. By the aforementioned definition, it is obvious that ζ (t,t) > t for all 0 < t < 1. Definition 4.15. Let (X, M, ∗) be a fuzzy metric space and f : X → X be a mapping. Suppose that there exists ζ ∈ Z such that M( f x, f y,t) ≥ ζ (M( f x, f y,t), M(x, y,t))

(4.23)

for all x, y ∈ X with f x ̸= f y and t > 0. Then f is called a fuzzy Z -contractive mapping with respect to the function ζ ∈ Z .

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Example 4.8. Every Tirado’s contraction with contractive constant k is a fuzzy Z contraction with respect to the function ζ f ∈ Z defined by ζ f (t, s) = 1 + k(s − 1) for all s,t ∈ (0, 1]. Example 4.9. Every fuzzy contractive mapping with contractive constant k is a fuzzy Z -contraction with respect to the function ζGS ∈ Z defined by ζGS (t, s) = s k+(1−k)s for all s,t ∈ (0, 1]. Example 4.10. In view of Remark 4.8, every H -contractive mapping with respect to η ∈ H is a fuzzy Z -contraction with respect to the function ζW ∈ Z defined by ζW (t, s) = η −1 (kη(s)) for all s,t ∈ (0, 1]. Example 4.11. Every ψ-contractive mapping is a fuzzy Z -contraction with respect to the function ζM defined by ζM (t, s) = ψ(s) for all s,t ∈ (0, 1]. Example 4.12. Let X = R and d be the usual metric on X. Then (X, Md , ∗m ) is t a complete fuzzy metric space, where Md = for all x, y ∈ X,t > 0, is t + d(x, y) the standard fuzzy metric induced by d (see [15]). Let f : X → X be Edelstein’s mapping (contractive mapping) on metric space (X, d), i.e., d( f x, f y) < d(x, y) for all x, y ∈ X, then f is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined by  s+t if t > s; 2 , ζm (t, s) = 1, otherwise. Indeed, the aforementioned fact remains true, if instead s+t 2 (i.e., the arithmetic mean of s and t) for t > s, one take geometric or harmonic mean of s and t. Remark 4.12. If (X, M, ∗) is an arbitrary fuzzy metric space and f : X → X be a Edelstein’s mapping on (X, M, ∗), i.e., M( f x, f y,t) > M(x, y,t) for all x, y ∈ X and t > 0. Then f is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined in the previous example. Therefore, we conclude that for any given fuzzy Edelstein’s mapping we always have ζ (= ζm ) ∈ Z such that the fuzzy Edelstein mapping is a fuzzy Z -contractive mapping and so the contractive mappings considered by Tirado [39], Gregori and Sapena [20], Wardowski [44] and Mihet [28] are included in this new class. Although there are fuzzy Z -contractive mapping which do not belong to any of these considered classes (see, e.g., Examples 4.13, 4.15 and 4.17). The following example shows that a fuzzy Z -contractive mapping may not have a fixed point even in an M-complete fuzzy metric space: Example 4.13. Let n n Xm=o N and define the fuzzy set M on X × X × (0, ∞) by M(n, m,t) = min , for all n, m ∈ X and t > 0. Then (X, M, ∗ p ) is an Mm n complete fuzzy metric space. Define a mapping f : X → X by f n = n + 1 for all n ∈ X. Then f is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined in Example 4.12. Notice that f is a fixed point free mapping on X.

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The aforementioned example motivates us for the consideration of a space having some additional property so that the existence of fixed point of fuzzy Z -contractive mapping can be ensured. Definition 4.16. Let (X, M, ∗) be a fuzzy metric space, f : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, f , ζ ) has the property (S) if, for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = f n x for all n ∈ N such that inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t) for all n ∈ N and t > 0 implies that m>n

m>n

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1

n→∞ m>n

for all t > 0. The following example shows that there exists a function ζ such that the mappings introduced by Tirado [39] forms a quadruple (X, M, f , ζ ) which satisfies the property (S), where (X, M, ∗) is an arbitrary fuzzy metric space: Example 4.14. Let (X, M, ∗) be an arbitrary fuzzy metric space and f : X → X be a fuzzy Tirado-contraction. Then the quadruple (X, M, f , ζ ) has the property (S) with ζ (t, s) = 1 + k(s − 1) for all t, s ∈ (0, 1]. Indeed, if x ∈ X and {xn } be a Picard sequence with initial value x such that inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t)

m>n

m>n

for all n ∈ N and t > 0, then lim inf M(xn , xm ,t) must exists for all t > 0. Suppose n→∞ m>n

that lim inf M(xn , xm ,t) = a(t) for all t > 0, then a(t) ≤ 1. By the definition of ζ , n→∞ m>n

for every t > 0, we have lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1 + k(a(t) − 1).

n→∞ m>n

Also, by the contractivity condition, we obtain 1 + ka(t) ≤ k + a(t) and so 1 ≤ a(t). It shows that a(t) = 1 for all t > 0, i.e., lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1.

n→∞ m>n

The following theorem generalizes Theorem 4.4 (see also Corollary 3.9 in Ref. [28]) for arbitrary t-norms: Theorem 4.19. Let (X, M, ∗) be an M-complete fuzzy metric space and f : X → X be a fuzzy Z -contraction. If the quadruple (X, M, f , ζ ) has the property (S), then f has a unique fixed point u ∈ X. Proof. First, we show that if the fixed point of f exists, then it is unique. Suppose that u, v are two distinct fixed point of f , i.e., f u = u and f v = v and there exists

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s > 0 such that M(u, v, s) < 1. Then, by the condition (4.35) and the definition of ζ , we have M(u, v, s) = M( f u, f v, s) ≥ ζ (M( f u, f v, s), M(u, v, s)) > M(u, v, s). This contradiction shows that M(u, v,t) = 1 for all t > 0 and so u = v. It proves the uniqueness. Now, we show the existence of fixed point of f . Let x0 ∈ X and define the Picard sequence {xn } by xn = f xn−1 for all n ∈ N. If xn = xn−1 for any n ∈ N, then f xn−1 = xn = xn−1 is a fixed point of f . Therefore, we assume that xn ̸= xn−1 for all n ∈ N, i.e., no consecutive terms of the sequence {xn } are equal. Further, if xn = xm for some n < m, then, as no consecutive terms of the sequence {xn } are equal from Equation (4.35), we have M(xn+1 , xn+2 ,t) ≥ ζ (M(xn+1 , xn+2 ,t), M(xn , xn+1 ,t)) > M(xn , xn+1 ,t), i.e., M(xn , xn+1 ,t) < M(xn+1 , xn+2 ,t). Similarly, one can prove that M(xn , xn+1 ,t) < M(xn+1 , xn+2 ,t) < · · · < M(xm , xm+1 ,t). Since xn = xm , we have xn+1 = f xn = f xm = xm+1 and so the aforementioned inequality yields a contradiction. Thus, we can assume that xn ̸= xm for all distinct n, m ∈ N. Now, for t > 0, let an (t) = inf M(xn , xm ,t). m>n

Then it follows from Equation (4.35) and the definition of ζ that M(xn+1 , xm+1 ,t) = M( f xn , f xm ,t) ≥ ζ (M( f xn , f xm ,t), M(xn , xm ,t)) > M(xn , xm ,t)

(4.24)

for each t > 0. Therefore, for all n < m, we have M(xn , xm ,t) < M(xn+1 , xm+1 ,t) for all n < m. Taking infimun over m(> n) in the aforementioned inequality, we obtain inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t),

m>n

m>n

i.e., an (t) ≤ an+1 (t) for all n ∈ N. Thus {an (t)} is bounded and monotonic for all t > 0. Suppose that lim an (t) = a(t) for all t > 0. We claim that a(t) = 1 for all t > 0. n→∞

If s > 0 and a(s) < 1, then, using the fact that the quadruple (X, M, f , ζ ) having the property (S), we obtain lim inf ζ (M(xn , xm , s), M(xn+1 , xm+1 , s)) = 1.

n→∞ m>n

(4.25)

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From the inequality (4.63), we have inf M(xn+1 , xm+1 , s) ≥ inf ζ (M( f xn , f xm , s), M(xn , xm , s)) ≥ inf M(xn , xm , s),

m>n

i.e.,

m>n

m>n

an+1 (s) ≥ inf ζ (M( f xn , f xm , s), M(xn , xm , s)) ≥ an (s). m>n

Letting n → ∞ and using (4.33) in the aforementioned inequality, we obtain lim inf M(xn , xm , s) = a(s) = 1.

n→∞ m>n

This contradiction verifies our claim. By the definition of an , we have lim M(xn , xm ,t) = 1 for all t > 0. Hence {xn } is an M-Cauchy sequence and, by

n,m→∞

M-completeness of X, there exists u ∈ X such that lim M(xn , u,t) = 1, ∀t > 0.

n→∞

(4.26)

Now, we show that u is a fixed point of f . Suppose that f u ̸= u. Without loss of generality, we can assume that xn ̸= u and xn ̸= Tu for all n ∈ N, and so, there exists s > 0 such that M(u, f u, s) < 1, M(xn , u, s) < 1 and M(xn+1 , f u, s) = M( f xn , f u, s) < 1 for all n ∈ N. Then we have M(xn , u, s) < ζ (M( f xn , f u, s), M(xn , u, s)) ≤ M( f xn , f u, s) = M(xn+1 , f u, s). Letting n → ∞ and using (4.34), we obtain 1 ≤ M(u, f u, s). This contradiction shows that M(u, f u,t) = 1 for all t > 0 and so f u = u. Thus the existence of fixed point follows. This completes the proof. Remark 4.13. Examples 4.8 and 4.14 show that the aforementioned theorem generalizes Theorem 4.4 for arbitrary t-norms. The following example shows that this generalization is proper: Example 4.15. Let {xn } be a strictly increasing sequence of real numbers such that 0 < xn ≤ 1 for all n ∈ N and limn→∞ xn = 1. Let X = {xn : n ∈ N} ∪ {1} and define a fuzzy set M on X × X × (0, ∞) by:  1, if x = y; M(x, y,t) = min{x, y}, otherwise, for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗m ) is an M-complete fuzzy metric space. Define a function ζ : (0, 1] × (0, 1] → R by  t, if t > s, ζ (t, s) = √ s, if t ≤ s, for all s,t ∈ (0, 1] and a mapping f : X → X by f xn = xn+1 and f 1 = 1

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for all n ∈ N. Then ζ ∈ Z and the quadruple (X, M, f , ζ ) has the property (S). Furthermore, the mapping f is a fuzzy Z -contractive mapping with respect to the function ζ . Thus, all the conditions of Theorem 4.21 are satisfied, and we can conclude the existence of fixed point of f . Indeed, x = 1 is the unique fixed point of f . Remark 4.14. In view of the aforementioned example, we can conclude that the mapping f is not Tirado’s contraction. For instance, take the sequence {xn } defined 1 by xn = 1 − 2 for all n ∈ N in the aforementioned example. Then we have 2n 1 − M( f xn , f xn+1 ,t) = 1 − M(xn+1 , xn+2 ,t) = 1 − xn+1 , 1 − M(xn , xn+1 ,t) = 1 − xn for all t > 0. Therefore, for sufficient large n, there exists no k such that k ∈ [0, 1) and 1 − M( f xn , f xn+1 ,t) ≤ k[1 − M(xn , xn+1 ,t)], ∀t > 0. Thus Theorem 4.21 is an actual generalization of the fixed point result of Tirado [39], i.e., Theorem 4.4. Next, we introduce another condition (S′ ) which is weaker than the condition (S). Definition 4.17. Let (X, M, ∗) be a fuzzy metric space, f : X → X be a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, f , ζ ) has the property (S′ ) if, for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = f n x for all n ∈ N such that inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t) for all n ∈ N and t > 0 and 0 < m>n

m>n

lim inf M(xn , xm ,t) < 1 for all t > 0, we have

n→∞ m>n

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1, ∀t > 0.

n→∞ m>n

The following example verifies the fact that condition (S′ ) is weaker than condition (S): Example 4.16. Let ε > 0 be fixed and X = [ε, ∞). Define a fuzzy set M on X × X × (0, ∞) by  if x = y;  1, 1 M(x, y,t) = , otherwise,  1 + max{x, y} for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗m ) is a fuzzy metric space. Define a mapping f : X → X by f x = 2x for all x ∈ X. Suppose that ζ : (0, 1] × (0, 1] → R is defined by ζ (t, s) = ψ(s) for all t, s ∈ (0, 1], where ψ ∈ Ψ is such that ψ(0) = 0. Then it is easy to see that the quadruple (X, M, f , ζ ) satisfies the condition (S′ ) trivially. On the other hand, the quadruple (X, M, f , ζ ) does not satisfy the condition (S). Indeed, for any x ∈ X and t > 0, we have inf M( f n x, f m x,t) = inf M(2n x, 2m x,t) = 0 < 1.

m>n

m>n

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Therefore, inf M( f n x, f m x,t) ≤ inf M( f n+1 x, f m+1 x,t) for all n ∈ N and t > 0, but we have

m>n

m>n

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = lim inf ψ(M(xn , xm ,t)) = 0 ̸= 1.

n→∞ m>n

n→∞ m>n

In the next theorem, we see that the condition (S′ ) enables us to extend the result of Mihet [28] for fuzzy Z -contraction, but with an additional assumption to Theorem 4.21: Theorem 4.20. Let (X, M, ∗) be an M-complete fuzzy metric space, f : X → X be a fuzzy Z -contraction and the quadruple (X, M, f , ζ ) has the property (S′ ). In addition, suppose that lim inf M( f n x, f m x,t) > 0 for all x ∈ X and t > 0. Then f has a n→∞ m>n

unique fixed point u ∈ X. Proof. Because of lim inf M( f n x, f m x,t) > 0 for all x ∈ X and t > 0, following the n→∞ m>n

lines of the proof of Theorem 4.21 and using the property (S′ ), we obtain the required result. This completes the proof. In the following example, we show that the class of fuzzy Z -contractions is wider than that of fuzzy ψ-contractions and verify the merit of fuzzy Z -contractive mappings over fuzzy ψ-contractive mappings. For this, we use the idea of Example 4.15. Example 4.17. Let X = {xn : n ∈ N}∪{1}, where {xn } is an arbitrary sequence such that xn ∈ (0, 1), xn < xn+1 for all n ∈ N and limn→∞ xn = 1. Define a fuzzy set M on X × X × (0, ∞) by  1, if x = y, M(x, y,t) = min{x, y}, otherwise, for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗m ) is an M-complete fuzzy metric space. Define a mapping f : X → X by f xn = xn+1 for all n ∈ N and f 1 = 1. Then we claim that T is not a fuzzy ψ-contraction. On the contrary, suppose that T is a fuzzy ψcontraction. Therefore, there exists ψ ∈ Ψ such that ψ(M(xn , xm ,t)) ≤ M( f xn , f xm ,t) for all n, m ∈ N with n < m, i.e., xn < ψ(xn ) ≤ xn+1 .

(4.27)

Since ψ ∈ Ψ, we can choose the sequence {xn } such that, for any x1 ∈ (0, 1), xn+1 = xn + ψ(xn ) for all n ∈ N. Then, by Equation (4.27), we obtain 2 xn < ψ(xn ) ≤

xn + ψ(xn ) . 2

The aforementioned inequalities contradict the definition of ψ. Therefore, f is not a fuzzy ψ-contraction. On the other hand, we have shown in Example 4.15 that the mapping f is a fuzzy Z -contractive mapping as well as that the condition (S′ ) is satisfied. Now the existence and uniqueness of fixed point of f are assured by Theorem 4.22. Indeed, 1 is the unique fixed point of f .

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Corollary 4.2. Let (X, M, ∗) be an M-complete fuzzy metric space, f : X → X be a fuzzy ψ-contractive mapping and lim inf M( f n x, f m x,t) > 0 for all x ∈ X, t > 0. n→∞ m>n

Then f has a unique fixed point u ∈ X.

Proof. In view of Example 4.11 we need only to show that the quadruple (X, M, f , ζ ) have the property (S′ ), where ζ (t, s) = ψ(s). Suppose that x ∈ X and {xn } is a Picard sequence with the initial value x such that inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t) m>n

m>n

and, for all t > 0, 0 < lim inf M(xn , xm ,t) = a(t) < 1. Then, by the definition of ψ, n→∞ m>n

it follows that, for all t > 0,

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = ψ(a(t)).

n→∞ m>n

Also, by the ψ-contractivity, we obtain ψ(a(t)) ≤ a(t) and so a(t) = 1, i.e., lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1, ∀t > 0.

n→∞ m>n

Therefore, the quadruple (X, M, f , ζ ) have the property (S′ ). This completes the proof. Remark 4.15. Since the class of fuzzy ψ-contractions consists of the class of fuzzy contractive mappings [20], Tirado’s contraction [39] and Wardowski’s contraction [44], therefore, fixed point results for these contractions can be obtained by the aforementioned corollary. Remark 4.16. It is clear from the definition that every fuzzy Z -contractive mapping is a fuzzy Edelstein’s mapping (contractive mapping). Also, Remark 4.12 shows that, for every fuzzy Edelstein’s mapping f , there exists a function ζmean ∈ Z such that f is a fuzzy Z -contractive mapping with ζmean ∈ Z . In view of existence of fixed point of mapping f , notice that, for a fuzzy Edelstein’s mapping, the quadruple (X, M, f , ζmean ) need not to have the property (S), e.g., in Example 4.13, f is a fuzzy Edelstein’s mapping but the quadruple (X, M, f , ζ ) does not possess the property (S). Indeed, in this example, for any Picard sequence {xn } with initial value x ∈ X, we have inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t)

m>n

m>n

for all n ∈ N and t > 0, but lim inf ζmean (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 0 ̸= 1, ∀t > 0.

n→∞ m>n

Therefore, the condition (S) of Theorem 4.21 is not satisfied. Also, one can see that the condition: lim inf M( f n x, f m x,t) > 0 for all x ∈ X and t > 0 of Theorem 4.22 is n→∞ m>n

not satisfied, while the condition (S’) is satisfied. Remark 4.17. Motivated by the results of Tirado [39] and Mihet [27], we introduced the class of fuzzy Z -contractive mappings and showed that the mappings of this

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new class have a unique fixed point on an arbitrary M-complete fuzzy metric space having the properties (S) and (S′ ). With suitable examples, we showed that the class of fuzzy Z -contractive mappings is weaker than the existing ones in the literature. Further, it will be interesting to apply this new approach in general settings, e.g., in fuzzy metric-like setting (see [33,34]) as well as it will be interesting to generalize the class of fuzzy Z -contractive mappings for weaker contractive conditions, e.g., (ε, δ )-type contractive conditions (see [29]).

4.8

SUZUKI TYPE FUZZY Z -CONTRACTIVE MAPPINGS AND FIXED POINT

In this section, we discuss the fixed point results of Suzuki type fuzzy Z -contractive mappings given in Ref. [18]. Definition 4.18. Let (X, M, ∗) be a fuzzy metric space and T : X → X be a mapping. Suppose, there exists ζ ∈ Z and q ∈ (0, 1) such that M(x, T x,t) > qM(x, y,t) =⇒ M(T x, Ty,t) ≥ ζ (M(T x, Ty,t), M(x, y,t))

(4.28)

for all x, y ∈ X, T x ̸= Ty,t > 0. Then T is called a Suzuki type fuzzy Z -contractive mapping with respect to the function ζ ∈ Z . The following example shows that a Suzuki type fuzzy Z -contractive mappings may not have a fixed point even in a G-complete fuzzy metric space. Example 4.18. Let X = N. Define a ∗m b = min{a, b} for all a, b ∈ [0, 1] and   h |x − y| i−1 M(x, y,t) = exp t for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗m ) is a fuzzy metric space. Define the mapping T : X → X by T x = 2x for all x ∈ X. Then, it is easy to see that T is a Suzuki type fuzzy Z -contractive mapping with respect to function ζ (t, s) = η −1 (kη(s)) for 1 all s,t ∈ (0, 1], where η : (0, 1] → [0, ∞) is defined by η(t) = − 1 for all t ∈ (0, 1]. t Note that T is a fixed point free mapping on X. The aforementioned example motivates us for the consideration of a space having some additional properties so that the existence of fixed point of Suzuki type fuzzy Z -contractive mapping can be ensured. We need the following properties to prove our main theorems. Definition 4.19. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (P), if for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = T n x for all n ∈ N such that limn→∞ M(xn , xn+1 ,t) = l where l ∈ (0, 1], we have, lim inf ζ ((M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1 for all t > 0.

n→∞ m>n

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Definition 4.20. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (N), if for any sequence {xn } in X such that limn→∞ M(xn , x∗ ,t) = 1, we have M(xn , T xn ,t) ≥ M(x∗ , T xn ,t) for all t > 0 and x∗ ∈ X. Now, we are ready to prove our main theorems. Theorem 4.21. Let (X, M, ∗) be a G-complete fuzzy metric space and T : X → X be a Suzuki type fuzzy Z -contractive mapping. If the quadruple (X, M, T, ζ ) have the properties (P) and (N) respectively, then T has a unique fixed point u ∈ X. Proof. First, we show that if the fixed point of T exists, then it is unique. Suppose, u, v be two distinct fixed point of T, i.e., Tu = u and T v = v and there exists s > 0 such that M(u, v, s) < 1. Since M(u, Tu, s) > qM(u, v, s) then by the condition (2) and definition of ζ , we have M(u, v, s) = M(Tu, T v, s) ≥ ζ (M(Tu, T v, s), M(u, v, s)) > M(u, v, s). This contradiction shows that M(u, v,t) = 1 for all t > 0, and so, u = v. It proves the uniqueness. Now, we shall show the existence of fixed point of T. Let x0 ∈ X and define the Picard sequence {xn } by xn = T xn−1 for all n ∈ N. Being M a GV -fuzzy metric, we have M(x0 , T x0 ,t) > qM(x0 , T x0 ,t), for all t > 0. Then, by condition (2), we have M(T x0 , T x1 ,t) ≥ ζ (M(T x0 , T x1 ,t), M(x0 , x1 ,t)) i.e. M(x1 , x2 ,t) ≥ ζ (M(x1 , x2 ,t), M(x0 , x1 ,t)) > M(x0 , x1 ,t). Similarly, we get M(x2 , x3 ,t) > M(x1 , x2 ,t). In general, we have M(xn , xn+1 ,t) > M(xn−1 , xn ,t). Therefore, (M(xn , xn+1 ,t)) is a strictly non-decreasing sequence of positive real numbers in (0,1] and denote limn→∞ M(xn , xn+1 ,t) by l. We claim that l = 1. If l = 1 we finish, otherwise since M(xn , xn+1 ,t) > qM(xn , xn+1 ,t), then, by condition (2), we have M(T xn , T xn+1 ,t) ≥ ζ (M(T xn , T xn+1 ,t), M(xn , xn+1 ,t)) i.e. M(xn+1 , xn+2 ,t) ≥ ζ (M(xn+1 , xn+2 ,t),t), M(xn , xn+1 ,t))

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taking limit as n → ∞, and using property (P), we get lim

inf

n→∞ (m=n+1)>n

M(xn+1 , xm+1 ,t) ≥ lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1 n→∞ m>n

i.e., l ≥ 1 =⇒ l = 1. Thus {xn } is G-Cauchy sequence. Since X is G-complete, there exists x∗ ∈ X such that limn→+∞ xn = x∗ . We shall show that x∗ is a fixed point of T . First, M(xn , T xn ,t) = M(xn , xn+1 ,t) > M(xn−1 , xn ,t) = M(xn−1 , T xn−1 ,t). By property (N), we have M(xn , T xn ,t) > M(xn−1 , T xn−1 ,t) ≥ M(x∗ , T xn−1 ,t) > qM(x∗ , xn ,t) and then we have, M(T xn , T x∗ ,t) ≥ ζ (M(T xn , T x∗ ,t), M(xn , x∗ ,t)) > M(xn , x∗ ,t), i.e.

M(xn+1 , T x∗ ,t) > M(xn , x∗ ,t).

Taking limit as n → ∞, we obtain M(T x∗ , x∗ ,t) ≥ 1 i.e.

M(T x∗ , x∗ ,t) = 1

for all t > 0 and so T x∗ = x∗ . Example 4.19. Let X = {1, 2, 4}, a ∗m b = min{a, b} for all a, b ∈ [0, 1] and M(x, y,t) =

t , t + d(x, y)

for all t > 0. Then (X, M, ∗) is a G- complete (as well as M- complete) fuzzy metric space. Define T : X → X by T (1) = 1 = T (2), T (4) = 2. Then T is a Suzuki type fuzzy Z -contractive mapping. Also, the quadruple (X, M, T, ζ ) possess properties (P) and (N) trivially, and hence, all the conditions of the aforementioned theorem are satisfied. Thus, T has a unique fixed point u = 1. Lemma 4.1. [8] Each M-complete non-Archimedean fuzzy metric space (X, M, ∗) with ∗ of Hadˇzi´c –type is G-complete. Corollary 4.3. Let (X, M, ∗) be a M-complete non-Archimedean fuzzy metric space with ∗ of Hadˇzi´c –type and T : X → X be a Suzuki type fuzzy Z -contractive mapping. If the quadruple (X, M, T, ζ ) have the properties (P) and (N) respectively, then T has a unique fixed point u ∈ X.

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Now we will try to replace in the previous result the Hadˇzi´c -typeness condition on ∗ by a new property, called (K1). Definition 4.21. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (K1), if for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = T n x for all n ∈ N such that limn→∞ M(xn , xn+1 ,t) = 1, there exists k0 ∈ N and q ∈ (0, 1) such that M(xn , xn+1 ,t) > qM(xn , xm ,t)

(4.29)

for all m > n ≥ k0 , t > 0. Theorem 4.22. Let (X, M, ∗) be an M-complete non-Archimedean fuzzy metric space and T : X → X be a Suzuki type fuzzy Z -contractive mapping. If the quadruple (X, M, T, ζ ) have the properties (P), (K1) and (N) respectively, then T has a unique fixed point u ∈ X. Proof. Following the same proof of Theorem 4.21, we obtain a G-Cauchy sequence {xn }. Now, we prove that {xn } is a M-Cauchy sequence. If the sequence {xn } is not M-Cauchy, then there are ε ∈ (0, 1) and t > 0 such that for each k ∈ N, there exist m(k), n(k) ∈ N with m(k) > n(k) ≥ k and M(xm(k) , xn(k) ,t) ≤ 1 − ε . Let, for each k, m(k) be the least positive integer exceeding n(k) satisfying the aforementioned property, that is M(xm(k)−1 , xn(k) ,t) > 1 − ε and M(xm(k) , xn(k) ,t) ≤ 1 − ε. Then, for each positive integer k, 1 − ε ≥ M(xm(k) , xn(k) ,t) ≥ M(xm(k) , xm(k)−1 ,t) ∗ M(xm(k)−1 , xn(k) ,t) ≥ M(xm(k) , xm(k)−1 ,t) ∗ (1 − ε)

Taking limit as k → ∞, we obtain (1 − ε) ≥ lim M(xm(k) , xn(k) ,t) ≥ (1 − ε) k→∞

and therefore

lim M(xm(k) , xn(k) ,t) = 1 − ε.

k→∞

Since limn→∞ M(xn , xn+1 ,t) = 1, using property (K1), there exists k0 ∈ N such that M(xn(k) , T xn(k) ,t) = M(xn(k) , xn(k)+1 ,t) > qM(xn(k) , xm(k) ,t)

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for all m(k) > n(k) ≥ k0 , t > 0. Now, for each positive integer k ≥ k0 , we have M(xm(k) , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ M(xm(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ M(xm(k)+1 , xn(k)+1 ,t) ∗ M(xn(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ M(T xm(k) , T xn(k) ,t) ∗ M(xn(k)+1 , xn(k) ,t) ≥ M(xm(k) , xm(k)+1 ,t) ∗ ζ (M(T xm(k) , T xn(k) ,t), M(xm(k) , xn(k) ,t)) ∗ M(xn(k)+1 , xn(k) ,t).

(by property (K1))

Taking limit as k → ∞, we obtain 1−ε ≥ 1∗1∗1 = 1

(by property (P))

which is a contradiction and so {xn } is a M-Cauchy sequence. The rest of the proof follows similar to the proof of Theorem 4.21. Remark 4.18. It is obvious that the set of our contractions in Theorem 3.8 includes that of the fuzzy Z -contractions. But converse is not true. The following example illustrates the aforementioned theorem and validates remark 3.9. Example 4.20. Let X = (0, 12 ] ∪ {1, 2}, a ∗ p b = ab for all a, b ∈ [0, 1] and min{x,y} M(x, y,t) = max{x,y} for all x, y ∈ X and for all t > 0. Clearly, (X, M, ∗ p ) is an Mcomplete non-Archimedean fuzzy metric space. Define the mapping T : X → X by   2 if x ∈ (0, 21 ], Tx =  1 if x = 1, 2 for all t > 0. It is easy to see that T is a Suzuki type fuzzy Z -contractive mapping with respect to the function ζ ∈ Z defined by ζ (t, s) = ψ(s). However, T is not a fuzzy Z -contractive mapping for any ζ ∈ Z (for this consider x ∈ (0, 12 ] and y = 1). Also, the quadruple (X, M, T, ζ ) possess properties (P), (K1) and (N) trivially, and hence, all the conditions of the aforementioned theorem are satisfied. Thus, T has a unique fixed point x∗ = 1. The next examples demonstrate that the properties (P), (K1) and (N) in Theorem 3.8 are not superfluous.

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Example 4.21. Let X = N, a ∗ b = ab for a, b ∈ [0, 1] and define the fuzzy set n all n mo for all n, m ∈ X and t > 0. Then M on X × X × (0, ∞) by M(n, m,t) = min , m n (X, M, ∗ p ) is an M-complete fuzzy metric space. Define a mapping T : X → X by T n = n + 1 for all n ∈ X. Then T is a fuzzy Z -contractive mapping and hence Suzuki type fuzzy Z -contractive mapping with respect to the function ζm ∈ Z (see [35]) but the quadruple (X, M, T, ζ ) does not satisfy the property (P) (for this, consider Picard sequence xn = n generated by T ). Notice that T is a fixed point free mapping on X. Example 4.22. Let X = (0, ∞), a ∗ p b = ab for all a, b ∈ [0, 1] and M(x, y,t) = min{x,y} for all x, y ∈ X and for all t > 0. Clearly, (X, M, ∗ p ) is an M-complete max{x,y} non-Archimedean fuzzy metric space. Define the mapping T : X → X by  √  x if x ̸= 1, Tx =  2 if x = 1 for all t > 0. If ζ defined from (0, 1] × (0, 1] into R by ζ (t, s) = ψ(s), where ψ : (0, 1] → (0, 1] is a function such that s < ψ(s) for all s ∈ (0, 1). Then, it is easy to see that T is a Suzuki type fuzzy Z -contractive mapping but the quadruple (X, M, T, ζ ) does not satisfy the property (N) (for this, consider xn = 2 + 1n , n ∈ N). Notice that T is a fixed point free mapping on X. To establish Theorem 3.8 for general fuzzy metric space and to generalize the main theorem of [35], we introduce the followings: Definition 4.22. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (K2), if for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = T n x for all n ∈ N and t > 0 such that M(xm , xm+1 ,t) > M(xn , xn+1 ,t), for all m > n, there exists k0 ∈ N and q ∈ (0, 1) such that M(xn , xn+1 ,t) > qM(xm , xn ,t) (4.30) for all m > n ≥ k0 . Definition 4.23. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (S), if for any Picard sequence {xn } with initial value x ∈ X, i.e., xn = T n x for all n ∈ N such that inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t) for all n ∈ N,t > 0 implies that

m>n

m>n

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = 1 for all t > 0.

n→∞ m>n

The following example verifies the fact that condition (P) is weaker than condition (S).

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Example 4.23. Let X = [1, ∞) , a ∗m b = min{a, b} for all a, b ∈ [0, 1]. Define a fuzzy set M on X × X × (0, ∞) by:  if x = y;  1, 1 M(x, y,t) = for all x, y ∈ X,t ∈ (0, ∞). , otherwise  1 + max{x, y} Then, (X, M, ∗m ) is a fuzzy metric space. Define T : X → X by T x = 2x for all x ∈ X. Suppose ζ : (0, 1] × (0, 1] → R be defined by ζ (t, s) = ψ(s) for all t, s ∈ (0, 1], where ψ ∈ Ψ is such that ψ(0) = 0. Then, it is easy to see that the quadruple (X, M, T, ζ ) satisfies the condition (P) trivially. On the other hand, the quadruple (X, M, T, ζ ) does not satisfy the condition (S). Indeed, for any x ∈ X, t > 0 we have inf M(T n x, T m x,t) = inf M(2n x, 2m x,t) = 0 < 1.

m>n

m>n

Therefore, inf M(T n x, T m x,t) ≤ inf M(T n+1 x, T m+1 x,t) for all n ∈ N,t > 0. But, m>n

m>n

lim inf ζ (M(xn+1 , xm+1 ,t), M(xn , xm ,t)) = lim inf ψ(M(xn , xm ,t)) = 0 ̸= 1.

n→∞ m>n

n→∞ m>n

Theorem 4.23. Let (X, M, ∗) be an M-complete fuzzy metric space and T : X → X be a Suzuki type fuzzy Z -contractive mapping. If the quadruple (X, M, T, ζ ) have the properties (S), (K2) and (N) respectively, then T has a unique fixed point u ∈ X. Proof. Same proof of Theorem 4.21 is valid for uniqueness. We shall show the existence of fixed point of T. Let x0 ∈ X and define the Picard sequence {xn } by xn = T xn−1 for all n ∈ N. If xn = xn−1 for any n ∈ N, then T xn−1 = xn = xn−1 is a fixed point of T . Therefore, we assume that xn ̸= xn−1 for all n ∈ N, i.e., no consecutive terms of the sequence {xn } are equal. As no consecutive terms of the sequence {xn } are equal and M(xn , xn+1 ,t) > qM(xn , xn+1 ,t), from Equation (4.2), we have M(xn+1 , xn+2 ,t) ≥ ζ (M(xn+1 , xn+2 ,t), M(xn , xn+1 ,t)) > M(xn , xn+1 ,t) i.e., M(xn , xn+1 ,t) < M(xn+1 , xn+2 ,t). Fix n < m. Similarly one can prove that M(xn , xn+1 ,t) < M(xn+1 , xn+2 ,t) < · · · < M(xm , xm+1 ,t).

(4.31)

Suppose, now that xn = xm , we have xn+1 = T xn = T xm = xm+1 , and so, the aforementioned inequality yields a contradiction. Thus, we can assume that xn ̸= xm for all distinct n, m ∈ N. Now, for t > 0, let an (t) = inf M(xn , xm ,t). m>n

By Equation (4.31) and property (K2) there exists k0 ∈ N such that M(xn , xn+1 ,t) > qM(xm , xn ,t)

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for all m > n ≥ k0 . Then it follows from Equation (4.2) and by definition of ζ that M(xn+1 , xm+1 ,t) = M(T xn , T xm ,t) ≥ ζ (M(T xn , T xm ,t), M(xn , xm ,t)) (4.32)

> M(xn , xm ,t) for each t > 0 and m > n ≥ k0 . Therefore, we have M(xn , xm ,t) < M(xn+1 , xm+1 ,t) for all k0 ≤ n < m. Taking infimum over m(> n) in the aforementioned inequality we obtain inf M(xn , xm ,t) ≤ inf M(xn+1 , xm+1 ,t)

m>n

m>n

i.e., an (t) ≤ an+1 (t) for all n ≥ k0 . Thus, {an (t)} is bounded and monotonic for each t > 0. Suppose, lim an (t) = a(t),t > 0. We claim that a(t) = 1 for each t > 0. If s > 0 n→∞

and a(s) < 1, then, using the fact that the quadruple (X, M, T, ζ ) have the property (S), we obtain lim inf ζ (M(xn , xm , s), M(xn+1 , xm+1 , s)) = 1.

(4.33)

n→∞ m>n

From inequality (4.63) we have inf M(xn+1 , xm+1 , s) ≥ inf ζ (M(T xn , T xm , s), M(xn , xm , s)) ≥ inf M(xn , xm , s)

m>n

m>n

m>n

i.e., an+1 (s) ≥ inf ζ (M(T xn , T xm , s), M(xn , xm , s)) ≥ an (s). m>n

Letting n → ∞ and using (4.33) in the aforementioned inequality we obtain lim inf M(xn , xm , s) = a(s) = 1.

n→∞ m>n

This contradiction verifies our claim. By definition of an we have lim M(xn , xm ,t) = n,m→∞

1 for all t > 0. Hence, {xn } is an M-Cauchy sequence and by M-completeness of X there exists x∗ ∈ X such that lim M(xn , x∗ ,t) = 1 for all t > 0.

n→∞

(4.34)

Finally, the same proof of Theorem 4.21 is valid for x∗ = T x∗ . Remark 4.19. In view of Theorems 3.8 and 3.16, Example 3.15 it seems that property(K2) is weaker than property(K1). It will be interesting to have an example illustrating this fact.

Theory of Fuzzy Contractive Mappings and Fixed Points

4.9

95

OBSERVATIONS

1. The following definition can be considered as a variant of Suzuki type fuzzy Z contractive mapping. Definition 4.24. Let (X, M, ∗) be a fuzzy metric space and T : X → X be a mapping. Suppose, there exists ζ ∈ Z such that t M(x, T x,t) ≥ M(x, y, ) =⇒ M(T x, Ty,t) ≥ ζ (M(T x, Ty,t), M(x, y,t)) 2

(4.35)

for all x, y ∈ X, T x ̸= Ty,t > 0. Then T is called a Suzuki type fuzzy Z -contractive mapping with respect to the function ζ ∈ Z . 2. The following examples demonstrate that Definitions 3.1 and 4.1 are independent of each other. Example 4.24. Consider the M-complete non-Archimedean fuzzy metric space as defined in 4.22 and define the mapping T : X → X by  √  x if x ̸= 1, Tx =  5 if x = 1 for all t > 0. If ζ defined from (0, 1] × (0, 1] into R by ζ (t, s) = ψ(s), where ψ : (0, 1] → (0, 1] is a function such that s < ψ(s) for all s ∈ (0, 1). Then, it is easy to see that T is a Suzuki type fuzzy Z -contractive mapping with respect to condition (9) but T does not satisfy condition (2)(for this, consider x = 1 and y = 2.6). Example 4.25. Let X = {0, 1, 2}. Define a ∗ b = ab for all a, b ∈ [0, 1] and   h |x − y| i−1 M(x, y,t) = exp t for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗) is a fuzzy metric space. Define the mapping T : X → X by   2 if x = 0, Tx =  1 if x = {1, 2} for all t > 0. If ζ defined from (0, 1] × (0, 1] into R by ζ (t, s) = ψ(s), where ψ : (0, 1] → (0, 1] is a function such that s < ψ(s) for all s ∈ (0, 1). Then, it is easy to see that T is a Suzuki type fuzzy Z -contractive mapping with respect to condition (2) but T does not satisfy condition (9)(for this, consider t = 1 and x = 0, y = 1). 3. Further, we observe that all the aforementioned theorems will remain true if we replace Suzuki type fuzzy Z -contractive mapping with respect to condition (2) by condition (9). For it, we replace (4.29) and (4.30) by M(xn , xn+1 ,t) > M(xn , xm ,t/2).

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4. Examples 3.12 and 4.2 demonstrate that Suzuki type fuzzy Z -contractive mappings need not be continuous, whereas the known classes of fuzzy contractive mappings [20,28,35,44] are necessarily continuous. 5. It is well known that the contractive mappings given by Suzuki [38] in the sense of classical metric space characterize metric completeness. Thus, it will be interesting to have a kind of Suzuki type fuzzy Z -contractive mapping which could characterize completeness of underline fuzzy metric space. 4.9.1

ˇ C´ CIRI ´ C ´ OPERATORS AND UNIFIED FUZZY-PRESI FIXED POINT THEOREMS

´ c operators In 2016, Shukla et al. [4] introduced the notion of Fuzzy- Preˇsi´c-Ciri´ and established unified fixed point theorems for such operators and also discussed some new properties of fuzzy contractive sequences in the framework of George and Veeramani fuzzy metric space. Definition 4.25. Let (X, M, ∗) be a fuzzy metric space, k a positive integer and ´ c operator if f : X k → X be a mapping. Then f is called a fuzzy-Preˇsi´c-Ciri´   1 1 − 1 ≤ λ · max −1 , M( f (x1 , . . . , xk ), f (x2 , . . . , xk+1 ),t) 1≤i≤k M(xi , xi+1 ,t)

(4.36)

for all x1 , . . . , xk , xk+1 ∈ X and t > 0, where λ ∈ (0, 1). Alternatively, the aforementioned condition may be written as  −1 1 −1 +1 , 1≤i≤k M(xi , xi+1 ,t) (4.37) for all x1 , . . . , xk , xk+1 ∈ X and t > 0, where λ ∈ (0, 1).   M( f (x1 , . . . , xk ), f (x2 , . . . , xk+1 ),t) ≥ λ · max

Remark 4.20. Taking M as Md in condition (4.36) (or (4.37)), we get d( f (x1 , . . . , xk ), f (x2 , . . . , xk+1 )) ≤ λ max{d(xi , xi+1 ) : 1 ≤ i ≤ k}, for all x1 , . . . , xk , xk+1 ∈ X, where λ ∈ (0, 1). The following definition will be needed to prove a fixed point theorem for the ´ c operators in M-complete fuzzy metric spaces. fuzzy-Preˇsi´c-Ciri´ Let ∗ be a given t-norm. For a1 , a2 , . . . , an ∈ [0, 1], we use the notation n

∗ ai = a1 ∗ a2 ∗ · · · ∗ an .

i=1

Let a ∈ [0, 1]. Then we can define the sequence {∗n a}n∈N by ∗1 a = a and ∗n+1 a = (∗n a) ∗ a, for n ≥ 1.

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Definition 4.26 (Hadˇzi´c and Pap [24]). A t-norm ∗ is said to be of H-type if the sequence {∗n a}n∈N is equicontinuous at 1, that is, for all ε ∈ (0, 1), there exists δ ∈ (0, 1) such that, a ∈ (1 − δ , 1] implies ∗n a > 1 − ε for all n ∈ N. An important H-type t-norm is ∗m . Some other examples of H-type t-norms can be found in Ref. [24]. We denote the class of all Hadˇzi´c-type t-norms by H . Theorem 4.24. Let (X, M, ∗) be an M-complete fuzzy metric space, k a positive inte´ c operator. Suppose that one of the following ger and f : X k → X a fuzzy-Preˇsi´c-Ciri´ conditions holds: (H1) ∗ ∈ H and there exist x1 , x2 . . . , xk ∈ X such that inf M(xi , xi+1 ,t) > 0, i = 1, 2, . . . , k − 1,

t>0

inf M(xk , f (x1 , . . . , xk ),t) > 0.

t>0

(H2) There exist x1 , x2 . . . , xk ∈ X such that the following property holds: for each ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that, for m, n > n0 , with m > n, h i j −1 (n) (n) 1 k we get ∗m−1 > 1 − ε, where s j = 2 j−n+1 , j = n, . . . , m − 2, j=n 1 + µ(ts j )λ (n)

(n)

sm−1 = sm−2 and µ(z) := max



max

1≤i≤k−1

1 i

λk



   1 1 1 −1 , −1 . M(xi , xi+1 , z) λ M(xk , f (x1 , . . . , xk ), z)

Then f has a fixed point in X. If, in addition, we suppose that, on the diagonal ∆ ⊂ X k, M( f (u, . . . , u), f (v, . . . , v),t) > M(u, v,t),

∀t >0

(4.38)

holds for u, v ∈ X with u ̸= v, then f has a unique fixed point. Proof. Let x1 , x2 , . . . , xk be the points in X given by hypothesis. Define a sequence {xn } by xn+k = f (xn , xn+1 , . . . , xn+k−1 ), ∀ n ∈ N. For notational convenience,  set  M(xn , xn+1  ,t) = Mn (t),  for all n ∈ N and t > 0 and consider µ(t) := 1 1 1 max − 1 : 1 ≤ i ≤ k , where θ = λ k . By mathematical induction, we i θ Mi (t) show that 1 − 1 ≤ µ(t)θ n , ∀ n ∈ N, ∀ t > 0. (4.39) Mn (t) By the definition of µ(t), it is obvious that (4.39) is true for n = 1, 2, . . . , k. 1 1 Let the following k inequalities, for t > 0, − 1 ≤ µ(t)θ n , −1 ≤ Mn (t) Mn+1 (t) 1 µ(t)θ n+1 , . . . , − 1 ≤ µ(t)θ n+k−1 be the induction hypothesis. Mn+k−1 (t)

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Since θ = λ k < 1, from Equation (4.36) we have 1 Mn+k (t)

1 −1 M( f (xn , xn+1 , . . . , xn+k−1 ), f (xn+1 , xn+2 , . . . , xn+k ),t)   1 −1: 1 ≤ i ≤ k ≤ λ max M(xn+i−1 , xn+i ,t)   1 = λ max −1: 1 ≤ i ≤ k Mn+i−1 (t)  ≤ λ max µ(t) θ n+i−1 : 1 ≤ i ≤ k ≤ λ µ(t)θ n = µ(t)θ n+k , t > 0.

−1 =

Hence, by induction, (4.39) is true for all n ∈ N. Next, we show that {xn } is an M-Cauchy sequence. Consider ε ∈ (0, 1) and t > 0 fixed. For n, m ∈ N with m > n, we have, using (4.39), that M(xn , xm ,t) ≥M(xn , xn+1 ,t/2) ∗ M(xn+1 , xm ,t/2) ≥M(xn , xn+1 ,t/2) ∗ M(xn+1 , xn+2 ,t/22 ) ∗ · · · ∗ M(xm−2 , xm−1 ,t/2m−n−1 ) ∗ M(xm−1 , xm ,t/2m−n−1 )   j−(n−1) = ∗m−2 M (t/2 ) ∗ Mm−1 (t/2m−n−1 ) j j=n  h i−1   −1 j−(n−1) j ≥ ∗m−2 1 + µ(t/2 )θ ∗ 1 + µ(t/2m−n−1 )θ m−1 j=n  h i−1   −1 j−(n−1) n ≥ ∗m−2 1 + µ(t/2 )θ ∗ 1 + µ(t/2m−n−1 )θ n . j=n Under condition (H1), it is easy to check that µ := sup µ(t) ∈ [0, ∞), since t>0

sup µ(t) ≤ max

1≤i≤k

t>0



     1  1 1 1   sup − 1 = max − 1  θ i t>0 Mi (t) 1≤i≤k  θ i inf Mi (t) t>0

and inf Mi (t) = inf M(xi , xi+1 ,t) > 0, i = 1, 2, . . . , k − 1, t>0

t>0

inf Mk (t) = inf M(xk , xk+1 ,t) = inf M(xk , f (x1 , . . . , xk ),t) > 0.

t>0

t>0

t>0

In these conditions, for n, m ∈ N with m > n, we have that M(xn , xm ,t) ≥ [1 + µθ n ]−1 ∗ · · · ∗ [1 + µθ n ]−1 = ∗m−n [1 + µθ n ]−1 .

(4.40)

Since ∗ ∈ H , there exists δ ∈ (0, 1) such that: if [1 + µθ n ]−1 ∈ (1 − δ , 1], then ∗m−n [1 + µθ n ]−1 > 1 − ε, for m > n. As 0 < θ < 1, given δ > 0 there exists n0 ∈ N such that [1 + µθ n ]−1 ∈ (1 − δ , 1], for all n > n0 (it suffices to take n0 ∈ N such that 1 1 + µθ n0 < 1−δ ). With this choice, we obtain ∗m−n [1 + µθ n ]−1 > 1 − ε, ∀ n > n0 , m > n. The aforementioned inequality with (4.40) and the properties of M give M(xn , xm ,t) > 1 − ε, ∀ n, m > n0 , ∀ t > 0.

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On the other hand, if (H2) holds, there exists n0 ∈ N such that, for m, n > n0 , with m > n, we get  h i−1   −1 m−2 j−(n−1) j M(xn , xm ,t) ≥ ∗ j=n 1 + µ(t/2 )θ ∗ 1 + µ(t/2m−n−1 )θ m−1 m−1

= ∗

j=n

h i−1 (n) 1 + µ(ts j )θ j > 1 − ε.

Thus, in both situations, {xn } is an M-Cauchy sequence. By M-completeness of X, there exists u ∈ X such that lim M(xn , u,t) = 1, ∀ t > 0.

n→∞

(4.41)

Now, we show that u is a fixed point of f . Indeed, for any n ∈ N and t > 0, we have M(xn+k , f (u, . . . , u),t) = M( f (xn , . . . , xn+k−1 ), f (u, . . . , u),t) ≥ M( f (xn , . . . , xn+k−1 ), f (xn+1 , . . . , xn+k−1 , u),t/2) ∗ M( f (xn+1 , . . . , xn+k−1 , u), f (xn+2 , . . . , xn+k−1 , u, u),t/22 ) ∗ · · · ∗ M( f (xn+k−2 , xn+k−1 , u, . . . , u), f (xn+k−1 , u, . . . , u),t/2k−1 ) ∗ M( f (xn+k−1 , u, . . . , u), f (u, . . . , u),t/2k−1 ).

(4.42)

Using (4.37), (4.39) and θ ∈ (0, 1), we have M( f (xn , . . . , xn+k−1 ), f (xn+1 , . . . , xn+k−1 , u),t)      −1 1 1 −1 , −1 +1 ≥ λ · max max M(xn+k−1 , u,t) 1≤i≤k−1 M(xn+i−1 , xn+i ,t)      −1 1 1 = λ · max max −1 , −1 +1 M(xn+k−1 , u,t) 1≤i≤k−1 Mn+i−1 (t)  −1    1 n+i−1 , −1 +1 ≥ λ · max max µ(t)θ M(xn+k−1 , u,t) 1≤i≤k−1    −1 1 ≥ λ · max µ(t)θ n , −1 +1 . M(xn+k−1 , u,t) Using (4.41) and the fact that 0 < θ < 1, it follows from the aforementioned inequality that lim M( f (xn , . . . , xn+k−1 ), f (xn+1 , . . . , xn+k−1 , u),t) = 1, ∀ t > 0. Similarly, n→∞

lim M( f (xn+1 , . . . , xn+k−1 , u), f (xn+2 , . . . , xn+k−1 , u, u),t) = 1, ∀ t > 0, . . . ,

n→∞

lim M( f (xn+k−1 , u, . . . , u), f (u, . . . , u),t) = 1, ∀ t > 0.

n→∞

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

The aforementioned properties with (4.42) imply that lim M(xn+k , f (u, . . . , u),t) = 1, ∀ t > 0.

n→∞

(4.43)

Therefore, for any n ∈ N and t > 0, we have M(u, f (u, u, . . . , u),t) ≥ M(u, xn+k ,t/2) ∗ M(xn+k , f (u, . . . , u),t/2), which, together with (4.41) and (4.43), gives M(u, f (u, u, . . . , u),t) = 1, ∀ t > 0. Thus, f (u, u, . . . , u) = u, that is, u is a fixed point of f . Finally, for uniqueness, suppose that v ∈ X is another fixed point of f with u ̸= v. Then, from Equation (4.38), we have M(u, v,t) = M( f (u, . . . , u), f (v, . . . , v),t) > M(u, v,t), for every t > 0. This contradiction shows that u = v. Thus, under condition (4.38), the fixed point of f is unique. Remark 4.21. The uniqueness condition (4.38) in Theorem 4.24 is reduced, for M = Md , to d(T (u, . . . , u), T (v, . . . , v)) < d(u, v), ∀ u, v ∈ X with u ̸= v. From the proof of Theorem 4.24, it is obvious that uniqueness is also derived just considering the following weaker hypothesis: for each u, v ∈ X fixed with u ̸= v, there exists t > 0 such that M( f (u, . . . , u), f (v, . . . , v),t) > M(u, v,t).

(4.44)

Corollary 4.4. Let (X, M, ∗) be an M-complete fuzzy metric space and f : X → X be a fuzzy contractive mapping (see Gregori and Sapena [20]), that is,   1 1 −1 ≤ λ − 1 , ∀x, y ∈ X, ∀t > 0, M( f x, f y,t) M(x, y,t) where λ ∈ (0, 1). Suppose that one of the following conditions holds: (h1) ∗ ∈ H and there exists x1 ∈ X such that inf M(x1 , f (x1 ),t) > 0. t>0

(h2) There exists x1 ∈ X such that the following property holds: for each ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that, for m, n > n0 , with m > n, we get h i−1 (n) (n) (n) 1 j , j = n, . . . , m − 2, sm−1 = ∗m−1 1 + µ(ts )λ > 1 − ε, where s j = 2 j−n+1 j=n j   1 1 (n) sm−2 and µ(z) := −1 . λ M(x1 , f (x1 ), z) Then f has a unique fixed point in X. Proof. Take k = 1 in Theorem 4.24, then the existence of a fixed point u ∈ X 1 follows. Further, for x, y ∈ X fixed with x ̸= y, we get M(x,y,t) − 1 > 0 for some

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t > 0, thus, since f is a fuzzy contractive mapping, we have

1 −1 < M( f x, f y,t)

1 − 1, for some t > 0, that is, M( f x, f y,t) > M(x, y,t), for some t > 0, where M(x, y,t) x, y ∈ X are fixed with x ̸= y. Hence, the uniqueness condition (4.44) of Remark 4.21 is satisfied. Therefore, the fixed point of f is unique.

Remark 4.22. Note that, if we select M = Md , conditions (H1) in Theorem 4.24 and (h1) in Corollary 4.4 are valid only if the mapping f has fixed points. Int = 1, if f (x1 ) = x1 and deed, for k = 1, inf Md (x1 , f (x1 ),t) = inf t>0 t>0 t + d(x1 , f (x1 )) inf Md (x1 , f (x1 ),t) = 0, if f (x1 ) ̸= x1 . Similarly, in the general case k ∈ N, for the

t>0

validity of (H1) it is obliged that x1 = x2 = · · · = xk = f (x1 , x2 , . . . , xk ). Thus, for the standard fuzzy metric induced by the metric d, the restriction (H1) is not useful, since it requires to start the process with a fixed point. However, condition (H2) is interesting in the general case and, in particular, for the standard fuzzy metric induced by the metric d. Indeed, for k = 1, M = Md and any x1 ∈ X, we have   µ(z) :=

1  λ

1  d(x1 , f (x1 )) − 1 = z λz z + d(x1 , f (x1 ))

and, thus, for t > 0 and m, n ∈ N with m > n,  −1 h i−1 m−1 j−1 d(x , f (x )) λ (n) 1 1  . ∗ 1 + µ(ts j )λ j = ∗ 1 + (n) j=n j=n t s

m−1

j

Note that, if f (x1 ) = x1 , then µ is null and, for every ε > 0, it is satisfied that h i−1 m−1 (n) = 1 > 1 − ε, for all m, n ∈ N with m > n. Consider the gen∗ 1 + µ(ts j )λ j j=n

eral case f (x1 ) ̸= x1 or f (x1 ) = x1 . If we take ∗ = ∗m , then, for t > 0 and m, n ∈ N with m > n, i−1 h m−1 (n) ∗ 1 + µ(ts j )λ j = j=n

−1 j−1 d(x , f (x )) λ 1 1  min 1 + (n) n≤ j≤m−1 t s 

j

−1 j−1 d(x , f (x )) λ 1 1  . = 1 + max n≤ j≤m−1 s(n) t 

j

1 Replacing s j = 2 j−n+1 , j = n, . . . , m − 2 and sm−1 = sm−2 , we have for t > 0 and m, n ∈ N with m > n, (n)

(n)

(n)

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

max

n≤ j≤m−1

λ j−1 (n) sj

 = max 2λ n−1 , 4λ n , . . . , 2m−n−1 λ m−3 , 2m−n−1 λ m−2  = max (2λ )n−1 2−n+2 , (2λ )n 2−n+2 , . . . , (2λ )m−3 2−n+2 , (2λ )m−2 2−n+1  ≤ max (2λ )n−1 2−n+2 , (2λ )n 2−n+2 , . . . , (2λ )m−3 2−n+2 , (2λ )m−2 2−n+2  ≤ max (2λ )n−1 , (2λ )n , . . . , (2λ )m−3 , (2λ )m−2 2−n+2 .

If λ ∈ (0, 21 ], since m > n, we get 0 ≤

max

n≤ j≤m−1

λ j−1 (n) sj

≤ (2λ )n−1 2−n+2 = 2λ n−1 →

0, as n → ∞, so that, for ε ∈ (0, 1) and t > 0 fixed, there exists n0 ∈ N ε λ j−1 t and, such that, for m, n > n0 , with m > n, max < (n) n≤ j≤m−1 s d(x1 , f (x1 )) 1 − ε j  −1 h i−1 j−1 d(x1 , f (x1 )) λ (n) j  > 1 − ε. hence, ∗m−1 = 1 + max j=n 1 + µ(ts j )λ n≤ j≤m−1 s(n) t j

This proves that condition (h2) holds for M = Md and ∗ = ∗m if λ ∈ (0, 21 ], independently of the choice of x1 ∈ X. Moreover, for a general k ∈ N, taking M = Md and any x1 , . . . , xk ∈ X, we get " ( # #) " 1 1 1 1 −1 , −1 µ(z) := max max i z z λ z+d(x , f (x 1≤i≤k−1 λ k z+d(xi ,xi+1 ) 1 ,...,xk )) k   1 1 = max max d(xk , f (x1 , . . . , xk )) i d(xi , xi+1 ), λz 1≤i≤k−1 λ k z   1 d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) max . = max , i z λ 1≤i≤k−1 λk i h j −1 (n) k is equal to Hence, for t > 0 and m, n ∈ N with m > n, ∗m−1 j=n 1 + µ(ts j )λ −1  j d(x , x ) d(x , f (x , . . . , x )) i i+1 1 k k max λ k . ∗ 1 + (n) max , i j=n λ 1≤i≤k−1 λk ts j 

m−1

1



Taking ∗ = ∗m , we have, for t > 0 and m, n ∈ N with m > n, that the previous expression is  −1   j d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) 1 + 1 max 1 max max , λ k . i t n≤ j≤m−1 s(n) λ 1≤i≤k−1 λk j

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Now, since λ > 0 and s j > 0, for every j = n, . . . , m − 1, we get 1



 j d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) max (n) max λk , max i n≤ j≤m−1 s λ 1≤i≤k−1 λk j   j  j  d(xi , xi+1 )λ k d(xk , f (x1 , . . . , xk ))λ k , = max max max (n) i (n) 1≤i≤k−1  n≤ j≤m−1 sj λ k sj λ   j  j  d(xk , f (x1 , . . . , xk ))λ k d(xi , xi+1 )λ k , max ≤ max max max (n) i (n)  1≤i≤k−1 n≤ j≤m−1 n≤ j≤m−1 sj λ k sj λ   j  j  λk d(xi , xi+1 ) λ k d(xk , f (x1 , . . . , xk )) max (n) = max max max (n) , i 1≤i≤k−1 n≤ j≤m−1 n≤ j≤m−1 λ λk s s  j

= max



j

d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) , i λ 1≤i≤k−1 λk max



max

λ

j k

n≤ j≤m−1 s(n) j

.

1 For s j = 2 j−n+1 , j = n, . . . , m − 2 and sm−1 = sm−2 , we have for t > 0 and m, n ∈ N with m > n, (n)

(n)

(n)

j

max

λk

n≤ j≤m−1 s(n) j

n n o n+1 m−2 m−1 = max 2λ k , 4λ k , . . . , 2m−n−1 λ k , 2m−n−1 λ k n 1 1 1 = max (2λ k )n 2−n+1 , (2λ k )n+1 2−n+1 , . . . , (2λ k )m−2 2−n+1 , o 1 (2λ k )m−1 2−n n o 1 1 1 1 ≤ max (2λ k )n , (2λ k )n+1 , . . . , (2λ k )m−2 , (2λ k )m−1 2−n+1 . 1

If λ ∈ (0, 21k ], then 2λ k ≤ 1 and, since m > n, we get j

0≤

max

λk

n≤ j≤m−1 s(n) j

1

n

≤ (2λ k )n 2−n+1 = 2λ k → 0, as n → ∞,

so that, for ε ∈ (0, 1) and t > 0 fixed, there exists n0 ∈ N such that, for m, n > n0 , with m > n,   j 1 d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) ε max (n) max max , λk 1 − ε. Therefore, (H2) holds for M = Md and ∗ = ∗m and ∗m−1 1 + µ(ts )λ j=n j if λ ∈ (0, 21k ], independently of the choice of x1 , . . . , xk ∈ X.

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Next, we give an example that illustrates Theorem 4.24. 2α + 2n−1 − 1 1 , n ∈ N, and consider the set , xn = 2 2n 2 X = {1} ∪ {xn : n ∈ N} . Define the fuzzy set M : X × (0, ∞) → [0, 1] by ( x ∗m y if x ̸= y, M(x, y,t) = ∀x, y ∈ X, ∀t > 0. 1 if x = y, Example 4.26. Let 0 < α
0, where λ = . Therefore, f is a fuzzy-Preˇsi´c-Ciri´ 2 1 operator with λ = . 2 We prove that condition (H1) is satisfied. Indeed, ∗m ∈ H and, besides, starting with the points x1 = α < x2 = 41 + α2 in X, we get inf M(x1 , x2 ,t) = min{x1 , x2 } = x1 = α > 0 and

t>0



1 α + , 2α,t 4 2

inf M(x2 , f (x1 , x2 ),t) = inf M(x2 , 2 min{x1 , x2 },t) = inf M t>0 t>0  if α = 16 ,  1 = > 0.  min{ 41 + α2 , 2α} if α ̸= 16

t>0



Also, by definition of f , for x, y ∈ X with x ̸= y, we have M( f (x, x), f (y, y),t) = M(1, 1,t) = 1 > M(x, y,t), ∀t > 0. Hence, all the conditions of Theorem 4.24 are satisfied and, thus, we can conclude the existence of a unique fixed point of f . In fact, 1 is the unique fixed point of f . Next, we give a sufficient condition for the validity of condition (4.38) under hypothesis (4.36) (or, equivalently, (4.37)) provided that k ≥ 2. This condition is related to M and the t-norm selected ∗ and allows to establish the following corollary of Theorem 4.24. Corollary 4.5. Let (X, M, ∗) be an M-complete fuzzy metric space, k an integer with ´ c operator. Suppose that one of the condik ≥ 2 and f : X k → X a fuzzy-Preˇsi´c-Ciri´ tions (H1) or (H2) holds. Then f has a fixed point in X. If, in addition, we suppose that

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for each u, v ∈ X fixed with u ̸= v, there exists t > 0 such that    −1 1 k ∗ λ· −1 +1 > M(u, v,t), i=1 zi

(4.45)

where zi = M(u, v,t/2i ), for i = 1, . . . , k − 1, and zk = zk−1 , then f has a unique fixed point. Proof. The first part of the corollary follows from the proof of Theorem 4.24. Now, for the uniqueness of fixed point, suppose that u, v ∈ X are fixed points of f with u ̸= v. Then, for any t > 0, we have M(u, v,t) = M( f (u, . . . , u), f (v, . . . , v),t) ≥ M( f (u, . . . , u), f (u, . . . , u, v),t/2) ∗ M( f (u, . . . , u, v), f (v, . . . , v),t/2) ≥ M( f (u, . . . , u), f (u, . . . , u, v),t/2) ∗ M( f (u, . . . , u, v), f (u, . . . , u, v, v),t/22 ) ∗ · · · ∗ M( f (u, u, v, . . . , v), f (u, v, . . . , v),t/2k−1 ) ∗ M( f (u, v, . . . , v), f (v, . . . , v),t/2k−1 ). (4.46) Using (4.37), we get, for every t > 0,   M( f (u, . . . , u), f (u, . . . , u, v),t/2) ≥ λ ·

 −1 1 −1 +1 M(u, v,t/2)

and, similarly,   M( f (u, . . . , u, v), f (u, . . . , u, v, v),t/22 ) ≥ λ ·

 −1 1 − 1 + 1 , ..., M(u, v,t/22 )

 −1 1 · · · M( f (u, u, v, . . . , v), f (u, v, . . . , v),t/2 −1 +1 M(u, v,t/2k−1 )  −1   1 k−1 M( f (u, v, . . . , v), f (v, . . . , v),t/2 ) ≥ λ · −1 +1 . M(u, v,t/2k−1 ) k−1

  )≥ λ·

In consequence, by Equation (4.46), for every t > 0, the following inequality holds  −1    −1 1 1 −1 +1 ∗ λ· − 1 + 1 M(u, v,t/2) M(u, v,t/22 )    −1    −1 1 1 ∗···∗ λ · −1 +1 ∗ λ· −1 +1 . M(u, v,t/2k−1 ) M(u, v,t/2k−1 )

  M(u, v,t) ≥ λ ·

From the previous inequality and (4.45), we can affirm that there exists t > 0 such that M(u, v,t) > M(u, v,t), which is a contradiction, so that u = v and the fixed point of f is unique.

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Next, we present the following extension of Theorem 4.24. Theorem 4.25. Let (X, M, ∗) be an M-complete fuzzy metric space, k a positive inte´ c operator. Suppose that one of the following ger and f : X k → X a fuzzy-Preˇsi´c-Ciri´ conditions holds: (H1*) Condition (H1). (H2*) There exist x1 , x2 . . . , xk ∈ X such that the following property holds: for each ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that, for m, n > n0 , with m > i h j −1 (n,m) k > 1 − ε, for some collection of values n, we get ∗m−1 1 + µ(ts )λ j=n j (n,m)

(n,m)

sj > 0, j = n, . . . , m − 1, with ∑m−1 j=n s j statement of Theorem 4.24.

≤ 1, where µ(z) is given in the

Then f has a fixed point in X. If, in addition, we suppose that, on the diagonal ∆ ⊂ X k , condition (4.38) holds for u, v ∈ X with u ̸= v, then f has a unique fixed point. Proof. It is similar to the proof of Theorem 4.24. To prove that {xn } is an M(n,m) Cauchy sequence, we consider the choice for s j given in the statement. Using the nondecreasing character of M(x, y, ·) for every x, y ∈ X and following the proof of Theorem 4.8 [20], we have, for t > 0 and n, m ∈ N with m > n, (n,m)

M(xn , xm ,t) ≥ M(xn , xn+1 ,tsn

(n,m)

(n,m)

) ∗ M(xn+1 , xn+2 ,tsn+1 ) (n,m)

∗ · · · ∗ M(xm−2 , xm−1 ,tsm−2 ) ∗ M(xm−1 , xm ,tsm−1 ) h i−1 m−1 m−1 (n,m) (n,m) = ∗ M j (ts j ) ≥ ∗ 1 + µ(ts j )θ j . j=n

j=n

The case (H1*) is analogous to the proof of Theorem 4.24. Under condition (H2*), given ε > 0, there exists n0 ∈ N such that, for m, n > n0 with m > n, we get h i−1 (n,m) M(xn , xm ,t) ≥ ∗m−1 )θ j > 1 − ε. The proof is completed similarly j=n 1 + µ(ts j to that of Theorem 4.24. Corollary 4.6. Let (X, M, ∗) be an M-complete fuzzy metric X → X be  space and f :  1 1 a fuzzy contractive mapping, that is, −1 ≤ λ − 1 , ∀x, y ∈ M( f x, f y,t) M(x, y,t) X, ∀t > 0, where λ ∈ (0, 1). Suppose that one of the following conditions holds: (h1*) Condition (h1) is satisfied. (h2*) There exists x1 ∈ X such that the following property holds: for each ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that, for m, n > n0 , with m > n, we get h i−1 (n,m) (n,m) j ∗m−1 1 + µ(ts )λ > 1 − ε, for some collection of values s j > 0, j=n j (n,m)

j = n, . . . , m − 1, with ∑m−1 j=n s j

Then f has a unique fixed point in X.

≤ 1, where µ is given in Corollary 4.4.

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Remark 4.23. Theorem 4.25 and Corollary 4.6 are more general than Theorem 4.24 and Corollary 4.4, respectively. Conditions (H2*) and (h2*) show that the rel(n,m) (n,m) evant point in the choice of the values s j is the fact that s j > 0, for every m−1 (n,m)

j = n, . . . , m − 1, and ∑ s j j=n

(n,m)

≤ 1. Hence, we can select different values of s j

as

long as these requirements are fulfilled. Note also that it is possible to take the ex(n,m) (n) pressions of s j to be independent of m, that is, in the form s j , if we select them ∞

(n)

as positive numbers such that ∑ s j ≤ 1. Moreover, we can take the expressions j=n

(n,m)

of s j

to be independent of n and m, that is, in the form s j , if we select them as ∞

positive numbers such that ∑ s j ≤ 1. j=1

Remark 4.24. It is important to note that, in the case where M = Md and ∗ = ∗m , condition (H2*) (resp. (h2*)) is trivially valid for arbitrary choices of x1 , . . . , xk (resp., (n,m) 1 x1 ) and for any value of λ ∈ (0, 1), since we can choose s j = j( j+1) , for j = ∞

n, . . . , m − 1, which are positive and such that ∑ j=1

1 j( j+1)

= 1. (n,m)

We start with the case k = 1. For M = Md , ∗ = ∗m , any x1 ∈ X and taking s j = j−1 λ 1 = j( j+1) , j = n, . . . , m−1, we have, for t > 0 and m, n ∈ N with m > n, n≤max j≤m−1 s(n,m) j max j( j + 1)λ j−1 . We study the function ϕ(x) := x(x + 1)λ x−1 , whose derivative

n≤ j≤m−1

is ϕ ′ (x) = (2x + 1)λ x−1 + x(x + 1) log(λ )λ x−1 = λ x−1 (2x + 1 + x(x + 1) log(λ )). Since λ ∈ (0, 1), the quadratic function ψ(x) := log(λ ) x2 + (log λ + 2)x + 1 is con1 cave and has its vertex at x = − 12 − log(λ ) , which is arbitrarily large if λ is arbitrarily close to zero. However, there exists n1 ∈ N large enough (depending just on λ ) such that, for every x > n1 , ψ(x) < 0. Therefore, for every x > n1 , ϕ ′ (x) < 0 and, thus, ϕ is decreasing on (n1 , +∞). In consequence, if we take t > 0 and λ j−1 m, n ∈ N with m > n > n1 , then max (n,m) = n(n + 1)λ n−1 → 0, as n → ∞. n≤ j≤m−1 s j Hence, for ε ∈ (0, 1) and t > 0 fixed, there exists n0 ∈ N with n0 ≥ n1 such that, λ j−1 t ε for m, n > n0 , with m > n, max (n,m) < and, in consequence, n≤ j≤m−1 s d(x1 , f (x1 )) 1 − ε j  −1 h i−1 d(x1 , f (x1 )) (n,m) m−1 m−1 j j−1 ∗ j=n 1 + µ(ts j )λ = ∗ j=n 1 + j( j + 1)λ > 1 − ε. Here, t we have considered that d(x1 , f (x1 )) > 0 since the condition f (x1 ) = x1 leads to a trivial case. Therefore, since λ ∈ (0, 1), condition (h2*) holds for M = Md and ∗ = ∗m , independently of the choice of x1 .

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Now, we consider the general case k ∈ N. For M = Md , ∗ = ∗m and any x1 , . . . , xk ∈ (n,m) 1 X, we get, for t > 0 and m, n ∈ N with m > n, taking s j = j( j+1) , j = n, . . . , m − 1, that j

max

λk

n≤ j≤m−1 s(n,m) j

n o n m−2 m−1 = max n(n + 1)λ k , . . . , (m − 2)(m − 1)λ k , (m − 1)mλ k . 1

e We consider the function ϕ(x) := x(x + 1)ν x , being ν = λ k , where ϕe′ (x) = x ν (2x + 1 + x(x + 1) log(ν)). The sign of ϕe′ coincides with the sign of the function e given by ψ(x) e ψ, := log(ν) x2 + (log ν + 2)x + 1. Since λ ∈ (0, 1), also ν ∈ (0, 1) 1 e is a concave parabola with vertex at x = − 12 − log(ν) . Similarly and the graph of ψ to the case k = 1, there exists ne1 ∈ N large enough (depending on λ ) such that, for e every x > ne1 , ψ(x) < 0, hence, for every x > ne1 , ϕe′ (x) < 0 and ϕe is decreasing on (ne1 , +∞). Therefore, for fixed t > 0 and taking m, n ∈ N with m > n > ne1 , we get n λ j−1 max (n,m) = n(n + 1)λ k → 0, as n → ∞. This proves that, for ε ∈ (0, 1) and n≤ j≤m−1 s j t > 0 fixed, there exists n0 ∈ N with n0 ≥ ne1 such that, for m, n > n0 , with m > n, j

max

λk


0 and s j > 0, for every j = n, . . . , m − 1, we have, following the calculations in Remark 4.22, for t > 0 and m, n ∈ N with m > n, that m−1

∗

j=n

h i j −1 (n,m) 1 + µ(ts j )λ k 

1 ≥ 1 + max t



d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) max , i λ 1≤i≤k−1 λk



max

−1 j λk  .

n≤ j≤m−1 s(n,m) j

Hence, we have proved that, for ε ∈ (0, 1) and t > 0 fixed, there exists n0 ∈ N such h i j −1 (n,m) k that, for m, n > n0 with m > n, ∗m−1 1 + µ(ts )λ > 1 − ε. In the previous j j=n inequalities, we have assumed that 

d(xi , xi+1 ) d(xk , f (x1 , . . . , xk )) max , max i λ 1≤i≤k−1 λk



> 0,

since the opposite situation leads to a trivial case. Again, since λ ∈ (0, 1), condition (H2*) holds for M = Md and ∗ = ∗m , independently of the choice of x1 , . . . , xk .

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Corollary 4.7. Let (X, M, ∗) be an M-complete fuzzy metric space, k an integer with ´ c operator. Suppose that one of the condik ≥ 2 and f : X k → X a fuzzy-Preˇsi´c-Ciri´ tions (H1*) or (H2*) holds. Then f has a fixed point in X. If, in addition, we suppose that for each u, v ∈ X fixed with u ̸= v, there exists t > 0 such that  −1   1 k −1 +1 > M(u, v,t), (4.47) ∗ λ· i=1 zi where zi = M(u, v,t ri ), for i = 1, . . . , k, for some sequence of values ri > 0, i = k

1, . . . , k, with

∑ ri ≤ 1, then f has a unique fixed point.

i=1

Proof. The existence of fixed points follows from the proof of Theorem 4.25. For the uniqueness of fixed point, we suppose that u, v ∈ X are fixed points of f with u ̸= v. Then, using the nondecreasing character of M(x, y, ·), for all x, y ∈ X, we have, for any t > 0, M(u, v,t) = M( f (u, . . . , u), f (v, . . . , v),t) ≥ M( f (u, . . . , u), f (u, . . . , u, v),t r1 ) ∗ M( f (u, . . . , u, v), f (u, . . . , u, v, v),t r2 ) ∗ · · · ∗ M( f (u, u, v, . . . , v), f (u, v, . . . , v),t rk−1 ) ∗ M( f (u, v, . . . , v), f (v, . . . , v),t rk ).

(4.48)

Similarly to the proof of Corollary 4.5, using (4.37), we get, for every t > 0,  −1   1 −1 +1 M( f (u, . . . , u), f (u, . . . , u, v),t r1 ) ≥ λ · M(u, v,t r1 ) and, similarly,    −1 1 M( f (u, . . . , u, v), f (u, . . . , u, v, v),t r2 ) ≥ λ · −1 +1 , ..., M(u, v,t r2 )    −1 1 M( f (u, u, v, . . . , v), f (u, v, . . . , v),t rk−1 ) ≥ λ · −1 +1 M(u, v,t rk−1 )    −1 1 M( f (u, v, . . . , v), f (v, . . . , v),t rk ) ≥ λ · −1 +1 . M(u, v,t rk ) Therefore, the previous inequalities and (4.48) imply, for every t > 0, that M(u, v,t) ≥    −1 1 ∗ki=1 λ · −1 +1 . Hence, from Equation (4.47), there exists t > 0 M(u, v,t ri ) such that M(u, v,t) > M(u, v,t), and we obtain a contradiction again, so that the fixed point of f is unique. Remark 4.25. Concerning condition (4.47), we note that the expression of the val(n,m) ues ri , i = 1, . . . , k, can be of similar type to s j in (H2*) or different, provided that k

the requirements ri > 0, i = 1, . . . , k, and ∑ ri ≤ 1 are fulfilled. i=1

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Remark 4.26. As final remark concerning the fixed point results, instead of (X, M, ∗) an M-complete fuzzy metric space, if we consider the hypothesis that (X, M, ∗) is a G-complete fuzzy metric space, then we can remove the restrictions (H1), (H2) in Theorem 4.24 and Corollary 4.5 and (h1), (h2) in Corollary 4.4. This comes from the proof of Theorem 4.24; in this case, we can start with arbitrary points x1 , . . . , xk in X and, to prove that the sequence defined is G-Cauchy, we just note that, for t > 0 and p ∈ N fixed, we have M(xn , xn+p ,t) ≥ =

n+p−1

∗

j=n n+p−1

∗

j=n

(n)

M(x j , x j+1 ,ts j ) (n)

M j (ts j ) ≥

n+p−1

∗

j=n

h i−1 (n) , 1 + µ(ts j )θ n

1 (n) where s j = j−n+1 , j = n, . . . , n + p − 1. 2 Note that the last term in the previous inequality consists of a fixed number of terms (for every n), that is, p terms, each of one tends to 1 as n → ∞ due to θ ∈ (0, 1) (n) and the fact that s j represents a constant sequence for each j fixed, in the sense that sn = 21 , for every n, sn+1 = (n)

(n)

1 , 22

(n)

for every n, . . . , sn+p−1 =

1 2p ,

for every n. Hence

p)

lim M(xn , xn+p ,t) = 1∗ · · · ∗1 = 1, for each t > 0 and p > 0, and {xn } is G-Cauchy.

n→∞

We include some conclusions on fuzzy contractive sequences that are derived from the proof of the main results in the previous section. Definition 4.27. Let (X, M, ∗) be a fuzzy metric space and k a positive integer. We say that {xn } ⊂ X is a fuzzy contractive sequence if there exists λ ∈ (0, 1) such that   1 1 − 1 ≤ λ · max −1 , (4.49) M(xn+k , xn+k+1 ,t) 1≤i≤k M(xn+i−1 , xn+i ,t) for all t > 0 and n ∈ N. Condition (4.49) can also be written as   M(xn+k , xn+k+1 ,t) ≥ λ · max 1≤i≤k

1 M(xn+i−1 , xn+i ,t)

−1 −1 +1 , 

for all t > 0 and n ∈ N, where λ ∈ (0, 1). This notion is a generalization  of Definition 3.8[20] since, for k = 1, it is reduced 1 1 to −1 ≤ λ · − 1 , for all t > 0 and n ∈ N, where M(xn+1 , xn+2 ,t) M(xn , xn+1 ,t) λ ∈ (0, 1). For the case k = 1, it is proposed in Ref. [20] the following open question: Is a fuzzy contractive sequence a Cauchy sequence in George and Veeramani’s sense (that is, an M-Cauchy sequence). We study this problem for an arbitrary k ∈ N, by imposing sufficient conditions which guarantee the validity of this assertion.

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For a given sequence {xn }, consider µ(z) := max

1≤i≤k

the hypotheses:

1 i

λk



 1 − 1 , and M(xi , xi+1 , z)

(HS1) ∗ ∈ H and inf M(xi , xi+1 ,t) > 0, for all i = 1, 2, . . . , k. t>0

(HS2) For each ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that, for m, n > n0 , h i j −1 (n,m) with m > n, we get ∗m−1 )λ k > 1 − ε, for some collection j=n 1 + µ(ts j (n,m)

of values s j

(n,m)

> 0, j = n, . . . , m − 1, such that ∑m−1 j=n s j

≤ 1.

Theorem 4.26. Let (X, M, ∗) be a fuzzy metric space and k a positive integer. Let {xn } ⊂ X be a fuzzy contractive sequence. Suppose that one of the conditions (HS1) or (HS2) holds. Then {xn } is an M-Cauchy sequence. Proof. As in the proof of Theorem  4.24,  we denote  Mn (t) := M(xn , xn+1 ,t), for n ∈ N 1 1 1 and t > 0 and µ(t) = max1≤i≤k − 1 , where θ = λ k . Similarly to the θ i Mi (t) proof of Theorem 4.24, by induction, we prove that 1 − 1 ≤ µ(t)θ n , Mn (t)

∀ n ∈ N, ∀ t > 0.

(4.50)

Indeed, it is true for n = 1, 2, . . . , k. Assuming that it is true for n, n + 1, . . . , n + k − 1, we have, from Equation (4.49),   1 1 1 −1 = − 1 ≤ λ max −1 Mn+k (t) M(xn+k , xn+k+1 ,t) 1≤i≤k M(xn+i−1 , xn+i ,t)    1 − 1 ≤ λ max µ(t) θ n+i−1 ≤ λ µ(t)θ n = µ(t)θ n+k , = λ max 1≤i≤k 1≤i≤k Mn+i−1 (t) 1

for t > 0, where we have used that θ = λ k < 1. To check that {xn } is an M-Cauchy sequence, we take ε ∈ (0, 1) and t > 0 fixed. Then, by Equation (4.50), using the nondecreasing character of M(x, y, ·) for every x, y ∈ X and following the proof of Theorem 4.8 [20], we have, for n, m ∈ N h i−1 (n,m) (n,m) m−1 j with m > n, that M(xn , xm ,t) ≥ ∗m−1 M (ts ) ≥ ∗ 1 + µ(t s )θ , for j j=n j j=n j (n,m)

(n,m)

any collection of values s j > 0, j = n, . . . , m − 1, with ∑m−1 ≤ 1. j=n s j If (HS1) holds, then µ := sup µ(t) ∈ [0, ∞), therefore, for n, m ∈ N with m > n, we t>0

get M(xn , xm ,t) ≥ ∗m−n [1 + µθ n ]−1 . Since ∗ ∈ H , the proof is complete similarly to the proof of Theorem 4.24. On the other hand, (HS2) provides trivially the character of M-Cauchy sequence for {xn }. Remark 4.27. If M = Md , condition (HS1) is satisfied only for constant sequences {xn }.

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Remark 4.28. The role of specific class of t-norm i.e. H-type t-norm can easily be observed from the proof of Theorems 4.24 and 4.25. Then the following question naturally arises. Question. Does Theorems 4.24 and 4.25 remain true if H-type t-norm is replaced by any arbitrary t-norm ?

4.10

CARISTI TYPE MAPPINGS AND FIXED POINT

In 2016, Abbasi et al. [1] presented an interesting generalization of Caristi’s [6] fixed point theorem and Ekeland’s [13] variational principle in the setting of fuzzy metric spaces. Theorem 4.27. Let (X, M, ∗) be a complete fuzzy metric space with ∗ is continuous and archimedean, T : X → X be a self-mapping, ϕ : X → [0, 1] be such that ϕ is nontrivial (i.e. x ∈ X such that ϕ(x) ̸= 0) and upper semi-continuous functions. Assume that M(x, T x,t) ∗ ϕ(T x) ≥ ϕ(x) (4.51) for all x ∈ X and t > 0. Then T has a fixed point in X Very recently, Moreno et al. [18] extended the results of Abbasi et al. [1] and also proved completeness characterization of corresponding fuzzy metric space. Definition 4.28. Let η : [0, 1] → [0, 1]. • η is said to be amenable if η −1 ({1}) = {1}. • η is said to be ∗-superadditive if η(t ∗ s) ≥ η(t) ∗ η(s), for all t, s ∈ [0, 1]. Lemma 4.2. Let η : [0, 1] → [0, 1] be a continuous and nondecreasing mapping. If η(t) = 1 for some t ∈ (0, 1) and ∗ is archimedean, then η(s) = 1, for all s ∈ [0, 1]. Proof. If t ∈ (0, 1), since ∗ is archimedean, there exists n with ∗nt < s. Then η(s) ≥ ∗n η(t) = 1. By continuity, η(1) = η(0) = 1. Let (X, M, ∗) be a fuzzy metric space, ϕ : X → [0, 1] and η : [0, 1] → [0, 1]. For any x ∈ X such that ϕ(x) ̸= 0, set the Caristi- Kirk balls C(x) := {y ∈ X : η(M(x, y,t)) ∗ ϕ(y) ≥ ϕ(x), ∀t > 0}. Theorem 4.28. Let (X, M, ∗) be a fuzzy metric space with ∗ is continuous and archimedean, T, S : X → X be two self-mappings, ϕ : X → [0, 1] be such that ϕ is nontrivial on S (i.e. ∃x ∈ X such that ϕ(Sx) ̸= 0) and upper semi-continuous functions. Let η : [0, 1] → [0, 1] be a continuous, nondecreasing mapping such that η(t ∗ s) ≥ η(t) ∗ η(s) and η −1 ({1}) = {1} and satisfying η(M(Sx, T x,t)) ∗ ϕ(T x) ≥ ϕ(Sx) for all x ∈ X and t > 0. If S(X) is complete, then T and S have a common point in X.

(4.52)

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Proof. For any x ∈ X such that ϕ(x) ̸= 0 set the Caristi–– Kirk balls C(x) := {y ∈ X : η(M(x, y,t)) ∗ ϕ(y) ≥ ϕ(x), ∀t > 0} and

α(x) := sup ϕ(y). y∈C(x)

Then for each y ∈ C(x) we have 1 ≥ α(x) ≥ ϕ(y). Observe that C(Sx) ̸= 0/ for all x, becacuse by Equation (4.52) T x ∈ C(Sx). Let x1 = x, and choose inductively xn+1 such that Sxn+1 ∈ C(Sxn ) and 1 ϕ(Sxn+1 ) ≥ α(Sxn ) − , n

∀t ≥ 0.

Since Sxn+1 ∈ C(Sxn ), ϕ(Sxn+1 ) ≥ η(M(Sxn , Sxn+1 ,t)) ∗ ϕ(Sxn+1 ) ≥ ϕ(Sxn ), for all t > 0. So {ϕ(Sxn )} is an increasing sequence, and in a consequence, it converges. On the other hand, 1 α(Sxn ) ≥ ϕ(Sxn+1 ) ≥ α(Sxn ) − . n So limn→∞ α(Sxn ) = limn→∞ ϕ(Sxn ) exists. Setting k := lim α(Sxn ) = lim ϕ(Sxn ). n→∞

n→∞

(4.53)

For any n ∈ N, by induction on m, we prove that the following inequality holds: η(M(Sxn , Sxm ,t)) ∗ ϕ(Sxm ) ≥ ϕ(Sxn ),

∀t > 0, ∀m > n.

(4.54)

For m = n + 1, it is true because Sxn+1 ∈ C(Sxn ). Suppose (4.54) is valid for m > n and prove for m + 1: t t η(M(Sxn , Sxm+1 ,t)) ∗ ϕ(Sxm+1 ) ≥ η(M(Sxn , Sxm , )) ∗ η(M(Sxm , Sxm+1 , )) ∗ ϕ(Sxm+1 ) 2 2 t t ≥ η(M(Sxn , Sxm , )) ∗ η(SM(xm , Sxm+1 , )) ∗ ϕ(Sxm+1 ) 2 2 t ≥ η(M(Sxn , Sxm , )) ∗ ϕ(Sxm ) 2 ≥ ϕ(Sxn ). Therefore (4.54) is true for m + 1. Now, we claim that {Sxn } is Cauchy. Suppose {Sxn } is not a Cauchy sequence, so there exist 0 < ε < 1 and t > 0 such that for each n ∈ N there exists m ∈ N such that M(Sxn , Sxm ,t) ≤ 1 − ε. By Equation (4.53), for each 0 < ε ′ < 1, there exists N ∈ N

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such that k ≥ ϕ(Sxn ) ≥ k(1 − ε ′ ), for all n > N. From Equation (4.54) and properties of η, we can conclude k ∗ η((1 − ε)) ≥ η(M(Sxn , Sxm ,t)) ∗ k ≥ η(M(Sxn , Sxm ,t)) ∗ ϕ(Sxm ) ≥ ϕ(Sxn ) ≥ k(1 − ε ′ ), valid for all m > n > N i.e. k ∗ η((1 − ε)) ≥ k(1 − ε ′ ) which is a contradiction with archimedean condition due to amenability of η. So {Sxn } is a Cauchy sequence. Since S(X) is complete, {Sxn } converges to v = Su ∈ S(X). Since ϕ is u.s.c. and by Equation (4.53), we have k = lim supn→∞ ϕ(Sxn ) ≤ ϕ(Su). On the other hand, by taking limit from both sides of Equation (4.54), we obtain ϕ(Sxn ) ≤ lim sup(η(M(Sxn , Sxm ,t)) ∗ ϕ(Sxm )) ≤ η(M(Sxn , u,t)) ∗ ϕ(Su), (4.55) m→∞

for all t > 0. Thus, Su ∈ C(Sxn ). Therefore, α(Sxn ) ≥ ϕ(Su). So by Equation (4.53), k ≥ ϕ(Su) and so k = ϕ(Su) = ϕ(v). Since Su ∈ C(Sxn ) and (4.52) holds, Tu ∈ C(Su). Note that, η(M(Sxn , Tu,t)) ∗ ϕ(Tu) ≥ η(M(Sxn , Su, 2t )) ∗ η(M(Su, Tu, 2t )) ∗ ϕ(Tu) ≥ η(M(Sxn , Su, 2t )) ∗ ϕ(Su) ≥ ϕ(Sxn ). for all t > 0 hence Tu ∈ C(Sxn ) for all n ∈ N. This implies that ϕ(Tu) ≤ αn (xn ) for all n ∈ N. Hence by Equation (4.53), we get ϕ(Tu) ≤ k. Since (4.52) holds and ϕ(Su) = k, we have that ϕ(Su) = k ≥ ϕ(Tu) ≥ ϕ(Su). Thus ϕ(Su) = ϕ(Tu) = k. Also (4.53) shows that k ∗ η(M(Su, Tu,t)) ≥ k,

(4.56)

for all t > 0. It means that η(M(Su, Tu,t)) = 1 and for Lemma 4.2 M(Su, Tu,t) = 1, for all t > 0, and so Su = Tu. Taking S the identity, we obtain a fixed point result. Corollary 4.8. Let (X, M, ∗) be a complete fuzzy metric space with ∗ is continuous and archimedean, T : X → X be a self-mapping, ϕ : X → [0, 1] be such that ϕ is non-trivial (i.e. x ∈ X such that ϕ(x) ̸= 0) and upper semi-continuous functions. Let η : [0, 1] → [0, 1] be a continuous, nondecreasing mapping such that η(t ∗ s) ≥ η(t) ∗ η(s) and η −1 ({1}) = {1} and satisfying η(M(x, T x,t)) ∗ ϕ(T x) ≥ ϕ(x) for all x ∈ X and t > 0. Then T has a fixed point in X.

(4.57)

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The following example illustrates superiority over Abassi Theorem 3.1 [1] Example 4.27. Let {xn } be a strictly increasing sequence of real numbers such that 0 < xn ≤ 1 for all n ∈ N and limn→∞ xn = 1. Let X = {xn : n ∈ N} ∪ {1} and a ∗ 1 b = {2ab} {1+a+b−ab}

2

for all a, b ∈ [0, 1], define a fuzzy set M on X × X × (0, ∞) by: M(1, 1,t) = 1 = M(xn , xn ,t);

for each n ∈ N; and

1 M(x1 , x2 ,t) = M(x2 , x1 ,t) = , 7

1 . 49 Then it is easy to check that (X, M, ∗) is a fuzzy metric space. Define a mapping T : X → X by T xn = xn+1 for all n ∈ N and T 1 = 1 and if ϕ and η are defined by ϕ(r) = r and η(s) = 1. Then T satisfies all the conditions of Corollary 3.4 and consequently has a unique fixed point u = 1. M(x1 , x3 ,t) = M(x3 , x1 ,t) = ..... = M(x2 , x3 ,t) = M(x3 , x2 ,t) = M(x2 , x4 ,t)..... =

However, T does not satisfy the Abassi condition (3.1) (see [1] page 933). Suppose to the contrary that T satisfy the Abassi condition (3.1), then lim sup M(xn , T xn ,t) ∗ lim sup ϕ(T xn ) ≥ lim sup ϕ(xn ) n→∞

i.e. or

n→∞

n→∞

1 ∗ lim sup ϕ(xn+1 ) ≥ ϕ(xn ) 49 n→∞

1 ∗ lim sup ϕ(xn+1 ) ≥ ϕ(xn ) 7 n→∞ Since ϕ is u.s.c. and hence k = lim supn→∞ ϕ(xn+1 ) ≤ ϕ(u), where limn→∞ xn = u. So the aforementioned inequality reduces to 1 ∗k ≥ k 49 ( or 71 ∗ k ≥ k) which is a contradiction to the archimedean condition of ∗. If we consider η and S to be identities maps in Theorem 6.2, then we have following result of Abbasi et al. [1] Corollary 4.9. Let (X, M, ∗) be a complete fuzzy metric space with ∗ is continuous and archimedean, T : X → X be a self-mapping, ϕ : X → [0, 1] be such that ϕ is nontrivial (i.e. x ∈ X such that ϕ(x) ̸= 0) and upper semi-continuous functions. Assume that M(x, T x,t) ∗ ϕ(T x) ≥ ϕ(x) (4.58) for all x ∈ X and t > 0. Then T has a fixed point in X

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Lemma 4.3. If η : [0, 1] → [0, 1] is continuous, nondecreasing, ∗-superadditive, amenable and η(t) > 0, for all t > 0, then η is a fuzzy-metric-preserving. Proof. Let (X, M, ∗) be a fuzzy metric space. We want to check that M ′ = η ◦ M is a fuzzy metric space. Properties (GV1), (GV2), (GV3) and (GV5) are easy to prove. For (GV4), let x, y, z ∈ X and t, s > 0, then M ′ (x, z,t + s) = η(M(x, z,t + s)) ≥ η(M(x, y,t) ∗ M(y, z,t)) ≥ η(M(x, y,t)) ∗ η(M(y, z, s)) = M ′ (x, y,t) ∗ M ′ (y, z, s).

Remark 4.29. Obviously, Corollary 6.1 is a direct consequence of Corollary 4.8. From Lemma 4.3, adding the condition η(t) > 0, for all t > 0, we can easily deduce that Corollaries 6.1 and 4.8 are equivalent. In the following example, we exhibit a kind of archimedean t-norm ∗ such that a ∗m b > a ∗ b > ab. {2ab} Example 4.28. Let X = {1, 12 , 72 }, a ∗ 1 b = {1+a+b−ab} for all a, b ∈ [0, 1] and M : 2 X × X × (0, ∞) → [0, 1] defined for each t > 0 as:

1 1 2 2 M(1, 1,t) = M( , ,t) = M( , ,t) = 1; 2 2 7 7 2 1 2 2 1 1 2 M(1, ,t) = M( , 1,t) = M( , ,t) = M( , ,t) = ; 7 7 2 7 7 2 2 and

1 2 1 M(1, ,t) = M( , 1,t) = . 2 2 7 Then it is easy to check that (X, M, ∗) is a fuzzy metric space. Define the mapping 4 T : X → X by T (1) = 1, T ( 21 ) = 72 , T ( 27 ) = 27 and ϕ(1) = 1, ϕ( 21 ) = 19 , ϕ( 27 ) = 27 . Then it is easy to see that the map T satisfies all the conditions of Corollary 3.5 and consequently have fixed point u = 1 and u = 72 . Note that the t-norm ∗ is an archimedean such that a ∗ 1 b > ab. 2

In the next theorem, we obtain another generalization of the Abassi Theorem 3.1 [1]. Theorem 4.29. Let (X, M, ∗) be a complete fuzzy metric space with ∗ is continuous, archimedean and T : X → X be a k- continuous self-mapping satisfying the condition M(x, T x,t) ∗ ϕ(T x) ≥ ϕ(x)

(4.59)

for all x ∈ X and t > t0 , for some t0 > 0, where ϕ : X → [0, 1] be such that ϕ is non-trivial (i.e. x ∈ X such that ϕ(x) ̸= 0). Then T has a fixed point in X

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Proof. Considering η to be an identity map in the proof of Theorem 3.3, with similar arguments, it follows that {xn } is Cauchy. Since X is complete, there exists a point u ∈ X, such that limn→∞ (xn ) = u and limn→∞ (T p xn ) = u for each p ≥ 1. Then kcontinuity of T implies that limn→∞ (T k xn ) → Tu. Hence, Tu = u as limn→∞ (T k xn ) → u. Therefore, u is fixed point of T. The following theorem characterizes the completeness of an archimedean type fuzzy metric spaces. Theorem 4.30. Let (X, M, ∗) be a fuzzy metric space, with ∗ is continuous and archimedean. If every k- continuous self-mappings of X satisfying the hypothesis of Theorem 4.29 has a fixed point and the condition M(T x, T 2 x,t) > M(x, T x,t) =⇒ M(T x, T 2 x,t)2 ≥ M(x, T x,t)

(4.60)

for all x ̸= T x and t > 0 holds, then X is complete. Proof. Suppose that every k- continuous self-mappings of X satisfying hypothesis of Theorem 4.29 possesses a fixed point. We assert that X is complete. If possible, suppose that X is not complete. Then there exists a Cauchy sequence in X, say S = {u1 , u2 , u3 , . . .}, consisting of distinct points which does not converges. Let v ∈ X be given. Then, since v is not a limit point of the Cauchy sequence S, there exists a least positive integer N(v), such that v ̸= uN (v), and for each m ≥ N(v), and t > 0, we have M(v, uN (v),t) < M(uN (v), vm ,t) (4.61) Let us define a mapping T : X → X by T (v) = uN (v). Then T v ̸= v for each v and, using (9), for any v ∈ X and t > 0, we get M(T v, (T 2 v),t) = M(uN (v), uN (T v),t) > M(uN (v), vm ,t) = M(v, (T v),t)

(4.62)

then by Equation (4.60), we have M(T v, T 2 v,t)2 ≥ M(v, T v,t). Setting ϕ(v) = M(v, T v,t0 )2 , we have M(v, T v,t0 ) ∗ ϕ(T v) = M(v, T v,t0 ) ∗ M(T v, T 2 v,t0 )2 ≥ M(v, T v,t0 ) ∗ M(v, T v,t0 ) = ϕ(v).

(4.63)

M(v, T v,t) ∗ ϕ(T v) ≥ M(v, T v,t0 ) ∗ ϕ(T v) ≥ ϕ(v),

(4.64)

Moreover,

for all t ≥ t0 . So, the mapping T satisfies the contractive condition of the Theorem 4.29. Moreover, T is a fixed point free mapping, whose range is contained in the non-convergent Cauchy sequence S = {un }n∈N . Hence, there exists no sequence {xn }n∈N in X for which {T xn }n∈N converges, that is, there exists no sequence {xn }n∈N in X for which the condition (T xn ) → z =⇒ (T 2 xn ) → T z is violated. Therefore, T is 2- continuous mapping. Thus, we have a self-mapping T of X which satisfies all the conditions of Theorem 4.29, but does not possess a fixed point. This contradicts the hypothesis of the theorem. Hence, X is complete.

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The following example illustrates the aforementioned theorem. Example 4.29. Let X = (0, 1], a ∗ b = ab for all a, b ∈ [0, 1] and M(x, y,t) = min{x,y} for all x, y ∈ X and for all t > 0. Clearly, (X, M, ∗) is an M-complete fuzzy max{x,y} √ metric space. Define the mapping T : X → X by T x = x for all x ∈ X. If ϕ is defined by   x if x ∈ (0, 14 ], ϕ(x) =  1 if x ∈ ( 14 , 1] Then T satisfies all the conditions of Theorem 3.11 and consequently has a fixed point u = 1. Note that the mapping ϕ is not upper semi-continuous at x = 14 . The following theorem is another generalization of Abassi Theorem 3.1 [1]. Theorem 4.31. Let (X, M, ∗) be a fuzzy metric space with ∗ continuous and archimedean, T, S : X → X mappings. Suppose there exists a mapping ϕ : X → [0, 1] such that a. M(Sx, T x,t) ∗ ϕ(T x) ≥ ϕ(Sx), for all t ≥ 0 and x ∈ X. b. M(T x, Ty,t)2 > min{M(Sx, Sy,t)2 , M(Sx, Ty,t) ∗ M(T x, Sy,t)}, for all x ̸= y and t ≥ 0. c. T (X) ⊂ S(X). d. T (X) or S(X) are complete. then Then T and S have a common point in X. Proof. Let x1 = x, and choose xn such that Sxn = T xn−1 . Assume without loss of generality that Sxn ̸= Sxn for all n. In other case, T xn = Sxn . By a), ϕ(Sxn+1 ) ≥ M(Sxn , Sxn+1 ,t) ∗ ϕ(Sxn+1 ) ≥ ϕ(Sxn ). So {ϕ(Sxn )} is an increasing sequence and in a consequence it converges, for all t ≥ 0. Following the same argument in Theorem 6.2, we can conclude that {Sxn } is a Cauchy sequence. Since S(X) is complete, {Sxn } converges to Su = v ∈ X. Also, {T xn } converges to Su = v. We will prove that Su = Tu. Suppose that Su ̸= Tu, then using b) M(T xn , Tu,t)2 > min{M(Sxn , Su,t)2 , M(Sxn , Tu,t) ∗ M(T xn , Su,t)} Letting n → ∞, M(Su, Tu,t)2 ≥ M(Su, Tu,t). This is a contradiction. Hence, Su = Tu. Two maps T, S : X → X are called weakly compatible if they commute at coincidence points, i.e., T Su = STu, for all u such that Tu = Su.

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Theorem 4.32. Under the same conditions of Theorem 4.31. If T and S are weakly compatible mappings, then T and S have a unique common fixed point. Proof. Let v = Tu = Su. Since T and S are weakly compatible, Sv = STu = T Su = T v, i.e., v is another common point of T and S. Suppose that T v ̸= v, by b) M(T v, v,t)2

= M(T 2 u, Tu,t)2 > min{M(STu, Su,t)2 , M(STu, Tu,t) ∗M(T 2 u, Su,t)} . = min{M(STu, Su,t)2 , M(Sv, v,t)2 } = M(T v, v,t)2

This is a contradiction. Hence Sv = T v = v. Uniqueness of the common fixed point follows easily. Remark 4.30. It is to be noted that the associated t- norm in the definition of GV fuzzy metric space only needs to be continuous, and hence, the following problem remains open: Question: Does Theorem 4.28 remain valid if ∗ is any continuous t-norm ?

4.11

FUZZY MEIR-KEELER CONTRACTIVE MAPPINGS AND FIXED POINT

In 2019, Zheng and Wang [46] introduce the concept of Fuzzy Meir-Keeler contractive mappings and established corresponding fixed point results. Denote ω = {δ : (0, 1] → (0, 1] : δ is right continuous}, Definition 4.29. Let (X, M, ∗) be a fuzzy metric space. A mapping f : X → X is said to be a fuzzy Meir-Keeler contractive mapping with respect to δ ∈ ω if the following condition holds: ∀ε ∈ (0, 1), ε − δ (ε) < M(x, y,t) ≤ ε =⇒ M( f x, f y,t) > ε; for all x, y ∈ X,t > 0. Theorem 4.33. Let (X, M, ∗) be an M-complete fuzzy metric space and f : X → X be a fuzzy Meir-Keeler contractive mapping with respect to δ ∈ ω. Then f has a fixed V point if and only if there exists x0 ∈ X such that t>0 M(x0 , f (x0 ),t) > 0 for all t > 0. Remark 4.31. Every ψ-contractive mapping (and hence H -contractive mapping) is a fuzzy Meir-Keeler contractive mapping with respect to some δ ∈ ω

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The following example illustrates that fuzzy Meir-Keeler contractive mapping is more general than fuzzy ψ-contractive mapping (and hence H -contractive mapping). Example 4.30. Let X = [0, 1] ∪ {3, 4, . . . ., 3n, 3n + 1} with the Euclidean metric, and let f : X → X be defined as:  x if x ∈ [0, 1],  2,     0, if x = 3n f (x) =      1 1 − n+2 , if x = 3n + 1. 1 , then (X, M, .) is a complete stationary fuzzy metric with . 1 + d(x, y) is the product t-norm. It is not difficult to see that f is not a fuzzy ψ-contractive mapping. However, f is a fuzzy Meir-Keeler contractive mapping with respect to (2, 32 , δ ). Moreover, f satisfies all the conditions of Theorem 4.30 and admits a unique fixed point x = 0. Let M(x, y) =

4.12

CONCLUSIONS

The study of fixed points of mappings satisfying certain contraction conditions has many applications and has been at the centre of various research activities. Fuzzy fixed point theory is a fuzzy extension of fixed point theory. The notion of fuzzy metric was originally introduced by Kramosil and Mich´alek [25] and later was modified by George and Veeramani [15] in order to obtain a Hausdorff topology. An important and interesting topic in fuzzy metric spaces is the fixed point theory. A lot of fixed point theorems were obtained by introducing various fuzzy contractive mappings [14,16,17,20,26,44]. But due to the complexity exhibited in fuzzy metric spaces, researchers need to add various conditions to obtain fixed point theorems in fuzzy metric spaces (see[17,23,44,46]). Recently, many authors (see for examples [3–5,7,11,19,21,31,37] and referenced mentioned their in) observed that the various contraction mappings in metric spaces may be exactly translated into probabilistic or fuzzy metric spaces endowed with special t−norms, such as minimum t−norm. Starting with the famous Banach contraction principle, a huge number of mathematicians started to formulate better contractive condition for which fixed point exists. In this chapter, we have identified some of the first but no less important contraction conditions that have been formulated by well-known mathematicians, Grabiec, Gregori-Sapena, Vasuki and Veeramani, Tirado, Mihet, Wardowski, Gopal, Abbasi, and Zheng and made them in the framework of the fuzzy metric space. It is our hope that the material presented in this chapter will be enough to stimulate scientists and students to further investigate this challenging field.

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21. Gregori V., Almanzor S., Remarks to “on strong intuitionistic fuzzy metrics”. Journal of Nonlinear Sciences and Applications , 2018;11(2):316–322. 22. Gregori V., Minana J. J., Remarks on fuzzy contractive mappings. Fuzzy Sets and Systems, 2014;251:101–103. 23. Gregori V., Min˜nana J.-J., On fuzzy ψ-contractive mappings. Fuzzy Sets and Systems, 2016;300:93–101. 24. Klement E. P., Mesiar R., Pap E., Triangular Norms. Kluwer Academic Publishers, Trends in Logic 8, Dordrecht (2000). 25. Kramosil I., Michalek J., Fuzzy metric and statistical metric spaces. Kybernetica, 1975;11:336–344. 26. Mihet D., A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets and Systems, 2004;144:431–439. 27. Mihet D., On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 2007;158:915–921. 28. Mihet D., Fuzzy ψ-contractive mappings in non-archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 2008;159:739–744. 29. Mihet D., A class of contractions in fuzzy metric spaces. Fuzzy Sets and Systems, 2010;161:1131–1137. 30. Mihet D., A note on fuzzy contractive mappings in fuzzy metric. Fuzzy Sets and Systems, 2014;251:83–91. 31. O’Regan D., Abbas, M., Necessary and sufficient conditions for common fixed point theorems in fuzzy metric space. Demonstratio Mathematica, 2009;42(4):887–900. 32. Samet B., Vetro C., Vetro P., Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Analysis, 2012;75:2154–2165. 33. Shukla S., Abbas M., Fixed point results in fuzzy metric-like spaces. Iranian Journal of Fuzzy Systems, 2015;11(5):81–92. 34. Shukla S., Gopal D., De-Hierro A.-F. R.-L., Some fixed point theorems in 1-M complete fuzzy metric-like spaces. International Journal of General Systems, 2016;3:1–15. 35. Shukla S., Gopal D., Sintunavrat W., A new class of fuzzy contractive mappings and fixed points. Fuzzy Sets and Systems, 2018;350:85–95. 36. Subrahmanyam P.V., A common fixed point theorem in a fuzzy metric space. Information Sciences, 1995;83:109–112. 37. Sumalai P., Kumam P., Gopal D., Hasan M., Common fixed point theorems in fuzzy metric-like spaces employing common property (EA). Mathematical Methods in the Applied Sciences, 2019;42(17):5834–5844. 38. Suzuki T., A generalized Banach contraction principle which characterizes metric completeness. Proceedings of the American Mathematical Society, 2008;136(5):1861–1869. 39. Tirado P., On compactness and G-completeness in fuzzy metric spaces. Iranian Journal of Fuzzy Systems, 2012;9(4):151–158. 40. Tirado P., Contraction mappings in fuzzy quasi-metric spaces and [0, 1]-fuzzy posets. Fixed Point Theory, 2012;13(1):273–283. 41. Vasuki R., A common fixed point theorem in a fuzzy metric space. Fuzzy Sets and Systems, 1998;97(3):395–397. 42. Vetro C., Fixed points in weak non-archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 2011;162(1):84–90. 43. Wang S., Answers to some open questions on fuzzy ψ-contractions in fuzzy metric spaces. Fuzzy Sets and Systems, 2013;222:115–119.

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Fixed-Point 5 Common Theorems in Fuzzy Metric Spaces 5.1

INTRODUCTION AND PRELIMINARIES

Fixed point theory and its applications are very well-known concepts in functional analysis; it was initiated by S. Banach in 1922. After five decades, I. Kramosil and J. Michalek [25] introduced fuzzy metric space by using fuzzy sets, which were introduced by L. Zadeh [48] and proved some fixed point results. After almost 20 years, A. George and P. Veeramani [11] modified the definition of fuzzy metric space given by Kramosil and Michalek [25]; they also defined topology on this new fuzzy metric space and proved this metric space is Housdorff. On this definition given by George and Veeramani [11], lot of manuscripts were published in various directions to prove fixed points as well as common fixed points. Mishra et al. [31] (in the mid 1990s) and R. Vasuki [43] (in the late 1990s) proved some common fixed point results on fuzzy metric space defined by Kramosil and Michalek [25]. In the 21st century, lot of researchers proved and generalized common fixed point theorems using modified concept of fuzzy metric spaces given by George and Veeramani [11], for example S. H. Cho [6], M. Imdad and Javid Ali [19], Abbas et al. [2], Gopal et al. [15], D. Gopal and M. Imdad [14], Rao et al. [35], Javid et al. [20], Cho et al. [6] and etc. Definition 5.1. A point x ∈ X is said to be common fixed point for the pair of selfmappings ( f , g) on X is such that x = f x = gx.

5.2

SOME IMPORTANT RESULTS

In 1994, Mishra et al. [31] obtained some results on common fixed point for asymptotic maps on fuzzy metric space, using the definition of Kramosil and Michalek [25]. They gave some basic definitions and results for proving their main results. Lemma 5.1. [13] Let Yn be a sequence in a fuzzy metric space X. If there exists a positive number k < 1 such that M(Yn+2 ,Yn+l , kt) ≥ M(yn+l ,Yn ,t), t > O, n ∈ N, then Yn is a Cauchy sequence.

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Mishra et al. [31] used the following lemma. Lemma 5.2. If for two x, y ∈ X and for a positive k < 1, M(x, y, kt) ≥ M(x, y,t), then x = y. Definition 5.2. [31] Self-maps f and g of a fuzzy metric space X will be called z-asymptotically commuting (or simply asymptotically commuting) if for all t > 0 lim M( f gxn , g f xn ,t) = 1

n→∞

whenever xn is a sequence in X such that lim f xn = lim gxn = z for some z ∈ X.

n→∞

n→∞

Following Jungck’s nomenclature [23,24], asymptotically commuting maps may also be called compatible maps. Such maps are more general than commuting and weakly commuting maps [37] (see also [23]) both. Lemma 5.3. [31] If f and g are asymptotically commuting maps on a fuzzy metric space X with t ∗ t ≥ t for all t ∈ [0, 1] and f wn , gwn → z for some z in X, ({wn } being a sequence in X) then f gwn → gz provided g is continuous (at z). Mishra et al. [31] proved an equivalent formulation of the above lemma on a metric space, referring Jungck [23] Theorem 5.1. [31] Let (X, M, ∗) be a complete fuzzy metric space with t ∗t ≥ t ,t ∈ [0, 1], and f , g : X → X. If there exist continuous maps S, T : X → X and a constant k ∈ (0, 1) such that ST = T S

(5.1)

( f , S) and (g, T ) are asymptotically commuting pairs,

(5.2)

f T (X) ∪ gS(X) ⊂ ST (X), and M( f x, gy, kt) ≥ M(Sx, Ty,t), M ∗ ( f x, Sx,t), M(gy, Ty,t) ∗ M( f x, Ty, at) ∗ M(gy, Sx, (2 − α)t)

(5.3) (5.4)

for all x, y in X, t > 0 and α ∈ (0, 2); then f , g, S and T have a unique common fixed point. Proof. Pick x0 ∈ X. Construct a sequence {xn } as follows: f T x2n = ST x2n+1 , gSx2n+1 = ST x2n+2 , n = 0, 1, 2, . . .

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We can do this since (5.3) holds. Indeed such a sequence was first introduced in Ref. [41]. Let zn = ST xn . Then, for α = 1 − q, q(0, 1), by Equation (5.4), M(z2n+l , z2n+2 , kt) = M( f T x2n , gSx2n+1 , kt) ≥ M(z2n , z2n+1 ,t) ∗ M(z2n+l , z2n ,t) ∗ M(z2n+2 , z2n+1 ,t) ∗ M(z2n+l , z2n+1 , (1 − q)t) ∗ M(z2n+2 , z2n , (1 + q)t) ≥ M(z2n , z2n+1 ,t) ∗ M(z2n+l , z2n+2 ,t) ∗ M(z2n , z2n+1 , qt) Since the norm ∗ is continuous and M(x, y, .∗) is left-continuous, making q → 1 gives M(z2n+1 , z2n+2 , kt) ≥ M(z2n , z2n+1 ,t) ∗ M(z2n+1 , z2n+2 ,t) Similarly, taking x = T x2n+2 , y = Sx2n+1 , α = 1 + q′ , q′ (0, 1) in (4), simplifying and making q′ → 1, we have M(z2n+2 , z2n+3 , kt) ≥ M(z2n+1 , z2n+2 ,t) ∗ M(z2n+2 , z2n+3 ,t). So in general M(zm+1 , zm+2 , kt) ≥ M(zm , zm+1 ,t) ∗ M(zm+1 , zm+2 ,t) Consequently, M(zm+1 , zm+2 , kt) ≥ M(zm , zm+1 ,t) ∗ M(zm+1 , zm+2 ,tk−p ), m, p ∈ N. Since M(zm+1 , zm+2 ,tk−p ) → 1 as p → ∞, we have M(zm+1 , zm+2 , kt) ≥ M(zm , zm+1 ,t), m ∈ N. By Lemma 5.1, {zn } is a Cauchy sequence, and has a limit in X. Call it z. { f T z2n } and {gSz2n+1 } being the subsequences of {ST zn } also converge to z. Let Yn = T xn and wn = Sxn , n ∈ N. Then f y2n → z , Sy2n → z, Tw2n+l → z and Qw2n+l → z. So, for t > 0, by Equation (5.2), M( f Sy2n , S f y2n ,t) → 1 and M(gTw2n+1 , T gw2n+1 ,t) → 1. Moreover, by the continuity of T and Lemma 5.3, T Tw2n+1 → T z, T gw2n+1 → T z and gTw2n+1 → T z. By Equation (5.4) with α = 1, M( f y2n , gTw2n+1 , kt) ≥ M(Sy2n , T Tw2n+1 ,t) ∗ M( f y2n , Sy2n ,t)) ∗ M(gTw2n+l , T Tw2n+I ,t) ∗ M( f y2n , T Tw2n+1 ,t) ∗ M(gTw2n+1 , Sy2n ,t). This yields M(z, T z, kt) ≥ M(z, T z,t) ∗ 1 ∗ 1 ∗ M(z, T z,t) ∗ M(T z, z,t).

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So z = T z. Similarly z = Sz. Again by (4) with α = 1, M( f y2n , gz, kt) ≥ M(Sy2n , T z,t) ∗ M( f y2n , Sy2n ,t) ∗ M(gz, T z,t) ∗M( f y2n , T z,t), M(gz, Sy2n ,t) yields

M(z, gz, kt) ≥ 1 ∗ 1 ∗ M(gz, z,t) ∗ 1 ∗ M(gz, z,t).

So gz = z. Similarly f z = z. The uniqueness of z as the common fixed point of f , g, S and T follows easily from (4). Remark 5.1. [31] If ST and f , g, then Equations (5.1)–(5.3) say that f and S are asymptotically commuting and f (X) ⊂ S(X). In such a situation, the sequence {ST xn } constructed in the proof of Theorem reduces to { f xn , = Sxn+1 }’ and such a sequence was first introduced by Jungck [23]. Remark 5.2. [31] If S, T then Equation (5.1)–(5.3) say that S is asymptotically commuting with each of f , g, and f (X) ∪ g(X) ⊂ S(X). In this situation, the sequence {ST xn } is replaced by {Sxn } wherein f x2n = Sx2n+1 and gx2n+1 = Sx2n+2 , which was first introduced by Singh [38]. Theorem 5.1 with S = T the identity map is: Corollary 5.1. [31] Let (X, M, ∗) be a complete fuzzy metric space with t ∗t ≥ t, t ∈ [0, 1], and f , g : X → X. If there exists a constant k ∈ (0, 1) such that M( f x, gy, kt) ≥

(5.5)

M(x, y,t) ∗ M(x, f x,t) ∗ M(y, gy,t) ∗ M(y, f x,t) ∗ M(x, gy, (2 − α)t) for all x, y ∈ X, t > 0 and α ∈ (0, 2), then f and g have a unique common fixed point. In view of Remark 5.1, [31] furnished the following result. Corollary 5.2. Let (X, M, ∗) be a complete fuzzy metric space with t ∗ t ≥ t, t ∈ [0, 1], and f , S asymptotically commuting maps on X such that f (X) ⊂ S(X). If S is continuous and there exists a constant k ∈ (0, 1) such that M( f x, f y, kt) ≥

(5.6)

M(Sx, Sy,t) ∗ M(Sx, f x,t) ∗ M(Sy, f y,t) ∗ M(Sy, f x,t) ∗ M(Sx, f y, (2 − α)t) for all x, y ∈ X, t > 0 and α ∈ (0, 2), then f and S have a unique common fixed point. Corollary 5.3. (Grabiec’s fuzzy nanach contraction theorem). Let X be a complete fuzzy metric space with t ∗ t ≥ t, t ∈ [0, 1], and f : X → X such that M( f x, f y, kt) ≥ M(x, y,t) for all x, y ∈ X; 0 < k < 1. Then f has a unique fixed point.

(5.7)

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Proof. It follows from corollary (5.2) with f = g includes (5.7). However, Grabiec [13] does not require ”t ∗ t ≥ t” in his proof. The following metric version of the condition (5.5) is perhaps enough to elaborate this remark. For self-maps f and g of a metric space (Y, d) and 0 < k < 1, the metric version of Equation (5.5) is: d( f x, gy) < k max{d(x, y), d(x, f x), d(y, gy), [d(y, f x) + d(x, gy)]/2} for all x, y ∈ Y . Mishra et al. [31] also proved the following result using Corollaries 5.1 and 5.3. Theorem 5.2. Let (X, M, ∗) be a complete fuzzy metric space with t ∗t >≥ ,t ∈ [0, 1], and f , g two maps on the product X × X with values in X. If there exists a constant k ∈ (0, 1) such that M( f (x, y), g(u, v), kt) ≥ M( f (x, y), x,t) ∗ M(g(u, v), u,t) ∗ M(x, u,t) ∗ M(y, v,t) ∗M( f (x, y), u,t) ∗ M(g(u, v), x, (2 − α)t)

(5.8)

for all x, y, u, v ∈ X, t > 0 and α ∈ (0, 2), then there exists exactly one point w ∈ X such that f (w, w) = w = g(w, w). Remark 5.3. [31] If x, y ∈ X are such that x = f (x, y) and y = g(x, y) then it can be seen using (5.8) that x = y. At the end of 20th century, R. Vasuki [43] (1998) proved some results and generalized well-known results of Grabiec [13]; he also defined some new definitions which are given below: Definition 5.3. [13] A sequence {xn }, in a fuzzy metric space is said to be a Cauchy sequence if and only if lim M(xn+p , xn ,t) = 1, p > 0, t > 0. A sequence {xn }, in n→∞

a fuzzy metric space is converging to x ∈ X if lim M(x, , x,t) = 1 for each t > 0. A n→∞ fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent. Lemma 5.4. [13] M(x, y, ∗) is non-decreasing for all x, y ∈ X. The main result of Vasuki [43] is given below: Theorem 5.3. Let {Tn }n , be a sequence of mappings of a complete fuzzy metric space (X, M, ∗) into itself where ∗ is a continuous t-norm such that for any two mappings Ti , T j we have M(Tim x, T jm y, αi, j ,t) ≥ M(x, y,t) for some m and 0 < αi, j < k < 1, i, j = 1, 2, . . . , x, y ∈ X. Then the sequence {Tn }n has a unique common fixed point.

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Vasuki [43] also furnished an example in the support of his result, which shows this is genuine generalization of Grebiec [13] result for fuzzy metric space in sense of Kramosil and Michalek [25], Example 5.1. Let X = {1/n, n ∈ N} ∪ {0}. Define the sequence {Tn }n of mappings from X to X by Tn (x) = 21 x for all n ∈ N. Then under the fuzzy metric M(x, y,t) =

t , t + d(x, y)

for m = 2, αi, j = 14 . {Tn }n satisfies all the conditions of Theorem 5.3. Corollary 5.4. [13] Let (X, M, ∗) be a complete fuzzy metric space where ∗ is a continuous t-norm and T be a self-map of X such that M(T x, Ty, t) ≥ M(x, y,t) where 0 < α < 1 , x, y, z in X and all t > 0. Then T has a unique fixed point in X. Bijandra Singh and MS Chauhan [39] introduced the concept of compatibility in fuzzy metric space and prove two common fixed point theorems illustrating with an example. They used the newly introduced definition of fuzzy metric spaces by Goarge and Veeramani [47] . Definition 5.4. Self-mappings f and g of a fuzzy metric space (X, M, ∗) are said to be compatible if and only if M( f gxn , g f xn ,t) → 1, for all t > 0, whenever fixing is a sequence in X such that f xn , gxn → y for some y in X. The main result of Singh and Chauhan [39] is as follows: Theorem 5.4. Let f , g, S and T be self-maps of complete fuzzy metric space (X, M, ∗) with continuous t-norm defined by a ∗ b = min{a, b}, a, b ∈ [0, 1], satisfying the following conditions: I. II. III. IV.

f (X) ⊂ T (X), g(X) ⊂ S(X), S and T are continouos, ( f , S) and (g, T ) are compatible pairs of maps, for all x, y in X, k ∈ (0, 1), t > 0 M( f x, gy, kt) ≥ min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t), M(gy, Sx, 2t), M( f x, Ty,t)}

V. For all x, y ∈ X, M(x, y,t) → 1, as t → 1. Then f , g, S and T have a unique common fixed point in X. Theorem 5.5. Let f , g, S and T be self-maps of complete fuzzy metric space (X, M, ∗) with continuous t-norm defined by a ∗ b = min{a, b}, a, b ∈ [0, 1], satisfying conditions (II) and (V) of the aforementioned theorem and

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I. f S = S f , gT = T g II. f a (X) ⊂ T t (X), gb (X) ⊂ Ss (X), III. for all x, y in X, a, b, s,t ∈ N M( f a x, gb y, kt) ≥ min{M(Ss x, T t y,t), M( f a x, Ss x,t), M(gb y, T t y,t), M(gb y, Ss x, 2t), M( f a x, T t y,t)} Then f , g, S and T have a unique common fixed point in X. Singh and Chaohan [39] furnished an illustrative example in the support of their results. Example 5.2. Let (X, M, ∗) be a fuzzy metric space with X = [0; 1], t-norm ∗ defined by a ∗ b = min(a, b), a, b ∈ [0, 1] and M is the fuzzy set on X 2 × (0, ∞), defined by M(x, y,t) = [exp(|x − y|/t]−1 , for all x, y ∈ X, t > 0. Let us define self-maps f , g, S and T of X such that f x = x/16, T x = x/2, gx = x/8, Sx = x/4. Then for k ∈ [1/4, 1). In, 2006, Cho, [6] proved some common fixed point results using compatible maps in complete fuzzy metric space. Definition 5.5. Self-mappings f and g of a fuzzy metric space (X, M, ∗) is said to be compatible if lim M( f gxn , g f xn ,t) = 1 for all t > 0, whenever {xn } is a sequence nß∞

in X such that lim f xn = lim gxn = z for some z ∈ X. nß∞

nß∞

From now on, let (X, M, ∗) be a fuzzy metric space such that lim M(x, y,t) = 1 for all x, y ∈ X and s ∗ s ≥ s for all s ∈ [0, 1].

t→∞

Lemma 5.5. [6] Let (X, M, ∗) be a fuzzy metric space. Then for all x, y ∈ X, M(x, y, .) is nondecreasing. Lemma 5.6. [6] Let (X, M, ∗) be a fuzzy metric space. If there exists q ∈ (0, 1) such that M(x, y, qt) ≥ M(x, y,t) for all x, y ∈ X and t > 0, then x = y. Lemma 5.7. [6] Let (X, M, ∗) be a fuzzy metric space and let A and S be continuous self-mappings of X and ( f , S) be compatible. Let {xn } be a sequence in X such that f xn → z and Sxn → z. Then f Sxn → Sz. The main result of Cho [6] given as follows: Theorem 5.6. Let (X, M, ∗) be a fuzzy metric space and let f , g, S and T be selfmaps of complete fuzzy metric space (X, M, ∗) such that the following conditions are satisfied:

Common Fixed-Point Theorems in Fuzzy Metric Spaces

I. II. III. IV.

131

f (X) ⊂ T (X), g(X) ⊂ S(X), S and T are continouos, ( f , S) and (g, T ) are compatible pairs of maps, there exists q ∈ (0, 1) such that for every x, y in X, t > 0 M( f x, gy, qt) ≥ M(Sx, Ty,t) ∗ M( f x, Sx,t) ∗ M(gy, Ty,t) ∗ M(gy, Sx, 2t) ∗ M( f x, Ty,t)

Then f , g, S, and T have a unique common fixed point in X. Corollary 5.5. [6] Let (X, M, ∗) be a complete fuzzy metric space and let f , g, S and T be self-maps of complete fuzzy metric space (X, M, ∗) satisfying (I)-(III) of the aforementioned theorem and there exists q ∈ (0, 1) such that for every x, y in X, t > 0 M( f x, gy, qt) ≥ M(Sx, Ty,t) ∗ M( f x, Sx,t) ∗ M(gy, Ty,t) ∗ M(gy, Sx, 2t) ∗ M( f x, Ty,t) Then f , g, S and T have a unique common fixed point in X. Corollary 5.6. [6] Let (X, M, ∗) be a complete fuzzy metric space and let f , g, S and T be self-maps of complete fuzzy metric space (X, M, ∗) satisfying (I)-(III) of the aforementioned theorem and there exists q ∈ (0, 1) such that for every x, y in X, t > 0 M( f x, gy, qt) ≥ M(Sx, Ty,t) Then f , g, S and T have a unique common fixed point in X. Corollary 5.7. [6] Let (X, M, ∗) be a complete fuzzy metric space and let f , g, S and T be self-maps of complete fuzzy metric space (X, M, ∗) satisfying (I)-(III) of the aforementioned theorem and there exists q ∈ (0, 1) such that for every x, y in X, t > 0 M( f x, gy, qt) ≥ M(Sx, Ty,t) ∗ M(Sx, f x,t) ∗ M( f x, Ty,t) Then f , g, S and T have a unique common fixed point in X. Theorem 5.7. Let (X, M, ∗) be a complete fuzzy metric space. Then continuous selfmappings S and T of X have a common fixed point in X if and only if there exists a self-mapping f of X such that the following conditions are satisfied: I. f (X) ⊂ T (X), g(X) ⊂ S(X), II. ( f , S) and (g, T ) are compatible pairs of maps, III. there exists q ∈ (0, 1) such that for every x, y in X, t > 0 M( f x, gy, qt) ≥ M(Sx, Ty,t) ∗ M( f x, Sx,t) ∗ M( f y, Ty,t) ∗ M( f x, Ty,t). In fact f , S and T have a unique common fixed point in X.

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Cho [6] also furnished an illustrative example, which is given below: Example 5.3. Let X = [0, 1] and a ∗ b = min{a, b}. Let M be the standard fuzzy metric induced by d, where d(x, y) = |x − y| for x, y ∈ X. Then (X, M, ∗) is a complete x fuzzy metric space. Let f x = 16 , T x = 2x , gx = 8x and Sx = 4x . Then the conditions (I) and (II) of Theorem 5.6 are satisfied, and also for q = 1/2, condition (III) of Theorem 5.6 is satisfied and zero is the unique common fixed point of f , g, S and T . In 2006, M. Imdad and J. Ali [18] proved some common fixed point theorems in complete fuzzy metric spaces these results generalize earlier results. They also introduced the concept of R-weak commutativity of type (P) in fuzzy metric spaces. Some related results and illustrative examples are also discussed. The purpose of these results is to improve the main theorem of Vasuki [43] besides adopting R-weak commutativity of type (A f ), type (Ag) to fuzzy setting and to introduce R-weak commutativity of type (P) which are to be used to prove their results given below. Their main improvements are as follows: i. to relax the continuity requirement of maps completely, ii. to minimize the commutativity requirement of the maps to the point of coincidence, iii. to weaken the completeness requirement of the space to four alternative conditions, iv. to employ a more general contraction condition in proving our results. Definition 5.6. [18] A pair of self-mappings ( f , g) of a fuzzy metric space (X, M, ∗) is said to be i. weakly commuting, if M( f gx, g f x,t) ≥ M( f x, gx,t), ii. R-weakly commuting, if there exists some R > 0 such that M( f gx, g f x,t) ≥ M( f x, gx,t/R), iii. R-weakly commuting mappings of type (A f ) if there exists some R > 0 such that M( f gx, ggx,t) ≥ M( f x, gx,t/R), iv. R-weakly commuting mappings of type (Ag) if there exists some R > 0 such that M(g f x, f f x,t) ≥ M( f x, gx,t/R), v. R-weakly commuting mappings of type (P) if there exists some R > 0 such that M( f f x, ggx,t) ≥ M( f x, gx,t/R), for all x ∈ X and t > 0. Example 5.4. [18] X = R, the set of real numbers. Define a ∗ b = ab and ( |x−y| (e t )−1 , for all x, y ∈ X and t > 0, M(x, y,t) = 0, for all x, y ∈ X and t = 0, Then it is well known [43] that (X, M, ∗) is a fuzzy metric space. Define f x = 2x − 1 and gx = x2 Then by a straightforward calculation, one can show that M( f gx, g f x,t) = (e

2|x−1|2 t

)−1 = M( f x, gx,t/2)

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which shows that the pair ( f , g) is R-weakly commuting for R = 2. Note that the pair ( f , g) is not weakly commuting due to a strict increasing property of the exponential function. However, various kinds of the aforementioned “R-weak commutativity” notions are independent of one another and none implies the other. The earlier example can be utilized to demonstrate this inter-independence. To demonstrate the independence of ‘R-weak commutativity’ with “R-weak commutativity” of type (A f ) notice that M( f gx, ggx,t) = (e < (e

2|x4 −2x2 +1|2 t R|x−1|2 t

)−1 = (e

R|x−1|2 t

(x + 1)2 −1 ) R

)−1 = M( f x, gx,t/R, when x > 1

which shows that ‘R-weak commutativity’ does not imply ‘R-weak commutativity’of type (A f ). Second, in order to demonstrate the independence of ‘R-weak commutativity’ with ‘R-weak commutativity’ of type (P) note that  |x4 −4x+3|   R(x−1)2 | |x2 −2x+3|  −1 = e −1 t R M( f f x, ggx,t) = e t  R(x−1)2 |  −1 = M( f x, gx,t/R) for x > 1 < e t Finally, for a change, the pair ( f , g) is R-weakly commuting of type (Ag) as  |(2x−1)4 −4x+3|   4|x−1|2  −1 = e t −1 t M(g f x, f f x,t) = e = M( f x, gx,t/4) which shows that ( f , g) is R-weakly commutingof type (Ag) for R = 4. This situation may also be utilized to interpret that an R-weakly commuting pair of type (Ag) need not be R-weakly commutingpa ir of type (A f ) or type (P). It is not difficult to find examples to establish the independence of one of these definitions from the others which shows that there exist situations to suit a definition but not the others. Now, let (X, M, ∗) be a complete fuzzy metric space and let f , g, S and T be selfmappings of X satisfying the following conditions: f (X) ⊂ T (X) and g(X) ⊂ S(X),

(5.8)

M( f x, gy,t) ≥ φ (min{M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t)})

(5.9)

for all x, y ∈ X, where φ : [0, 1] → [0, 1] is a continuous function such that φ (s) > s for each 0 < s < 1. Then for any arbitrary point x0 ∈ X, by Equation (5.8), we choose a point x1 ∈ X such that f x0 = T x1 and for this point x1 , there exists a point x2 ∈ X such that Sx2 = gx1 and so on. Continuingin this way, we can construct a sequence {yn } in X such that y2n = T x2n+1 = f x2n , y2n+1 = Sx2n+2 = gx2n+1 for n = 0, 1, 2, . . .

(5.10)

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First, Imdad and Ali [18] proved the following lemma Lemma 5.8. Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗) satisfying the conditions (5.8) and (5.9). Then the sequence {yn } defined by Equation (5.10) is a Cauchy sequence in X. The main result of Imdad and Ali [18] Theorem 5.8. Let f , g, S and T be four self-mappings of a fuzzy metric space (X, M, ∗) satisfying the condition M( f x, gy,t) ≥ φ (min{M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t)}) for all x, y ∈ X and t > 0 where φ : [0, 1] → [0, 1] is a continuous function with φ (s) > s whenever 0 < s < 1. If f (X) ⊂ T (X) and g(X) ⊂ S(X) and one of f (X), g(X), S(X) and T (X) is a complete subspace of X, then i. f and S have a point of coincidence, ii. g and T have a point of coincidence. Moreover, if the pairs ( f , S) and (g, T ) are coincidentally commuting, then f , g, S and T have a unique common fixed point. Remark 5.4. [18] If we consider min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t)} = M(Sx, Ty,t), by setting f = g and S = T , one obtains a substantially improved version of [[43], Theorem 2] due to Vasuki as our result is proved under tight commutativity condition without any continuity requirement. Theorem 5.9. [18] Theorem 5.8 remains true if a ‘coincidentally commuting’ property is replaced by any one ( retaining the rest of the hypotheses ) of the following: i. ii. iii. iv. v.

R-weakly commuting property, R-weakly commuting property of type (A f ), R-weakly commuting property of type (Ag), R-weakly commuting property of type (P) weakly commuting property.

As an application of Theorem 5.8, we prove a common fixed point theorem for four finite families of mappings which runs as follows: Theorem 5.10. [18] Let { f1 , f2 , ..., fm }, {g1 , g2 , ..., gn }, {S1 , S2 , ..., S p } and {T1 , T2 , ..., Tq } be four finite families of self-mappings of a fuzzy metric space (X, M, ∗) such that f = f1 f2 ... fm , g = g1 g2 ...gn , S = S1 S2 ...S p and T = T1 T2 ...Tq satisfy condition (5.9) with f (X) ⊂ T (X) and g(X) ⊂ S(X). If one of f (X), g(X), S(X), or T (X) is a complete subspace of X, then

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i. f and S have a point of coincidence, ii. g and T have a point of coincidence. Moreover, if fi f j = f j fi , gk gl = gl gk , Sr Ss = Ss Sr , Tt Tu = Tu Tt , fi Sr = Sr fi and gk Tt = Tt Bk for all i, j ∈ I1 = {1, 2, ..., m}, k, l ∈ I2 = {1, 2, ..., n}, r, s ∈ I3 = {1, 2, ..., p} and t, u ∈ I4 = {1, 2, ..., q}, then (for all i ∈ I1 , k ∈ I2 , r ∈ I3 and t ∈ I4 ) fi , Sr , gk and Tt have a common fixed point. In 2009, Abbas et al. [2] proved common fixed point theorems for the class of four non-compatible mappings in fuzzy metric spaces These results are proved without exploiting the notion of continuity and without imposing any condition on t-norm. Definition 5.7. [2] Let f and g be self-maps on a fuzzy metric space (X, M, ∗). A pair ( f , g) is said to be: f. compatible of type (I) if for all t > 0, lim M( f gxn , x,t) ≥ M(gx, x,t)

n→∞

whenever {xn } is a sequence in X such that lim f xn = lim gxn = x, for some n→∞ n→∞ x ∈ X. g. compatible of type (II) if the pair (g, f ) is compatible of type (I). Definition 5.8. [2] Mappings f and g from a fuzzy metric space (X, M, ∗) into itself are weakly compatible if they commute at their coincidence point, that is f x = gx implies that f gx = g f x. It is known that a pair ( f , g) of compatible maps is weakly compatible but converse is not true in general. Definition 5.9. [2] Let f and g be self-maps on a fuzzy metric space (X, M, ∗). They are said to satisfy property (EA) if there exists a sequence {xn } in X such that lim f xn = lim gxn = x for some x ∈ X. n→1

n→1

Definition 5.10. [2] Let f , g, S and T be self-maps on a fuzzy metric space (X, M, ∗). They are said to satisfy common property (EA) if there exists a sequence {xn } in X such that lim f xn = lim Sxn = lim gxn = lim T xn = x for some x ∈ X. n→1

n→1

n→1

n→1

For more on (EA) and common (EA) properties, we refer to Refs. [1,28]. Note that compatible, noncompatible, compatible of type (I) and compatible of type (II) satisfy (EA) property but converse is not true in general. Example 5.5. [2] Let (X, M, ∗) be a fuzzy metric space, where X = [0, 2] with mint imum t−norm, and M(x, y,t) = t+d(x,y) for all t > 0 and for all x, y ∈ X. Define the self-maps f and g as follows: ( 2, when x ∈ [0, 1], fx = x 2 , when 1 < x ≤ 2,

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gx =

( 0, x+3 5 ,

when x = 1, otherwise.

Example 5.6. [2] Let (X, M, ∗) be a fuzzy metric space, where X = [0, 1] with mint imum t−norm, and M(x, y,t) = t+d(x,y) for all t > 0 and for all x, y ∈ X. Define the self-maps f and g as follows: ( fx =

1 2,

1

when 0 ≤ x < 1/2 or x = 1, when 12 ≤ x < 1.

Let a class of implicit relations be the set of all continuous functions φ : [0, 1] × [0, 1] → [0, 1] which are increasing in each coordinate and φ (t,t,t,t,t) > t for all t ∈ [0, 1). For examples of implicit relations, we refer to Ref. [7] and references therein. The following result [2] provides necessary conditions for the existence of common fixed point of four noncompatible maps in a fuzzy metric space. Theorem 5.11. Let (X, M, ∗) be a fuzzy metric space. Let f , g, S and T be maps from X into itself with f (X) ⊆ T (X) and g(X) ⊆ S(X) and there exists a constant k ∈ (0, 12 ) such that  M( f x, gy, kt) ≥ φ M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t),

 M( f x, Ty, αt), M(gy, Sx, (2 − α)t) , for all x, y ∈ X, α ∈ (0, 2), t > 0 and φ ∈ ψ. Then f , g, S and T have a unique common fixed point in X provided the pair ( f S) or (g, T ) satisfies (EA) property, one of f (X), T (X), g(X), S(X) is a closed subset of X and the pairs (g, T ) and ( f , S) are weakly compatible. Abbas et al. [2] gave an example to illustrate the fact that the aforementioned theorem is applicable to a larger class of mappings than those given in Ref. [7] as we do not require the assumptions of continuity of mappings and restriction on t-norm as t ∗ t = t. Example 5.7. [2] Let X = [2, 5000] and a ∗ b = ab (product t-norm. Let M be the standard fuzzy metric induced by d, where d(x, y) = |x − y| for x, y ∈ X. Then (X, M, ∗) is a complete fuzzy metric space. Define the self-maps f , g, S, and T on X as follows:

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( 2, when x = 2, fx = 3, when x > 2, ( 2, when x = 2, or x > 5, gx = 24, when 2 < x ≤ 5, ( 2, when x = 2, Sx = 24, when x > 2,   when x = 2, 2, T x = 24, when 2 < x ≤ 5,   x − 3, when x > 5. In our next result, Abbas et al. [2] proved common fixed point theorem for mappings satisfying common property (E.A.). Theorem 5.12. Let (X, M, ∗) be a fuzzy metric space. Let f , g, S and T be maps from X into itself such that  M( f x, gy, kt) ≥ φ M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t),  M( f x, Sy, αt), M(gy, Sx, (2 − α)t) , for all x, y ∈ X, k ∈ (0, 21 ) α ∈ (0, 2), t > 0 and φ ∈ ψ. Then f , g, S and T have a unique common fixed point in X provided the pair ( f S) or (g, T ) satisfies (EA) property, T (X) and S(X) is a closed subset of X and the pairs (g, T ) and ( f , S) are weakly compatible. Observe that the Corollaries 3.4–3.8 in Ref. [7] can be easily improved in the light of Theorems 5.11 and 5.12. For example: Corollary 5.8. [2] Let (X, M, ∗) be a fuzzy metric space. where ∗ is any continuous t-norm. Let f , g, R, S, H and T be maps from X into itself with f (X) ⊆ T H(X) and g(X) ⊆ SR(X) and there exists a constant k ∈ (0, 12 ) such that  M( f x, gy, kt) ≥ φ M(SRx, T Hy,t), M( f x, SRx,t), M(gy, T Hy,t),  M( f x, T Hy, αt), M(gy, SRx, (2 − α)t) , for all x, y ∈ X, α ∈ (0, 2), t > 0 and φ ∈ ψ. Then f , g, R, S, H and T have a unique common fixed point in X provided the pair ( f SR) or (g, T H) satisfies (EA) property, one of f (X), T H(X), g(X), SR(X) is a closed subset of X and the pairs (g, T H) and ( f , SR) are weakly compatible.

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Corollary 5.9. [2] Let (X, M, ∗) be a fuzzy metric space., where ∗ is any continuous t-norm. Let f , g, R, S, H and T be maps from X into itself and there exists a constant k ∈ (0, 21 ) such that  M( f x, gy, kt) ≥ φ M(SRx, T Hy,t), M( f x, SRx,t), M(gy, T Hy,t),  M( f x, T Hy, αt), M(gy, SRx, (2 − α)t) , for all x, y ∈ X, α ∈ (0, 2), t > 0 and φ ∈ ψ. Then f , g, S and T have a unique common fixed point in X provided the pair ( f SR) or (g, T H) satisfies (EA) property, T H(X) and SR(X) is a closed subset of X and the pairs (g, T H) and ( f , SR) are weakly compatible. In 2011, Gopal et al. [15] proved some common fixed results for fuzzy metric spaces, they observed that the notion of common property (E.A) relaxes the required containment of range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. As a consequence, a multitude of recent fixed point theorems of the existing literature are sharpened and enriched. Definition 5.11. [15] Two finite families of self-mappings fi and g j are said to be pairwise commuting if i. fi f j = f j fi , i, j ∈ {1, 2, ..., m}, ii. gi g j = g j gi , i, j ∈ {1, 2, ..., n}, iii. fi g j = g j fi , i ∈ {1, 2, ..., m} and j ∈ {1, 2, ..., n}. The following definitions will be utilized to state various results below Definition 5.12. [15] Let (X, M, ∗) be a fuzzy metric space and f , g : X → X a pair of mappings. The mapping f is called a fuzzy contraction with respect to g if there exists an upper semicontinuous function r : [0, ∞) → [0, ∞) with r(τ) < τ for every τ > 0 such that   1 1 −1 ≤ r −1 , M( f x, f y,t) m( f , g, x, y,t) for every x, y ∈ X and each t > 0, where m( f , g, x, y,t) = min{M(gx, gy,t), M( f x, gx,t), M( f y, gy,t)}. Definition 5.13. [15] Let (X, M, ∗) be a fuzzy metric space and f , g : X → X a pair of mappings. The mapping f is called a fuzzy k-contraction with respect to g if there exists k ∈ (0, 1) such that   1 1 −1 ≤ k −1 , M( f x, f y,t) m( f , g, x, y,t) for every x, y ∈ X and each t > 0, where m( f , g, x, y,t) = min{M(gx, gy,t), M( f x, gx,t), M( f y, gy,t)}.

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Definition 5.14. [15] Let f , g, S, and T be four self-mappings of a fuzzy metric space (X, M, ∗). Then, the mappings f and g are called a generalized fuzzy contraction with respect to S and T if there exists an upper semicontinuous function r : [0, ∞) → [0, ∞) with r(τ) < τ for every τ > 0 such that for each x, y ∈ X and t > 0,   1 1 −1 , −1 ≤ r M( f x, f y,t) min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t)} Gopal et al. [15] proved the following as a main result. Theorem 5.13. Let f , g, S, and T be self-mappings of a fuzzy metric space (X, M, ∗) such that the mappings f and g are a generalized fuzzy contraction with respect to mappings S and T . Suppose that the pairs ( f , S) and (g, T ) share the common property (E.A.) and S(X) and T (X) are closed subsets of X. Then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , g, S and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible. The following example is utilized to highlight the utility of the aforementioned theorem over earlier relevant results. Example 5.8. [15] Let X = [2, 20] and (X, M, ∗) be a fuzzy metric space defined as M(x, y,t) =

t , t > 0, x, y ∈ X t + |x − y|

Define f , g, S, T : X → X by ( 2, if x = 2, fx = 3, if x > 2, ( 2, if x = 2, Sx = 6, if x > 2,   2, if x = 2, gx = 6, if 2 < x ≤ 5,   3, if x > 5.   2, if x = 2, T x = 18, if 2 < x ≤ 5,   12, if x > 5. t In the foregoing theorem, if we set r(τ) = kτ, k ∈ (0, 1) and M(x, y,t) = t+|x−y| , then we get the following result which improves and generalizes the result of Jungck [22] Corollary 3.2 in metric space.

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Corollary 5.10. [15] Let f , g, S and T be self-mappings of a metric space (X, d) such that d( f x, gy) ≤ k max{d(Sx, Ty), d( f x, Sx), d(gy, Ty)}, for every x, y ∈ X, k ∈ (0, 1). Suppose that the pairs ( f , S) and (g, T ) share the common property (E.A) and S(X) and T (X) are closed subsets of X. Then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , gS and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible. By choosing f , g, S and T suitably, one can deduce corollaries for a pair as well as for two different trios of mappings. For the sake of brevity, we deduce, by setting f = g and S = T a corollary for a pair of mappings which is an improvement over the result of C. Vetro and P. Vetro [[45], Theorem 2]. Corollary 5.11. [15] Let ( f , S) be a pair of self-mappings of a fuzzy metric space (X, M, ∗) such that ( f , S) satisfies the property (E.A), f is a fuzzy contraction with respect to S and S(X) is a closed subset of X. Then, the pair ( f , S) has a point of coincidence, whereas the pair ( f , S) has a unique common fixed point provided that it is weakly compatible. Now, we know that a fuzzy k-contraction with respect to S implies f fuzzy contraction with respect to S. Thus, Gopal et al. [15] obtained the following corollary which sharpens Ref. [[45], Theorem 4]. Corollary 5.12. Let f and S be self-mappings of a fuzzy metric space (X, M, ∗) such that the pair ( f , S) enjoys the property (E.A), f is a fuzzy k-contraction with respect to S, and S(X) is a closed subset of X. Then, the pair ( f , S) has a point of coincidence. Further, f and S have a unique common fixed point provided that the pair ( f , S) is weakly compatible. Gopal et al. [15] recalled the following two implicit functions defined and studied in Refs. [19,40], respectively. First, following Singh and Jain [40], let Φ be the set of all real continuous functions φ : [0, 1]4 → R, non-decreasing in first argument, and satisfying the following conditions: i. for u, v ≥ 0, φ (u, v, u, v) ≥ 0, or φ (u, v, v, u) ≥ 0 implies that u ≥ v, ii. φ (u, u, 1, 1) ≥ 0 implies that u ≥ 1. Example 5.9. Define φ (t1 ,t2 ,t3 ,t4 ) = 15t1 − 13t2 + 5t3 − 7t4 . Where φ ∈ Φ Second, following Imdad and Ali [19], let Ψ denote the family of all continuous functions F : [0, 1]4 → R, satisfying the following conditions: i. F1 : for every u, > 0, v ≥ 0, F(u, v, u, v) ≥ 0, or F(u, v, v, u) ≥ 0 we have u > v, ii. F2 : F(u, u, 1, 1) < 0, for each 0 < u < 1.

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The following examples of functions F ∈ Ψ are essentially contained in Ref. [19]. Example 5.10. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t1 − φ (min{t2 ,t3 ,t4 }), where φ : [0, 1] → [0, 1] is a continuous function such that φ (s) > s for 0 < s < 1. Example 5.11. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t1 − k(min{t2 ,t3 ,t4 }), where k > 1. Example 5.12. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t1 − kt2 − (min{t3 ,t4 }), where k > 0. Example 5.13. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t1 − at2 − bt3 − ct4 , where a > 1 and b, c ≥ 0 (b, c ̸= 1). Example 5.14. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t1 − at2 − b(t3 +t4 ), where a > 1 and 1 ≤ b < 1. Example 5.15. Define F : [0, 1]4 → R as F(t1 ,t2 ,t3 ,t4 ) = t13 − kt2t3t4 , where k > 1. Gopal et al. [15] defined the following before proving their results; it may be noted that aforementioned classes of functions Φ and Ψ are independent classes as the implicit function F(t1 ,t2 ,t3 ,t4 ) = t1 − k(min{t2 ,t3 ,t4 }), where k > 1 (belonging to Ψ) does not belong to Φ as φ (u, u, 1, 1) < 0, for each 0 < u < 1., whereas implicit function φ (t1 ,t2 ,t3 ,t4 ) = 15t1 − 13t2 + 5t3 − 7t4 (belonging to Φ) does not belong to Ψ as F(u, u, 1, 1) = 0 implies u = v instead of u > v. Lemma 5.9. Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗). Assume that there exists F ∈ Ψ such that   F M( f x, gy,t), M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t) ≥ 0, for all x, y ∈ X and t > 0. Suppose that pair ( f , S) (or (g, T )) satisfies the property (E.A), and ( f (X) ⊂ T (X) (or (g(X) ⊂ S(X)). If for each {xn }, {yn } in X such that lim f xn = lim Sxn (or lim gyn = lim Tyn ), we have lim M( f xn , gyn ,t) > 0 for all n→∞

n→∞

n→∞

n→∞

n→∞

t > 0, then, the pairs ( f , S) and (g, T ) share the common property (E.A). With a view to generalize some fixed point theorems contained in Imdad and Ali [18,19], Gopal et al. [15] proved the following fixed point theorem which in turn generalizes several previously known results. Theorem 5.14. Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗). Assume that there exists F ∈ Ψ such that   F M( f x, gy,t), M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t) ≥ 0, for all x, y ∈ X and t > 0. Suppose that pair ( f , S) and (g, T ) satisfies the common property (E.A), and S(X) and T (X) are closed subsets of X. Then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , g, S and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible.

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Example 5.16. In the setting of Example 2.2, of Ref. [15] retain the same mappings 4 f , g, S and √ T and define F : [0, 1] → R as F(t1 ,t2 ,t3 ,t4 = t1 − φ (min{t2 ,t3 ,t4 }) with φ (r) = r. Then, f , g, S and T satisfy all the conditions of the aforementioned theorem and have a unique common fixed point x = 2 which also remains a point of discontinuity Further, we remark that Theorem 5.2 of Imdad and Ali [19] cannot be used in the context of this example, as the required conditions on containment in respect of ranges of the involved mappings are not satisfied. Corollary 5.13. i. M( f x, gy,t) ≥ φ (min{M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t)}), where φ : [0, 1] → [0, 1] is a continuous function such that φ (s) > s for 0 < s < 1. ii. M( f x, gy,t) ≥ k(min{M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty,t)}), where k > 1. iii. M( f x, gy,t) ≥ kM(Sx, Ty,t) − (min{M(Sx, f x,t), M(gy, Ty,t)}), where k > 0. iv. M( f x, gy,t) ≥ aM(Sx, Ty,t) − bM(Sx, f x,t) − cM(gy, Ty,t), where a > 1 and b, c ≥ 0 (b, c ̸= 1). v. M( f x, gy,t) ≥ aM(Sx, Ty,t) − b[M(Sx, f x,t) + M(gy, Ty,t)], where a > 1 and 0 ≤ b > 0. vi. M( f x, gy,t) ≥ kM(Sx, Ty,t)M(Sx, f x,t)M(gy, Ty,t), where k > 1. Remark 5.5. The aforementioned corollary corresponding to condition (i) is a result due to Imdad and Ali [18] where as the aforementioned corollary corresponding to various conditions presents a sharpened form of Corollary 5.2 of Imdad and Ali [19]. Similar to this corollary, one can also deduce generalized versions of certain previuos results. The following theorem generalizes a theorem contained in Singh and Jain [40]. Theorem 5.15. [15], Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗). Assume that there exists F ∈ Ψ such that   F M( f x, gy, kt), M(Sx, Ty,t), M(Sx, f x,t), M(gy, Ty, kt) ≥ 0,   F M( f x, gy, kt), M(Sx, Ty,t), M(Sx, f x, kt), M(gy, Ty,t) ≥ 0, for all x, y ∈ X k ∈ (0, 1) and t > 0. Suppose that pair ( f , S) and (g, T ) satisfy the common property (E.A), and S(X) and T (X) are closed subsets of X. Then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , g, S and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible. Example 5.17. In the setting of Example 2.2, of Ref. [15], we define φ (t1 ,t2 ,t3 ,t4 ) = 15t1 − 13t2 + 5t3 − 7t4 , besides retaining the rest of the example as it stands. Gopal et al. [15] concluded their manuscript by deriving the following results of integral type.

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Theorem 5.16. Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗). Assume that there exists a Lebesgue integrable function ϕ : R → R and a function φ : [0, 1]4 → R such that Z φ (u,1,u,1 0

ϕ(s)ds ≥ 0,

Z φ (u,1,1,u 0

ϕ(s)ds ≥ 0 or

Z φ (u,u,1,1 0

ϕ(s)ds ≥ 0

implies u = 1. Suppose that pair ( f , S) and (g, T ) satisfies the common property (E.A), and S(X) and T (X) are closed subsets of X. if
Z φ M( f x,gy,t),M(Sx,Ty,t),M(Sx, f x,t),M(gy,Ty,t) 0

ϕ(s)ds ≥ 0, ∀ x, y ∈ X and t > 0,

then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , g, S and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible. Corollary 5.14. [15], Let f , g, S and T be self-mappings of a fuzzy metric space (X, M, ∗). Assume that there exists a Lebesgue integrable function ϕ : R → R and a function φ : [0, 1]4 → R such that
Z φ M( f x,gy,t),M(Sx,Ty,t),M(Sx, f x,t),M(gy,Ty,t) 0

Z φ (u,u,1,1 0

ϕ(s)ds ≥ 0, ∀ x, y ∈ X and t > 0,

ϕ(s)ds ≥ 0 ∀ u ∈ (0, 1).

Suppose that pair ( f , S) and (g, T ) satisfy the common property (E.A), and S(X) and T (X) are closed subsets of X. Then, the pair ( f , S) as well as (g, T ) have a point of coincidence each. Further, f , g, S and T have a unique common fixed point provided that both the pairs ( f , S) and (g, T ) are weakly compatible. In 2011, Gopal and Imdad [14] proved some new common fixed point theorems in (GV)-fuzzy metric spaces. While proving our results, we utilize the idea of compatibility due to Jungck [23] together with subsequentially continuity due to Bouhadjera and Godet-Thobie [5] respectively (also alternately reciprocal continuity due to Pant [33] together with subcompatibility due to Bouhadjera and Godet-Thobie [5] as patterned in Imdad et al. [21] wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect of any one of the involved maps are relaxed. Their results substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in metric as well as fuzzy metric spaces which include some relevant results due to Imdad et al. [19], Mihet [29], Mishra [30], and several others. In metric space, the notion of weak compatibility (or coincidently commuting property or partially commuting property) coincides with pointwise R-weak commutativity which is a minimal commutativity condition for the existence of unique coincidence (or common fixed) point (see [33]).

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Definition 5.15. [14] A pair ( f , g) of self-mappings of a nonempty set X is said to be occasionally weakly compatible (O.W.C.) iff the pair ( f , g) commutes at least on one coincidence point (of the pair); i.e., there exists at least one point x in X such that f x = gx and f gx = g f x. Definition 5.16. [14] A pair of self-mappings ( f , g) defined on a fuzzy metric space (X, M, ∗) is said to be subcompatible if there exists a sequence {xn } in X with lim f xn = lim gxn = z,

n→∞

n→∞

for some z in X and lim M( f gxn , g f xn ,t) = 1, for all t > 0. n→∞

Obviously, every OWC pair is subcompatible but not conversely (see [5]). Most recently, Doric et al. [10] have shown that in respect of single-valued maps, the condition of occasionally weak compatibility reduces to weak compatibility in the presence of a unique point of coincidence (or a unique common fixed point) of the given pair of maps. Thus, no generalization can be obtained by replacing weak compatibility with OWC Property. Here, it is worth pointing out that Pant and Pant [34] utilized OWC Property in the context of Lipschitzian mappings. Thus, weak compatibility remains a minimal commutativity condition for the existence of unique common fixed of contractive type maps. In fact, Doric et al. Bouhadjera proved the following: Proposition 5.1. [10] Let a pair of mappings ( f , g) has a unique point of coincidence. Then it is weakly compatible if and only if it is occasionally weakly compatible. Definition 5.17. [14] A pair of self-mappings ( f , g) defined on a fuzzy metric space (X, M, ∗) is called reciprocally continuous if for sequences {xn } in X, lim f gxn = f z n→∞ and lim g f xn = gz, whenever n→∞

lim f xn = lim gxn = z

n→∞

n→∞

for some z in X. Clearly, every pair of continuous mappings is reciprocally continuous but not conversely. Definition 5.18. [14] A pair of self-mappings ( f , g) defined on a fuzzy metric space (X, M, ∗) is called rsubsequentially continuous if there exists sequences {xn } in X such that lim f xn = lim gxn = z n→∞

n→∞

for some z in X and lim f gxn = f z and lim g f xn = gz. n→∞

n→∞

In general, if the maps f and g are continuous or reciprocally continuous, then they are naturally subsequentially continuous. However, there exist subsequentially continuous pair of maps which are neither continuous nor reciprocally continuous. Following Ali and Imdad [3], let F6 be the set of all continuous functions F(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) : [0, 1]6 → R satisfying the following condition;

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(F1 ) : F(u, 1, u, 1, u, u) < 0, for all u ∈ (0, 1). Example 5.18. Define F(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) : [0, 1]6 → R as : F(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) = t1 − ψ(min{t2 ,t3 ,t4 ,t5 ,t6 }), where ψ : [0, 1] → [0, 1] is increasing and continuous function such that ψ(t) > t for all t ∈ (0, 1). Notice that (F1 ) : F(u, 1, u, 1, u, u) = u − ψ(u) < 0, for all u ∈ (0, 1). Example 5.19. Define F(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) : [0, 1]6 → R as F(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) =

Z t1 0

φ (t)dt − ψ

Z

min{t2 ,t3 ,t4 ,t5 ,t6 }

0

φ (t)dt



where ψ : [0, 1] → [0, 1] is an increasing and continuous function such that ψ(t) > t for all t ∈ (0, 1) and φ : R+ → R+ is a Lebesgue integrable function which is summable and satisfies 0
t for all t ∈ (0, 1) and φ : R+ → R+ is a Lebesgue integrable function which is summable and satisfies 0
0. Alternately, using reciprocal continuity (due to Pant [33]) together with subcompatibility (due to Bouhadjera and Godet-Thobie [5]), we have the following: Theorem 5.18. [14] Let f , g, S and T be four self-mappings of a (GV)-fuzzy metric space (X, M, ∗). If the pairs ( f , S) and (g, T ) are subcompatible and sreciprocally continuous mappings, then i. : the pair ( f , S) has a coincidence point, ii. : the pair ((g, T ) has a coincidence point. iii. : Further, f , g, S and T have a unique common fixed point provided the involved maps satisfy the following inequality  F M( f x, gy, s), M( f x, Sx, s), M(Sx, Ty, s),  M(gy, Ty, s), M(Sx, gy, s), M( f x, Ty, s) ≥ 0 for all x, y ∈ X, F ∈ F6 and s > 0. The earlier defined implicit relation enables us to derive a multitude of fixed point theorems as carried out by Ali and Imdad [3]. But, here we limit ourselves to only those conditions which correspond to Examples 2.1–2.3 yielding thereby the refined and sharpened versions of some fixed point theorems contained in Imdad et al. [19], Mishra et al. [30] besides some other ones which can be stated as under:

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Corollary 5.15. [14] The conclusions of Theorem 5.17 remain true if we replace the inequal- ity (iii) (of Theorem 5.17) by any one of the following (besides retaining rest of the hypotheses): i. : M( f x, gy, s) ≥ ψ(min{M( f x, Sx, s), M(Sx, Ty, s), M(gy, Ty, s), M(Sx, gy, s), M( f x, Ty, s)}), where ψ : [0, 1] → [0, 1] is increasing and continuous function such that ψ(t) > t for all t ∈ (0, 1). ii. : Z M( f x,gy,s)

φ (t)dt

0

≥ψ

Z

min{M( f x,Sx,s),M(Sx,Ty,s),M(gy,Ty,s),M(Sx,gy,s),M( f x,Ty,s)}

φ (t)dt

0



where ψ : [0, 1] → [0, 1] is an increasing and continuous function such that ψ(t) > t for all t ∈ (0, 1) and φ : R+ → R+ is a Lebesgue integrable function which is summable and satisfies 0
t for all t ∈ (0, 1) and φ : R+ → R+ is a Lebesgue integrable function which is summable and satisfies 0
0. Alternately, by setting S = T in Theorem 5.17, we can also derive yet another corollary for three mappings which runs as follows. Corollary 5.17. [14] Let f , g and T S be four self-mappings of a (GV)-fuzzy metric space (X, M, ∗). If the pairs ( f , S) and ( f , S) are compatible and subsequentially continuous mappings, then i. : the pair ( f , S) has a coincidence point, ii. : the pair ((g, S) has a coincidence point. iii. : Further, f , g and S have a unique common fixed point provided the involved maps satisfy the following inequality  F M( f x, gy, s), M( f x, Sx, s), M(Sx, Sy, s),  M(gy, Sy, s), M(Sx, gy, s), M( f x, Sy, s) ≥ 0 for all x, y ∈ X, F ∈ F6 and s > 0. Finally, by setting f = g and S = T in Theorem 5.17, we derive the following corollary for a pair of maps. Corollary 5.18. [14] Let f and S be four self-mappings of a (GV)-fuzzy metric space (X, M, ∗). If the pairs ( f , S) are compatible and subsequentially continuous mappings, then i. : the pair ( f , S) has a coincidence point, ii. : Further, f and S have a unique common fixed point provided the involved maps satisfy the following inequality  F M( f x, f y, s), M( f x, Sx, s), M(Sx, Sy, s),  M( f y, Sy, s), M(Sx, f y, s), M( f x, Sy, s) ≥ 0 for all x, y ∈ X, F ∈ F6 and s > 0.

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Gopal and Imdad [14] concluded this paper with two illustrative examples which demonstrate the validity of the hypotheses of Theorems 5.17 and 5.18. Example 5.21. [14] Let (X, M, ∗) be a (GV)-fuzzy metric space as defined as t M(x, y,t) = t+|x−y| , wherein X = (−3, ∞). Set f = g and S = T . Define f , S : X → X as follows:   if x ∈ (−3, 0), 0, f x = x/3, if x ∈ [0, 1],   2x − 1, if .x ∈ (1, ∞) ( x/2, if x ∈ (−3, 1], Sx = 3x − 2, if x ∈ (1, ∞). Example 5.22. [14] Let (X, M, ∗) be a (GV)-fuzzy metric space as defined as t M(x, y,t) = t+|x−y| , wherein X = R. Set f = g and S = T . Define f , S : X → X as follows: ( x + 1, if x ∈ (−∞, 1), fx = 2x − 1, if .x ∈ [1, ∞) ( x/2, if x ∈ (−∞, 1), fx = 3x − 2, if .x ∈ [1, ∞) Rao et al. [35] introduced the notion of M-maps w.r.to a single map and a pair of maps in fuzzy metric spaces and obtain common fixed point theorems for two pairs of subcompatible maps satisfying implicit relations. Our results generalize and extend several comparable results in existing literature like Abbas et al. [2] and Kumar et al. [26] to the setting of single and set-valued maps and also by relaxing some more conditions. They provided some background materials for future use. Lemma 5.10. [35] Let (X, M, ∗) be a fuzzy metric space and M(x, y,t) → 1 as t → ∞ for all x, y ∈ X. If M(x, y, kt) ≥ M(x, y,t) for all x, y ∈ X, t > 0 and for a number k ∈ (0, 1) then x = y. Definition 5.19. [35] A sequence {An } in B(X) is said to be convergent to a set A ∈ B(X) if δM (An , A,t) → 1 as n → ∞ for all t > 0. Lemma 5.11. [35] Let {An } and {Bn } are sequences in B(X) converging to A and B in B(X), respectively. Then δM (An , Bn , kt) → δM (A, B,t) as n → ∞ for all t > 0. Lemma 5.12. [35] If δM (A, B, kt) ≥ δM (A, B,t) for all A, B ∈ B(X) and for all t > 0, 0 < k < 1, then A = B = {singleton} provided M(x, y,t) → 1 as t → ∞ for all x, y ∈ X. Definition 5.20. [35] The maps F : X → B(X) and f : X → X are said to be weakly compatible or subcompatible if they commute at coincidence points, i.e., for each point u ∈ X such that Fu = { f u}, we have F f u = f Fu.

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Generally to prove common fixed point theorems for two pairs of maps or Gungck type maps using property (E.A), introduced in Ref. [1] one can tempt to assume that the range set of one mapping is closed or one of the mappings is surjective. here, Rao et al. [35] relax some conditions by introducing the following two definitions. Definition 5.21. [35] Let (X, M, ∗) be a fuzzy metric space and f : X → X and F : X → B(X). Then ( f , F) is said to be a pair of M-maps with respect to f if there exists a sequence {xn } in X such that for every t > 0, M( f xn , z,t) → 1 and δM (Fxn , {z},t) → 1 as n → ∞ for some z ∈ f (X). Definition 5.22. [35] Let (X, M, ∗) be a fuzzy metric space and f , g : X → X and F : X → B(X). Then ( f , F) is said to be a pair of M-maps with respect to ( f , g) if there exists a sequence {xn } in X such that for every t > 0, M( f xn , z,t) → 1 and δM (Fxn , {z},t) → 1 as n → ∞ for some z ∈ f (X) ∩ g(X). Let Φ denote the class of all continuous functions ϕ : [0, 1]6 → R+ satisfying ϕ(u, 1, 1, v, v, 1) ≥ 0 or ϕ(u, 1, v, 1, 1, v) ≥ 0 or ϕ(u, v, 1, 1v, v) ≥ 0 implies u ≥ v. These are some examples of implicit relations. Example 5.23. [35] ϕ(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) = t1 − min{t2 ,t3 ,t4 ,t5 ,t6 } Example 5.24. [35] ϕ(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) = t1 − min{t2 ,t3 ,t3t5 ,t4t6 } 4 +t5 t6 Example 5.25. [35] ϕ(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) = t1 − t3 t1+t 2

Example 5.26. [35] ϕ(t1 ,t2 ,t3 ,t4 ,t5 ,t6 ) = t1 − φ (t2 ,t3 ,t4 ,t5 ,t6 ), where φ is increasing in each coordinate and φ (t,t,t,t,t) > t, ∀ t ∈ [0, 1] The main result of Rao et al. [35] is as follows: Theorem 5.19. Let (X, M, ∗) be a fuzzy metric space and f , g : X → X and F, G : X → B(X) be maps satisfying  ϕ δM (Fx, Gy, kt), δM ( f x, gy,t), δM ( f x, Fx,t),  δM (gy, Gy,t), δM ( f x, Gy,t), δM (gy, Fx,t) ≥ 0 for all x, y ∈ X,t > 0 and k ∈ (0, 1) where ϕ ∈ Φ, the pairs ( f , F) and (g, G) are sub compatible, a. ( f , F) is a pair of M-maps with respect to f and Fx ⊆ g(X) for all x ∈ X or b. (g, G) is a pair of M-maps with respect to g and Gx ⊆ f (X) for all x ∈ X. Then f , g, F and G have a unique common fixed point z ∈ X such that Fz = Gz = {z} = { f z} = {gz}.

Common Fixed-Point Theorems in Fuzzy Metric Spaces

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Example 5.27. Let X = [0, 1] and d(x, y) = |x − y| and a ∗ b = min{a, b}, ∀ a, b ∈ t [0, 1]. Define M(x, y,t) = t+d(x,y) ∀ x, y ∈ X. Then (X, M, ∗) is a fuzzy metric space. Define F, G : X → B(X) and f , g : X → X as follows: ( ( 1 1 − x, if x ∈ (0, 12 ], , if x ∈ [0, 12 ], 2 gx = f x = x+1 1 0, if x ∈ ( 12 , 1] ∪ {0}, 4 , if x ∈ ( 2 , 1] n1o Fx = , 2

( { 1 }, if x ∈ (0, 12 ], gx = 32 1 [ 8 , 2 ], if x ∈ ( 12 , 1].

Two theorems (Theorem 2.1 and 2.3 of Ref. [2]) can be extended to two pairs of multi-valued and single valued maps and improved respectively as follows: Theorem 5.20. Let (X, M, ∗) be a fuzzy metric space and f , g : X → X and F, G : X → B(X) be maps satisfying  δM (Fx, Gy, kt) ≥ ϕ M( f x, gy,t), δM ( f x, Fx,t), δM (gy, Gy,t),  δM ( f x, Gy,t), δM (gy, Fx,t) for all x, y ∈ X,t > 0 and k ∈ (0, 1) where ϕ ∈ Φ, the pairs ( f , F) and (g, G) are sub compatible, a. ( f , F) is a pair of M-maps with respect to f and Fx ⊆ g(X) for all x ∈ X or b. (g, G) is a pair of M-maps with respect to g and Gx ⊆ f (X) for all x ∈ X. Then f , g, F and G have a unique common fixed point z ∈ X such that Fz = Gz = {z} = { f z} = {gz}. Theorem 5.21. Let (X, M, ∗) be a fuzzy metric space and f , g : X → X and F, G : X → B(X) be maps satisfying contraction condition, (a) and (b) of the aforementioned theorem and ( f , F) or (g, G) is a pair of M-maps with respect to ( f , g). Then f , g, F and G have a unique common fixed point z ∈ X such that Fz = Gz = {z} = { f z} = {gz}. In 2012, Imdad et al. [20] utilized the property (E.A.) and the common property (E.A.) to prove some existence results on common fixed point for contractive mappings in fuzzy metric spaces which include fuzzy metric spaces of two types, namely, Kramosil and Michalek fuzzy metric spaces along with George and Veeramani fuzzy metric spaces. Their results generalize and extend several relevant common fixed point theorems from the literature. They also furnished an illustrative example to validate of their main result.

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They begin with the following observation. Lemma 5.13. [20] Let f , g, S and T be four self-mappings of a KM-fuzzy metric space (X, M, ∗) satisfying the following conditions: i. the pair ( f , S) (or (g, T )) satisfies the property (E.A), ii. f (X) ⊂ T (X) (or g(X) ⊂ S(X)), iii. g(yn ) converges for every sequence {yn } in X whenever T (yn ) converges (or f (yn ) converges for every sequence {yn } in X whenever S(yn ) converges), iv. ∀ x, y ∈ X, x ̸= y, ∃ t > 0 : 0 < M(x, y,t) < 1, for some φ ∈ Φ M( f x, gy,t) ≥ φ (min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t), M( f x, Ty,t), M(Sx, gy,t)}),

(5.11)

then the pairs ( f , S) and (g, T ) share the common property (E.A). Remark 5.6. [20] The converse of the aforementioned lemma is not true in general. For a counterexample, one can see Example 3.1 of Ref. [20]. Their next result is a common fixed point theorem through common property (E.A) Theorem 5.22. [20] Let f , g, S and T be four self-mappings of a KM-fuzzy metric space (X, M, ∗) satisfying the condition (5.11). Suppose that i. the pairs ( f , S) and (g, T ) share the common property (E.A), ii. S(X) and T (X) are closed subsets of X. Then pair ( f , S) as well as (g, T ) have a coincidence point. Moreover, f , g, S and T have a unique common fixed point in X provided both the pairs ( f , S) and (g, T ) are weakly compatible. Remark 5.7. The aforementioned theorem generalizes the results of Mihet [29] for four mappings and others [18,19,44,46] to KM-fuzzy metric space as Theorem 2.1 never demands for continuity of mappings, completeness of space (or subspace) and containment of range sets while commutativity requirement is weakened to weak compatibility. Theorem 5.23. [20] The conclusions of Theorem 5.22 remain true if the condition (ii) (of Theorem 5.22) is replaced by following: ′

(II ) f (X) ⊂ T (X) and g(X) ⊂ S(X). As a corollary of Theorem 5.23, they [20] can have the following result which is also a variant of Theorem 5.22. Corollary 5.19. [20] The conclusions of Theorems 5.22 and 5.23 remain true if the ′ conditions (ii) and (ii ) are replaced by the following: ′′

(ii ) f (X) and g(X) are closed subsets of X provided f (X) ⊂ T (X) and g(X) ⊂ S(X).

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Theorem 5.24. [20] Let f , g, S and T be four self-mappings of a KM-fuzzy metric space (X, M, ∗) satisfying the conditions (i-iv) of Lemma 5.13. Suppose that (v) S(X) (or T (X)) is a closed subset of X. Then pair ( f , S) as well as (g, T ) have a coincidence point. Moreover, f , g, S and T have a unique common fixed point in X provided that the pairs ( f , S) and (g, T ) are weakly compatible. By choosing f , g, S and T suitably, one can deduce corollaries for a pair of mappings. Corollary 5.20. [20] Let f and S be two self-mappings of a KM-fuzzy metric space (X, M, ∗) satisfying the following conditions: i. the pair ( f , S) satisfies the property (E.A), ii. S(X) is a closed subset of X and iii. ∀ x, y ∈ X, x ̸= y, ∃ t > 0 : 0 < M(x, y,t) < 1, for some φ ∈ Φ M( f x, f y,t) ≥ φ (min{M(Sx, Sy,t), M( f x, Sx,t), M( f y, Sy,t), M( f x, Sy,t), M(Sx, f y,t)}). Then pair ( f , S) has a coincidence point. Moreover, f and S have a unique common fixed point in X provided that the pair ( f , S) is weakly compatible. Remark 5.8. The aforementioned corollary extends and generalizes certain relevant results involving pair of mappings from the existing literature (e.g. [18,19,32,44,46]). As an application of Theorem 5.22, Imdad et al. [20] have the following result for four finite families of self-mappings. Theorem 5.25. [20] Let { f1 , f2 , · · · , fm },{g1 , g2 , · · · , g p },{S1 , S2 , · · · , Sn } and {T1 , T2 , · · · , Tq } be four finite families of self mappings of a KM-fuzzy metric space (X, M, ∗) with f = f1 f2 · · · fm ,g = g1 g2 · · · g p ,S = S1 S2 · · · Sn and T = T1 T2 · · · Tq satisfying condition (5.11) and pairs ( f , S) and (g, T ) share the common property (E.A). If S(X) and T (X) are closed subsets of X, then i. the pairs ( f , S) and (g, T ) have a coincidence point each. Moreover, fi , Sk , gr and Tt , have a unique common fixed point provided the pairs of families ({ fi }, {Sk }) and ({gr }, {Tt }) commute pairwise, where i ∈ {1, . . . , m}, k ∈ {1, . . . , n}, r ∈ {1, . . . , p} and t ∈ {1, . . . , q}. By setting f1 = f2 = · · · = fm = G, g1 = g2 = · · · = g p = H, S1 = S2 = · · · = Sn = I and T1 = T2 = · · · = Tq = J in Theorem 5.25, we deduce the following result for iterates of mappings:

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Corollary 5.21. [20] Let G, H, I and J be four self-mappings of a KM-fuzzy metric space (X, M, ∗), pairs (Gm , I n ) and (H p , J q ) share the common property (E.A) and satisfying the condition ∀ x, y ∈ X, x ̸= y, ∃ t > 0 : 0 < M(x, y,t) < 1, for some φ ∈ Φ M(Gm x, H p y,t) ≥ φ (min{M(I n x, J q y,t), M(Gm x, I n x,t), M(H p y, J q y,t), M(Gm x, J q y,t), M(I n x, H p y,t)}) where m, n, p and q are positive integers. If I n (X) and J q (X) are closed subsets of X, then G, H, I and J have a unique common fixed point provided the pair (G, I) as well as (H, J) is commuting. Remark 5.9. Theorem 5.25 is a partial generalization of Theorem 5.22 as commutativity requirements in Theorem 5.25 are stronger than weak compatibility in Theorem 5.22. Furthermore, we can prove more results similar to Theorems 5.23 and 5.24 and Corollaries 5.19–5.20 in respect of Theorem 5.25 and Corollary 5.21. Next, we state and prove Grabiec-type common fixed point theorems for four self-mappings. First, we state the following lemma without proof as proof is an easy consequence of one of the properties of fuzzy metric. Lemma 5.14. [20] If M(x, y, kt) ≥ M(x, y,t) for all x, y ∈ X, t > 0 and for some k ∈ (0, 1), then x = y. Theorem 5.26. [20] Let f , g, S and T be four self-mappings of a KM-fuzzy metric space (X, M, ∗) satisfying the conditions (I) and (II) of Theorem 2.1, and for all x, y ∈ X, t > 0 and for some k ∈ (0, 1) M( f x, gy, kt) ≥ min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t), M( f x, Ty,t), M(Sx, gy,t)}.

(5.12)

Then pair ( f , S) as well as (g, T ) have a coincidence point. Moreover, f , g, S and T have a unique common fixed point in X provided both the pairs ( f , S) and (g, T ) are weakly compatible. Remark 5.10. Theorem 5.26 generalizes some common fixed point theorems due to Grabiec [13], Mishra et al. [31], Subrahmanyam [42], Vijayaraju and Sajath [46], and extends some relevant results of Ali and Imdad [3] to fuzzy metric spaces which also include quasi-contraction. If (X, M, ∗) is a GV-fuzzy metric space, then some of the hypotheses in the aforementioned results can be relaxed. Lemma 5.15. [20] Let f , g, S and T be four self-mappings of a GV-fuzzy metric space (X, M, ∗) satisfying the conditions (i)-(iii) of Lemma 5.13 and (iv) for some φ ∈ Φ and some t > 0

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M( f x, gy,t) ≥ φ (min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t), M( f x, Ty,t), (5.13)

M(Sx, gy,t)}), then the pairs ( f , S) and (g, T ) share the common property (E.A).

Theorem 5.27. [20] Let f , g, S and T be four self-mappings of a GV-fuzzy metric space (X, M, ∗) satisfying the condition (iv) of the aforementioned Lemma 5.15. Suppose that i. the pairs ( f , S) and (g, T ) share the common property (E.A), ii. S(X) and T (X) are closed subsets of X. Then pair ( f , S) as well as (g, T ) have a coincidence point. Moreover, f , g, S and T have a unique common fixed point in X provided both the pairs ( f , S) and (g, T ) are weakly compatible. Remark 5.11. Theorem 5.27 generalizes relevant results contained in Refs. [18,19, 29,46] and some others to GV-fuzzy metric spaces. Our results can also be viewed as a fuzzy version of some metric fixed point theorems contained in [3]. Now, we state a result similar to Theorem 5.26 in GV-fuzzy metric space under fewer conditions. Theorem 5.28. [20] Let f , g, S and T be four self-mappings of a GV-fuzzy metric space (X, M, ∗) satisfying the conditions (i) and (ii) of Theorem 5.27 and for all x, y ∈ X, for some k ∈ (0, 1) and some t > 0, M( f x, gy, kt) ≥ min{M(Sx, Ty,t), M( f x, Sx,t), M(gy, Ty,t), M( f x, Ty,t), (5.14)

M(Sx, gy,t)}.

Then pair ( f , S) as well as (g, T ) have a coincidence point. Moreover, f , g, S and T have a unique common fixed point in X provided both the pairs ( f , S) and (g, T ) are weakly compatible. Example 5.28. [20] Let (X, M, ∗) be a GV-fuzzy metric space wherein X = [0, 1], a ∗ b = ab with t M(x, y,t) = for all t > 0. t + |x − y| Define self-mappings f , g, S and T on X by ( ( 1 if x ∈ [0, 1] ∩ Q 1 f (x) = 1 g(x) = 1 if x ̸∈ [0, 1] ∩ Q, 2 4

if x ∈ [0, 1] ∩ Q if x ̸∈ [0, 1] ∩ Q,

( ( 1 if x ∈ [0, 1] ∩ Q 1 if x ∈ [0, 1] ∩ Q S(x) = and T (x) = 0 if x ̸∈ [0, 1] ∩ Q 0 if x ̸∈ [0, 1] ∩ Q.

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Recently, Choudhury et al. [8] studied coupled fixed point problems, which have attracted much attention. In their paper, they established coupled coincidence point and coupled fixed point results in the context of fuzzy metric spaces.The two mappings considered here are assumed to be compatible. Hadzic type t-norm is used. By an application of the coincidence point theorem in fuzzy metric spaces, a corresponding result is obtained in metric spaces. The main theorem of this paper is illustrated with an example. Their work extended some existing results. Choudhury et al. [8] proved a coupled coincidence point theorem for two mappings in a complete fuzzy metric space which has a partial order defined on it. Let (X, ⪯) be a partially ordered set and F be a mapping from X to itself. The mapping F is said to be non-decreasing if for all x1 , x2 ∈ X, x1 ⪯ x2 implies F(x1 ) ⪯ F(x2 ) and non-increasing if for all x1 , x2 ∈ X, x1 ⪯ x2 implies F(x1 ) ⪯ F(x2 ) [4]. Definition 5.23. [4] Let (X, ⪯) be a partially ordered set and F : X × X → X be a mapping. The mapping F is said to have the mixed monotone property if F is nondecreasing in its first argument and is non-increasing in its second argument, that is, if, for all x1 , x2 ∈ X, x1 ⪯ x2 implies F(x1 , y) ⪯ F(x2 , y), for fixed y ∈ X and, for all y1 , y2 ∈ X, y1 ⪯ y2 implies F(x, y1 ) ⪯ F(x, y2 ), for fixed x ∈ X. Definition 5.24. [27] Let (X, ⪯) be a partially ordered set and F : X × X → X and g : X → X be two mappings. The mapping F is said to have the mixed g-monotone property if F is g-non-decreasing in its first argument and is g-non-increasing in its second argument, that is, if, for all x1 , x2 ∈ X, g(x1 ) ⪯ g(x2 ) implies F(x1 , y) ⪯ F(x2 , y), for fixed y ∈ X and, for all y1 , y2 ∈ X, g(y1 ) ⪯ g(y2 ) implies F(x, y1 ) ⪯ F(x, y2 ), for fixed x ∈ X. Definition 5.25. [4] Let X be a non-empty set. An element (x, y) ∈ X × X is called a coupled fixed point of the mapping F : X × X → X if F(x, y) = x

and

F(y, x) = y

Further, Lakshmikantham and Ciric introduced the concept of coupled coincidence point. Definition 5.26. [27] Let X be a non-empty set. An element (x, y) ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F(x, y) = g(x)

and

F(y, x) = g(y)

The authors defined commuting mappings and used them to obtain a coupled coincidence point theorem in metric spaces. Definition 5.27. [27] Let X be a non-empty set. The mappings F : X × X → X and g : X → X are commuting if for all x, y ∈ X, g(F(x, y)) = F(g(x), g(y)).

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Compatibility between two mappings F : X × X → X and g : X → X, where (X, d) is a metric space was defined in [9]. It is an extension of the commuting condition. Compatibility was used to obtain a coupled coincidence point result in the same work. Definition 5.28. [9] Let (X, d) be a metric space. The mappings F and g where F : X × X → X and g : X → X, where (X, d), are said to be compatible if lim d(g(F(xn , yn )), F(g(xn ), g(yn )) = 0

n→∞

and lim d(g(F(yn , xn )), F(g(yn ), g(xn )) = 0

n→∞

whenever {xn } and {yn } are sequences in X such that lim F(xn , yn ) = lim g(xn ) = x and lim F(xn , yn ) = lim g(yn ) = y for some x, y ∈ X. n→∞

n→∞

n→∞

n→∞

Intuitively we can think that the functions F and g commute in the limit in the situations where the functional values tend to the same point This notion of compatibility was introduced in fuzzy metric spaces by Hu [17]. Definition 5.29. [17] Let (X, M, ∗) be a fuzzy metric space. The mappings F and g where F : X × X → X and g : X → X, where (X, d), are said to be compatible if lim M(g(F(xn , yn )), F(g(xn ), g(yn ),t) = 1

n→∞

and lim M(g(F(yn , xn )), F(g(yn ), g(xn ),t) = 1

n→∞

whenever {xn } and {yn } are sequences in X such that lim F(xn , yn ) = lim g(xn ) = x and lim F(xn , yn ) = lim g(yn ) = y for some x, y ∈ X. n→∞

n→∞

n→∞

n→∞

Given a metric space(X, d), consider the fuzzy metric space (X, M, ∗) constructed in Example 2.3 in Ref. [8]. Then {xn } converges to x in the metric space (X, d) if and only if{xn } converges to x in the fuzzy metric space (X, M, ∗). The equivalence between completeness of (X, d) and (X, M, ∗) was established by George and Veeramani in Result 2.9 of their paper [12]. In the following lemma, we establish that the compatibility in a metric space implies the compatibility in the corresponding fuzzy metric space of Example 2.3 in Ref. [8]. Then {xn }. Lemma 5.16. [8] Let (X, d) be a metric space. If the mappings F and g where F : X × X → X and g : X → X are compatible in (X, d) (according to Definition 5.28), then F and g are also compatible (according to Definition 5.29) in the corresponding fuzzy metric space (X, M, ∗) as described earlier.

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Choudhury et al. [8] used continuous Hadzic type t-norm in their theorem. Definition 5.30. [16] A t-norm ∗ is said to be Hadzic type t-norm if the family {∗ p } p ≥ 0 of its iterates defined foreach s ∈ [0, 1] by ∗0 (s) = 1, ∗ p+1 (s) = ∗(∗ p (s), s)

∀ p≥0

For an example of a non-trivial Hadzic type t-norm, we refer to Ref. [16]. The reason why we use continuous Hadzic type t-norm is that with this choice they [8] can ensure the existence of a function used in the statement of their main theorem. Also the proof depends on certain properties of this t-norm. These points are elaborated in the following. We will require the result of the following lemma to establish our main theorem. The speciality of its proof is that it utilizes equi-continuity of the iterates. Lemma 5.17. [8] Let (X, M, ∗) be a fuzzy metric space with a Hadzic type t-norm ∗ such that M(x, y,t) → 1 as t → ∞, for all x, y ∈ X. If the sequences {xn } and {yn } in X are such that, for all n ≥ 1, t > 0, t t M(xn , xn+1 ,t) ∗ M(yn , yn+1 ,t) ≥ M(xn−1 , xn , ) ∗ M(yn−1 , yn , ) k k where 0 < k < 1, then the sequences {xn } and {yn } are Cauchy sequences. Choudhury et al. [8] will require the following lemma to ensure the existence of the function γ which they use in their theorem below. Lemma 5.18. [16] Let ∗ be a t-norm such that the function c(x) = x ∗ x, x ∈ [0, 1] is right-continuous on an interval [b, 1) for b < 1. Then ∗ is a t-norm of Hadzic type if and only if there exists a sequence {bn }, n ∈ N from the interval (0, 1) of idempotents of ∗ such that lim bn = 1. n→∞

Choudhury et al. [8] main result as follows: Theorem 5.29. Let (X, M, ∗) be a complete fuzzy metric space with a Hadzic type t-norm such that M(x, y,t) → 1 as t → ∞, for all x, y ∈ X. Let ⪯ be a partial order defined on X. Let F : X × X → X and g : X → X be two mappings such that F has mixed g-monotone property and satisfies the following conditions: i. ii. iii. iv.

F(X × X) ⊆ g(X), g is continuous and monotonic increasing, (g, F) is a compatible pair, M(F(x, y), F(u, v), kt) ≥ γ(M(g(x), g(u),t) ∗ M(g(y), g(v),t)),

for all x, y, u, v ∈ X,t > 0 with g(x) ⪯ g(u) and g(y) ⪰ g(v), where 0 < k < 1, γ : [0, 1] → [0, 1] is a continuous function such that γ(a)∗γ(a) ≥ a for each 0 ≤ a ≤ 1. Also suppose that X has the following properties:

Common Fixed-Point Theorems in Fuzzy Metric Spaces

159

a. if a non-decreasing sequence {xn } → x, then xn ⪯ x for all n ≥ 0, b. if a non-decreasing sequence {yn } → y, then yn ⪰ y for all n ≥ 0. If there are x0 , y0 ∈ X such that g(x0 ) ⪯ F(x0 , y0 ), g(y0 ) ⪰ F(y0 , x0 ), then there exist x, y ∈ X such that g(x) = F(x, y) and g(y) = F(y, x), that is, g and F have a coupled coincidence point in X. Note. In the aforementioned theorem, Choudhury et al. [8] have defined a new fuzzy coupled contraction with the help of the function γ. The mapping γ in the statement of the Theorem 5.29 exists for the following reason. Since the t-norm ∗ is continuous Hadzic type, by Lemma 5.18, there exists an increasing sequence {bn } of distinct idempotents of ∗ in (0, 1) with lim bn = 1. Then, γ : [0, 1] → [0, 1] defined as n→∞

γ(s) = bn+1 whenever bn < s ≤ bn+1 , for all n, γ(s) = 1, if s = 1, is a function having the desired properties of γ in Theorem 5.29. Thus the statement of the Theorem 5.29 is meaningful for arbitrary continuous Hadzic type t-norms. The proof of the theorem is different from those of Refs. [17,36] due to the involvement of the mapping γ. Corollary 5.22. [8] Let (X, M, ∗) be a complete fuzzy metric space with a Hadzic type t-norm such that M(x, y,t) → 1 as t → ∞, for all x, y ∈ X. Let ⪯ be a partial order defined on X. Let F : X × X → X and g : X → X be two mappings such that F has mixed g-monotone property and satisfies the following conditions: i. ii. iii. iv.

F(X × X) ⊆ g(X), g is continuous and monotonic increasing, (g, F) is a commuting pair, M(F(x, y), F(u, v), kt) ≥ γ(M(g(x), g(u),t) ∗ M(g(y), g(v),t)),

for all x, y, u, v ∈ X,t > 0 with g(x) ⪯ g(u) and g(y) ⪰ g(v), where 0 < k < 1, γ : [0, 1] → [0, 1] is a continuous function such that γ(a) ∗ γ(a) ≥ a for each 0 ≤ a ≤ 1. Also suppose that X has the following properties: a. if a non-decreasing sequence {xn } → x, then xn ⪯ x for all n ≥ 0, b. if a non-decreasing sequence {yn } → y, then yn ⪰ y for all n ≥ 0. If there are x0 , y0 ∈ X such that g(x0 ) ⪯ F(x0 , y0 ), g(y0 ) ⪰ F(y0 , x0 ), then there exist x, y ∈ X such that g(x) = F(x, y) and g(y) = F(y, x), that is, g and F have a coupled coincidence point in X. Later, by an example, Choudhury et al. [8] will show that Corollary 5.22 is properly contained in Theorem 5.29.

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

The following corollary is a fixed point result. Corollary 5.23. [8] Let (X, ⪯) be a partially ordered set and let (X, M, ∗) be a complete fuzzy metric space with a Hadzic type t-norm such that M(x, y,t) → 1 as t → ∞, for all x, y ∈ X. Let ⪯ be a partial order defined on X. Let F : X ×X → X be a mapping such that F has mixed monotone property and satisfies the following conditions: M(F(x, y), F(u, v), kt) ≥ (M(x, u,t) ∗ M(y, v,t)) for all x, y, u, v ∈ X,t > 0 with x ⪯ u and y ⪰ v, where 0 < k < 1, γ : [0, 1] → [0, 1] is a continuous function such that γ(a) ∗ γ(a) ≥ a for each 0 ≤ a ≤ 1. Also suppose that X has the following properties: a. if a non-decreasing sequence {xn } → x, then xn ⪯ x for all n ≥ 0, b. if a non-decreasing sequence {yn } → y, then yn ⪰ y for all n ≥ 0. If there are x0 , y0 ∈ X such that x0 ⪯ F(x0 , y0 ), y0 ⪰ F(y0 , x0 ), then there exist x, y ∈ X such that x = F(x, y) and y = F(y, x), that is, F have a coupled fixed point in X Example 5.29. [8] Let (X, ⪯) is the partially ordered set with X = [0, 1] and the natural ordering ≤ of the real numbers as the partial ordering ⪯. Let for all t > 0, x, y ∈ X, M(x, y,t) = e−(|x−y|)/t . Let a ∗ b = min{a, b} for all a, b ∈ [0, 1]. Then (X, M, ∗) is a complete fuzzy metric space such that M(x, y,t) → 1 as t → ∞, for all x, y ∈ X. Let the mapping g : X → X be defined as g(x) = x2

for all

x∈X

and the mapping F : X × X → X be defined as F(x, y) =

x2 − y2 2 + . 3 3

Remark 5.12. In Example 5.30, the functions g and F do not commute. Hence Corollary 5.22 can not be applied to this example. This shows that Theorem 5.29 properly contains its Corollary 5.22. Theorem 5.29, and all its corollaries are valid for Hadzic type t-norm like minimum t-norm as is used in Example 5.30. But Theorem 5.29 can not be applied to other cases as for example, when a ∗ b = ab which is not a Hadzic type t-norm.

Common Fixed-Point Theorems in Fuzzy Metric Spaces

5.3

161

CONCLUSION

In 1976, Jungck’s remarkable result [22] promoted the study of common fixed point theorems for commuting mappings in classical metric space. After the appearance of this result, several authors tried to weaken the notion of commuting mappings by introducing several noncommuting mappings satisfying contractive or noncontractive conditions. Therefore, the study of common fixed point for a pair of self-mappings satisfying contractive or noncontractive conditions becomes interesting and practical because even commuting continuous mappings on such well-behaved entities as compact convex sets may fail to have a common fixed point. This theory was well considered in the setting of fuzzy metric spaces and the work on the topic is still going on. This chapter aims to provide the readers a systematic material on the topic so as it could be helpful to illustrate the direction of research over the last four decades up to the most recent contributions.

5.4

ACKNOWLEDGMENTS

The authors thank Dr Moh. Hasan, Department of Mathematics, Jazan University, Jazan, KSA for his contribution towards the completion of this work.

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10. Doric D., Kadelburg Z., Radenovic S., A note on occasionally weakly compatible mappings and common fixed point. Fixed Point Theory, 2012;13(2):475–480. 11. George A., Veeramani, P., On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 1994;64(3):395–399. 12. George A., Veeramani P., On some results of analysis of fuzzy metric spaces. Fuzzy Sets and Systems, 1997;90:365–368. 13. Grabiec M., Fixed points in fuzzy metric spaces. Fuzzy Sets Systems, 1988;27:385–389. 14. Gopal D., Imdad M., Some new common fixed point theorems in fuzzy metric spaces. Annali dell’Universita di Ferrara, 2011;57:303–316. 15. Gopal D., Imdad M., Vetro C., Impact of common property (E.A.) on fixed point theorems in fuzzy metric spaces. Journal of Fixed Point Theory and Applications , 2011;2011:14 pp. 16. Hadzic O., Pap E., Fixed Point Theory in Probabilistic Metric Space. Kluwer Academic Publishers, Dordrecht, 2001. 17. Hu X. Q., Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Journal of Fixed Point Theory and Applications, 2011;2011:8. 18. Imdad M., Ali J., Some common fixed point theorems in fuzzy metric spaces. Mathematical Communications, 2006;11:153–163. 19. Imdad M., Ali J., A general fixed point theorem in fuzzy metric spaces via an implicit functio. Journal of Applied Mathematics and Informatics, 2008;26:591–603. 20. Imdad M., Ali J., Hasan M., Common fixed point theorems in fuzzy metric spaces employing common property (E.A.). Mathematical and Computer Modelling, 2012;55:770– 778. 21. Imdad M., Ali J., Tanveer M., Remarks on some recent metrical fixed point theorems. Applied Mathematics Letters, 2011;24:1165–1169. doi:10.1016/j.aml.2011.01.045. 22. Jungck G., Commuting mappings and fixed points. The American Mathematical Monthly, 1976;83(4):261–263. 23. Jungck G., Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences, 1986;9:771–779. 24. Jungck G., Common fixed points for commuting and compatible maps on compacta. Proceedings of the American Mathematical Society, 1988;103:977–983. 25. Kramosil I., Michalek J., Fuzzy metrics and statistical metric spaces. Kybernetika, 1975;11(5):336–344. 26. Kumar S., Fisher B., A common fixed point theorem in fuzzy metric space using property (E.A) and implicit relation. Thai Journal of Mathematics, 2010;8(3):439–446. 27. Lakshmikantham V., Ciric L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis, 2009;70:4341–4349. 28. Lui W., Wu J., Li Z., Common fixed points of single-valued and multivalued maps. International Journal of Mathematics and Mathematical Sciences, 2005;19:3045–3055. 29. Mihet D., Fixed point theorems in fuzzy metric spaces using property E.A. Nonlinear Analysis, 2010;73:2184–2188. 30. Mishra U., Ranadive A. S., Gopal D., Some fixed points theorems in fuzzy metric spaces. Tamkang Journal of Mathematics, 2008;39(4):309–316. 31. Mishra S. N., Sharma N., Singh S. L., Common fixed points of maps on fuzzy metric spaces. International Journal of Mathematics and Mathematical Sciences, 1994;17(2): 253–258.

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to Fixed 6 Introduction Figure Problems in Fuzzy Metric Spaces 6.1

INTRODUCTION

Geometric and topological properties of the fixed point set have been extensively studied for various aspects in the fixed point theory. For example, in Ref. [7], it was proved that the fixed point set of a quasi-nonexpansive self-map of a metric space is always closed (see Lemma 1.1 on page 984). For more details, see [7,24] and the references therein. Recently, geometric aspects of the fixed point set of a selfoperator have been considered with various forms such as the fixed-circle, fixed-disc and fixed-ellipse problems. The most general form of these problems is the “fixed figure problem”. Briefly, if the fixed point set Fix(T ) = {u ∈ X : T u = u} of a self-map T contains some geometric figure (a circle, an ellipse or a Cassini curve, and so on), then this figure is called the fixed figure of T (a fixed circle, a fixed ellipse, and so on) (see [22] and the references therein). ¨ ur and Tas¸ examined the fixed circle issue in metric space. In Refs. [20,21], Ozg¨ This topic generated much interest recently to fixed point community. One of the motivations for this interest is the profound application of these results in neural network in terms of activation functions (see for instances [23,28]). Considering various advantages of generalized metric over regular metric, the fixed circle problems have been studied in various metric spaces including S-metric [19], rectangular metric [3], partial metric [27], and quasi-metric [4]. Fuzzy metric spaces [8,17] is one of the notable generalizations of regular metric concept. Indeed, fixed point theory of fuzzy metric spaces are more diverse than the regular metric fixed point theory. This is due to the pliability exhibited in the concept of fuzzy metric. But at the same time due to complexion involved in the nature of fuzzy metric, one might need to use or develop new fuzzy mathematical tools to establish new results in this field (see for example [2,13–16,26]). This pursuance of hunt make the new fuzzy results worthwhile. In this chapter, we discuss various aspects of fixed figure problems in fuzzy metric setting taken from [30]. Throughout this chapter, we consider the case Fix(T ) ̸= 0. / Here we quote some basic concepts and results, which will be needed for the development of the present topic.

✻

■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡ ❋✐❣✉#❡ -#♦❜❧❡♠1 ✐♥ ▼❡"#✐❝ ❙♣❛❝❡1

164

DOI: 10.1201/9781003427797-6

Introduction to Fixed Figure Problems in Fuzzy Metric Spaces

165

Let (X , d) be a metric space and define u ∗ v = uv for all u, v ∈ [0, 1]. Let Md be the function on X × X × (0, ∞) defined by Md (u, v, τ) =

τ . τ + d (u, v)

(6.1)

Then (X , Md , ∗) is a fuzzy metric space [8]. Md is called the standard fuzzy metric induced by d. Let X = (0, ∞) and u ∗ v = uv for all u, v ∈ [0, 1]. Two of the well-known fuzzy metric examples on (0, ∞) are defined by min{u, v} max{u, v}

(6.2)

min{u, v} + τ , max{u, v} + τ

(6.3)

M (u, v, τ) = and M (u, v, τ) =

for all u, v ∈ X and for all τ > 0. These fuzzy metrics have various advantages in front of classical metrics in the evaluation of images filtering process (for more details see [11,12]). Moreover, it has been pointed out in Ref. [8] that there exists no τ metric on X satisfying M(u, v, τ) = τ+d(u,v) , where M(u, v, τ) is defined by (2). The fuzzy metric space (X , M , ∗) is said to be non-Archimedean or strong if it satisfies the following condition: M (u, v, τ) ≥ M (u, w, τ) ∗ M(w, v, τ), for each u, v, w ∈ X and τ > 0. Example 6.1. Take X = N and define the fuzzy set M on X × X by M (u, v, τ) = 1 if u = v and M (u, v, τ) = 12 if u ̸= v, then (X , M , ∗m ) is a strong GV -fuzzy metric space. Remark 6.1. It is interesting to note that the example of this type of fuzzy metric is very useful in showing that the fuzzy distances are not necessarily equivalent to classical distances (see Example 6.18 given in page 14). The readers are referred to Ref. [1,8] for the definitions of a complete fuzzy metric space and an upper semi-continuous function. Theorem 6.1. [1] Let (X , M , ∗) be a complete fuzzy metric space with ∗ is continuous and Archimedean, T : X → X be a self-mapping, ϕ : X → [0, 1] be such that ϕ is non trivial (i.e. u ∈ X such that ϕ(u) ̸= 0) and upper semi-continuous function. Assume that M (u, T u, τ) ∗ ϕ(T u) ≥ ϕ(u) (6.4) for all u ∈ X and τ > 0. Then T has a fixed point in X .

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THE FIXED-CIRCLE PROBLEM ON FUZZY METRIC SPACES

We begin with following: Definition 6.1. Let (X , M , ∗) be a fuzzy metric space and let u0 ∈ X , 0 < r < 1, τ > 0. We define the circle of center u0 , radius r and parameter τ as Cr,τ (u0 ) := {u ∈ X : M (u0 , u, τ) = 1 − r}. For a self-mapping T : X → X , if T u = u for all u ∈ Cr,τ (u0 ) or Cr,τ (u0 ) ⊂ Fix(T ) then, we call the circle Cr,τ (u0 ) as a fixed circle of T . Example 6.2. Let X = (0, ∞), u ∗ v = uv for all u, v ∈ [0, 1] and let M (u, v, τ) be as in Equation (6.2). Consider the mapping T : X → X defined by  1 − u if u ∈ (0, 1),        u2   if u ∈ [1, 43 ) ∪ ( 43 , 2]   2 T u= 4  if u = 34 ,  3        6 if u ∈ (2, ∞)    Then the mapping T fixes the circle C 1 ,τ (1) = 12 , 2 . However, T does not fix the 2 3 4 circle C 1 ,τ (1) = 4 , 3 . Although the point u = 43 is a fixed point of the mapping 4  T . Note that Fix(T ) = 12 , 34 , 2, 6 . Let (X , M , ∗) be a fuzzy metric space and Cr,τ (u0 ) be a circle on X . Define a self-mapping as  u if u ∈ Cr,τ (u0 )    T u= . (6.5) u0 , otherwise    Clearly, T fixes the circle Cr,τ (u0 ). Considering the identity map IX , defined by IX u = u for each u ∈ X , together with the map T defined in Equation (6.5), we deduce that there exist at least two mappings that fix a given circle. On the other hand, there are self-mappings having fixed points but not fixed-circles. For example, for such a mapping, let us consider the set of complex numbers C with the usual metric d and let Md (u, v, τ) be the fuzzy metric defined in Equation (6.1). Define the self-mapping T by  u + 1 if |u| < 1    . Tu= 1 if |u| ≥ 1    u

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167

T has two fixed points −1 and 1. Although these fixed√points lie on each of the  r2 τ 2 −(1−r)2 rτ , where a = ± and τ ≥ 1−r circles Cr,τ (ai) = u ∈ C : |u − ai| = 1−r 1−r r , the other points of such circle are not fixed points of T . That is, T has no any fixedcircle but has two fixed points. For τ = 1 and r = 12 , we have the circle C 1 ,1 (0) = 2 {u ∈ C : |u| = 1} and observe that T maps the circle C 1 ,1 (0) onto itself. 2 Now, we want to determine some necessary conditions to ensure the fixed point set of a self-mapping contains a circle Cr,τ (u0 ) by the use of the given parameters u0 and τ. Theorem 6.2. Let (X , M , ∗) be a fuzzy metric space with ∗ as Archimedean, T : X → X be a self-mapping, and u0 ∈ X , τ > 0. Define the map ϕτ,u0 : X → (0, 1] by ϕτ,u0 (u) = M (u, u0 , τ), for all u ∈ X . Assume that there exists a number r with 0 < r < 1 such that P1) M (u, T u, τ) ∗ ϕτ,u0 (T u) ≥ ϕτ,u0 (u), P2) M (T u, u0 , τ) ≤ 1 − r, for each u ∈ Cr,τ (u0 ). Then, the set Fix(T ) contains the circle Cr,τ (u0 ), that is, Cr,τ (u0 ) is a fixed circle of T . Proof. Let us choose a number r satisfying the condition (P2). Consider the circle Cr,τ (u0 ) and let u ∈ Cr,τ (u0 ) be an arbitrary point. Then by (P1), (P2), and the monotonicity of ∗, we can write (1 − r) ∗ M (T u, u, τ) ≥ M (Tu, u0 , τ) ∗ M (T u, u, τ) ≥ M (u, u0 , τ) (by P1) = 1 − r, i.e. (1 − r) ∗ M (T u, u, τ) ≥ 1 − r. Since ♢ is Archimedean, therefore, we must have M (T u, u, τ) = 1, and this implies T u = u by (GV2). Hence, we obtain T u = u for each u ∈ Cr,τ (u0 ). This shows that Cr,τ (u0 ) ⊂ Fix(T ), that is, the circle Cr,τ (u0 ) is a fixed circle of the self-map T . Remark 6.2. 1. The converse statement of Theorem 6.2 is also true because of the condition (GV2) of Definition 1.2 and the Archimedean property of ∗. 2. In view of Example 6.2, Theorem 6.2 is stronger than Theorem 6.1 (in the sense that Theorem 6.2 can be used to produce self-mappings which fixes a given circle).

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However, Theorem 6.2 is a special case of Theorem 6.1 for the cases where the circle Cr,τ (u0 ) has only one element. 3. The condition (P1) of Theorem 6.2 guarantees that T u is not in the interior of the circle Cr,τ (u0 ) for each u ∈ Cr,τ (u0 ), while the condition (P2) guarantees that T u is not in the exterior of the circle Cr,τ (u0 ) for each u ∈ Cr,τ (u0 ). Combining these, we obtain T u ∈ Cr,τ (u0 ) for each u ∈ Cr,τ (u0 ). We note that the circle Cr,τ (u0 ) need not to be fixed even if T (Cr,τ (u0 )) = Cr,τ (u0 ) (see Example 6.5). 4. If the conditions (P1) and (P2) are satisfied by T for all x ∈ X , then it is clear from the proof of Theorem 6.2 that T u = u for each u ∈ X , that is, we have T = IX , the identity map on X . Now, we give some illustrative examples. Example 6.3. Let X = (0, ∞) , u ∗ v = uv for all u, v ∈ [0, 1] and M (u, v, τ) be as in Equation (6.3). Consider the mapping T : X → X defined by  u if u ≥ 14    . Tu = 1 if 0 < u < 14   4  Let u0 = 1, τ = 1. Then, T satisfies the conditions of Theorem 6.2 for r = 81 and  
we have the fixed circle C 1 ,1 (1) = 34 , 97 . Clearly, Fix(T ) = 41 , ∞ , and C 1 ,1 (1) ⊂ 8 8 Fix(T ). Now, let u0 = 2, τ = 1. Then, T satisfies the conditions of Theorem 6.2 for r = 16  and we get another fixed circle C 1 ,1 (2) = 32 , 13 of T . 5 6

Example 6.4. Let X =



 3 , 1, 2, 4 4

and (X , M , ∗) is same as in Example 6.2. Consider the self-mapping T : X → X defined by  3 if u ∈ { 43 , 4}    4 T u= . 2u otherwise    Then, T fulfils all the requirements of Theorem 6.2 only for the circle C 1 ,τ (1) = 4 3 4 . Hence, T has a unique fixed circle with one element. Example 6.5. Let X =



 1 , 1, 2, 3, 4 , 2

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and (X , M , ∗) is same as in Example 6.2. Consider the self-mapping T : X → X defined by  3 if u ∈ {1, 3, 4}    . T u= 1 if u ∈ { 12 , 2}    u    Consider the circle C 1 ,τ (1) = 12 , 2 . We have T C 1 ,τ (1) = C 1 ,τ (1). But the 2 2 2 points of the circle C 1 ,τ (1) is not fixed by T . 2

The following examples illustrate the necessity of the conditions (P1) and (P2) in Theorem 6.2. Example 6.6. Let X = {1, 2, 3} , and (X , M , ∗) is same as in Example 6.2. Consider the self-mapping T : X → X defined by  u + 1 if u ∈ {1, 2}    . T u= 1 if u = 3    Let u0 = 1 and τ > 0. Then, T satisfies the condition (P1) of Theorem 6.2 for the circle C 2 ,τ (1) = {3}. Indeed, for u = 3, we have 3

M (u, T u, τ) ∗ M (T u, u0 , τ) = M (3, 1, τ) ∗ M (1, 1, τ) =

1 3

1 ≥ M (u, u0 , τ) = M (3, 1, τ) = . 3 But, T does not satisfy the condition (P2) since we have M (T u, u0 , τ) = M (1, 1, τ) = 1 > 1 −

2 1 = . 3 3

Notice that T has no any fixed circle (resp. fixed point). This example shows the necessity of the condition (P2). Example 6.7. Let (X , M , ∗) and the self mapping T be same as in Example 6.6. Choose u0 = 2, r = 13 and τ > 0. Then (P2) holds for the circle C 1 ,τ (2) = {3}. 3 Indeed, for u = 3, we have M (T u, u0 , τ) = M (1, 2, τ) =

1 1 2 ≤ 1− = . 2 3 3

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However, for u = 3, we have 1 1 1 M (T u, u0 , τ) ∗ M (T u, u, τ) = M (1, 2, τ) ∗ M (1, 3, τ) = . = 2 3 6 2 < M (u, u0 , τ) = M (3, 2, τ) = , 3 i.e. (P1) is not satisfied by T . Examples 6.6 and 6.7 indicate that Theorem 6.2 characterizes the existence of fixed circles by means of the conditions (P1) and (P2). The next example elucidates that Theorem 6.2 may not true when ♢ is the minimum t-norm. Example 6.8. Take X = {1, 2, 3, . . .} and define the fuzzy set M on X × X by M (u, v, τ) = 1 if u = v and M (u, v, τ) = 13 if u ̸= v, then (X , M , ∗m ) is a strong GV fuzzy metric space. Let T be defined by T u = u + 1, ∀u ∈ X . Then the conditions (P1) and (P2) are satisfied for the circle C 2 ,τ (1) = {2, 3, . . .} but T does not fixes the 3 circle. For any circle with center u0 ̸= 1, the condition (P2) is not satisfied. Because, for the point u = u0 − 1, we have T u = u0 and so M (Tu, u0 , τ) = M (u0 , u0 , τ) = 1. In fact, T is a fixed point free mapping. The following example justifies superiority of Theorem 6.2 over Abbasi Theorem 3.1 [1]. Example 6.9. Let X = {un = (1 − 2uv 1+u+v−uv

1 n+1 ) :

n ∈ N} ∪ {1} and u ∗ 1 v = 2

for all u, v ∈ [0, 1], consider a fuzzy set M on X × X × (0, ∞) by: M (1, 1, τ) = 1 = M (un , un , τ),

for each n ∈ N, and

1 1 1 M (1, , τ) = M ( , 1, τ) = , 2 2 2 1 M (1, un , τ) = M (un , 1, τ) = , 4

for n = 2, 3, . . ., M (u1 , u3 , τ) = M (u3 , u1 , τ) = · · · = M (u2 , u3 , τ) = M (u3 , u2 , τ) 1 = M (u2 , x4 , τ) = · · · = . 4 Then (X , M , ∗) is a fuzzy metric space. Consider the circle C 1 ,τ ( 12 ) = {1} in X 2 and designate T : X → X by T un = un+1 for all n ∈ N and T 1 = 1.

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Then T fulfils all the requirements of Theorem 6.2, and consequently T fixes the circle C 1 ,τ ( 12 ). 2 However, T does not satisfy the Abbasi condition (3.1) (see [1] page 933). Suppose to the contrary that T satisfy the Abbasi condition (3.1), then lim sup M (un , T un , τ) ∗ lim sup ϕ(T un ) ≥ lim sup ϕ(un ) n→∞

i.e.

or

n→∞

n→∞

1 lim sup ϕ(un+1 ) ≥ ϕ(un ) 4 n→∞ 1 ∗ lim sup ϕ(un+1 ) ≥ ϕ(un ). 2 n→∞

Since ϕ is u.s.c. and hence k = lim supn→∞ ϕ(un+1 ) ≤ ϕ(u), where limn→∞ un = u. So the above inequality reduces to 1 ∗k ≥ k 4 (or 21 ∗ k ≥ k) which is a contradiction to the Archimedean condition of ♢. Obviously, we have Fix(T ) = X for the identity map IX . Now, we give a characterization of IX in the setting of a fuzzy metric space. We begin with the following definition. Definition 6.2. Let (X , M , ∗) be a fuzzy metric space. The fuzzy metric M is called Mh -triangular if there exists some h > 1 such that       1 1 1 h (6.6) −1 ≤ −1 +h −1 M (u, w, τ) M (u, v, τ) M (v, w, τ) for all u, v, w ∈ X such that v ̸= w and τ > 0. Example 6.10. Let X = {1, 2, 3}, u ∗ v = uv for all u, v ∈ [0, 1] and M (u, v, τ) = min{u,v} max{u,v} for all u, v ∈ X and for all τ > 0. Clearly, (X , M , ∗) is Mh -triangular for h = 4. Example 6.11. The fuzzy metric defined in Equation (6.1) is the Mh -triangular if d is a discrete metric. Remark 6.3. The fuzzy metrics M given in Examples 2.13 and 2.14 are actually strong fuzzy metrics. So, the following question arises naturally: Question 1: Does there exist a non-strong Mh -triangular fuzzy metric?

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Now, we are ready to state and prove our next result of this section. Theorem 6.3. Let (X , M , ∗) be a Mh -triangular fuzzy metric space and u0 is a point of X . Then, the self-mapping T : X → X satisfies the condition       1 1 1 −1 ≥ −1 +h −1 (6.7) M (T u, u0 , τ) M (u, u0 , τ) M (u, T u, τ) for all u ∈ X , τ > 0 and some h > 1 if and only if T = IX . Proof. Let u be any arbitrary point of X and assume that Tu ̸= u. By (GV2), we have M (u, T u, τ) ̸= 1. Then, using (6.7) and Mh -triangularity of fuzzy metric M , we get       1 1 1 −1 ≥ −1 +h −1 M (T u, u0 , τ) M (u, u0 , τ) M (u, T u, τ)   1 ≥h −1 , M (u0 , Tu, τ) a contradiction as h > 1. This means that T u = u for all u ∈ X , and so T = IX . Clearly, the identity map IX satisfies the condition (6.7). Remark 6.4. In Theorem 6.2, the fixed circle Cr,τ (u0 ) is not necessarily unique. Example 6.3 illustrates this situation. However, there are cases where the fixed circle is unique (see Example 6.4). Hence, the investigation of some uniqueness conditions for fixed circles of self-mappings appears a natural problem. In the following theorem, we give a uniqueness condition for fixed circles. Theorem 6.4. Let (X , M , ∗) be a fuzzy metric space and T : X → X be a selfmapping such that Cr,τ (u0 ) ⊂ Fix(T ). If the contractive condition M (u, T v, τ) > M (u, v, τ)

(6.8)

is satisfied for all u ∈ Cr,τ (u0 ) and v ∈ X \Cr,τ (u0 ) then Cr,τ (u0 ) is the unique fixed circle of T , that is, we have Fix(T ) = Cr,τ (u0 ). Proof. Assume that there exists a point v ∈ Fix(T ) \Cr,τ (u0 ). For any u ∈ Cr,τ (u0 ), there exists λ > 0 such that 0 < M (u, v, λ ) < 1. Using the condition (6.8), we have M (u, v, λ ) = M (u, T v, λ ) > M (u, v, λ ), a contradiction. This shows that Fix(T ) = Cr,τ (u0 ), that is, Cr,τ (u0 ) is the unique fixed circle.

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Example 6.12. Let X =



 3 4 , , 1, 2 4 5

and (X , M , ∗) is same as in Example 6.2. Consider the self-mapping T : X → X defined by T u = 2. 3 4 For u = 2 ∈ C 1 ,τ (1) and v ∈ 4 , 5 , 1 , we have 2

v M (u, T v, τ) = M (2, 2, τ) = 1 > M (u, v, τ) = M (2, v, τ) = . 2 Then, the condition (6.8) is satisfied, and hence the circle C 1 ,τ (1) is the unique fixed 2 circle of T .

6.3

THE FIXED-CASSINI CURVE PROBLEM ON FUZZY METRIC SPACES

Cassini curves have profound applications in various scientific disciplines such as nuclear physics, biosciences, and computational sciences, besides modeling human. On the other hand, fuzzy metric is a useful tool to describe imprecise information and process in terms of “degree of closedness”given by fuzzy metric M . These facts compelled us to look into the possibility of having fuzzy type Cassini Curves. In this section, we examine the same. Let (X , M , ∗) be a fuzzy metric space and let u1 , u2 ∈ X , 0 < γ < 1, τ > 0. We call the set Cγ,τ (u1 , u2 ) defined by Cγ,τ (u1 , u2 ) := {u ∈ X : M (u1 , u, τ) ∗ M (u2 , u, τ) = 1 − γ} as a Cassini curve on X . Then, the set Cγ (u1 , u2 ) is called a fixed Cassini curve of the self-mapping T : X −→ X if T u = u for all u ∈ Cγ,τ (u1 , u2 ) or Cγ,τ (u1 , u2 ) ⊂ Fix(T ). Theorem 6.5. Let (X , M , ♢) be a fuzzy metric space with ∗ as Archimedean, T : X → X be a self-mapping, and u1 , u2 ∈ X , τ > 0. Define the map ϕτ,u1 ,u2 : X → (0, 1] by ϕτ,u1 ,u2 (u) = M (u, u1 , τ) ∗ M (u, u2 , τ),

(6.9)

for all u ∈ X . Assume that there exists a number γ with 0 < γ < 1 such that C1) M (u, T u, τ) ∗ ϕτ,u1 ,u2 (T u) ≥ ϕτ,u1 ,u2 (u) and C2) ϕτ,u1 ,u2 (T u) ≤ 1 − γ, for each u ∈ Cγ,τ (u1 , u2 ). Then, the set Cγ,τ (u1 , u2 ) is a fixed Cassini curve of T .

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Proof. Choose a number γ satisfying the condition (C2) and consider the Cassini curve Cγ,τ (u1 , u2 ). Let u ∈ Cγ,τ (u1 , u2 ) be an arbitrary point. Using (C1) and (6.9), we have M (u, T u, τ)∗(M (T u, u1 , τ) ∗ M (T u, u2 , τ)) ≥ M (u, u1 , τ)∗M (u, u2 , τ) = 1−γ. (6.10) Then, by (C2), monotonicity of ∗ and (6.10), we obtain (1 − γ) ∗ M (u, T u, τ) ≥ (M (T u, u1 , τ) ∗ M (T u, u2 , τ)) ∗ M (u, T u, τ) ≥ 1 − γ, and so

(1 − γ) ∗ M (u, T u, τ) ≥ 1 − γ.

Since ∗ is Archimedean, we must have M (T u, u, τ) = 1 =⇒ T u = u, by (GV2). Therefore, we obtain T u = u for each u ∈ Cγ,τ (u1 , u2 ). Consequently, the self-map T fixes the Cassini curve Cγ,τ (u1 , u2 ). Now, we give two examples in which (C1) is satisfied but not (C2) and vice versa. Consequently, the Cassini curve is not fixed. Example 6.13. Let (X , M , ∗) and the self mapping T are same as in Example 6.6. Then, T satisfies the condition (C2) of Theorem 6.5 but T does not satisfy the condition (C1) for the Cassini curve C 1 ,τ (1, 2) = {1, 2}. Notice that T has no any 2 fixed point. This example shows the necessity of the condition (C1).  Example 6.14. Let X = 34 , 1, 98 , u ∗ v = uv for all u, v ∈ [0, 1] and consider the fuzzy metric defined in Equation (6.2). Consider theself-map T defined by T u = 1 for each u ∈ X and the Cassini curve C 1 ,τ (1, 89 ) = 43 . Then, T satisfies the con2 dition (C1) of Theorem 6.5 but does not satisfy the condition (C2) for the Cassini curve C 1 ,τ (1, 89 ). Clearly, the Cassini curve C 1 ,τ (1, 89 ) is not fixed by T . This exam2 2 ple shows the necessity of the condition (C2). Remark 6.5. Examples 6.13 and 6.14 indicate that Theorem 6.5 characterizes the existence of fixed Cassini curves by means of the conditions (C1) and (C2). Example 6.15. Let (X , M , ∗) be the fuzzy metric space defined in Example 6.2. Define the mapping T : X → X by T u = |u − 1| + |u − 2| + u − 1.

(6.11)

1 2

Then, for γ = and u1 = 1, u2 = 2, T satisfies the conditions (C1) and (C2) of Theorem 6.5 for the Cassini curve C 1 ,τ (1, 2) = [1, 2]. Clearly, we have Fix(T ) = 2 [1, 2] and the set C 1 ,τ (1, 2) is a fixed Cassini curve of T . 2

Remark 6.6. The uniqueness condition (6.8) given in Theorem 6.4 can also be used for a fixed Cassini curve, in general, for any geometric figure contained in the set Fix(T ).

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6.4

FIXED POINT SETS OF FUZZY QUASI-NONEXPANSIVE MAPS

In [7], Chaoha et al. established some interesting results concerning the topological properties of the fixed point sets of a quasi-nonexpansive mapping. Precisely, we quote the following amazing results: 1. (Lemma 1.1 on page 984 [7]) The fixed point set of a quasi-nonexpansive selfmap of a metric space is always closed. 2. (Theorem 2.1 on page 985 [7]) Let A be a nonempty subset of a CAT (0) space (X , d). Then there exists a continuous map f : X → X such that F( f ) = A. Motivated from these results, we launch quasi-nonexpansive self-mapping in fuzzy setting and establish a general result for the fixed point set of a fuzzy quasinonexpansive self-map. Recall that an open ball Br,τ (u0 ) with centre u0 ∈ X , radius r (0 < r < 1) and parameter τ > 0 is defined as Br,τ (u0 ) = {u ∈ X : M (u0 , u, τ) > 1 − r}, and a closed ball Br,τ [u0 ] is defined as Br,τ [u0 ] = {u ∈ X : M (u0 , u, τ) ≥ 1 − r}. The sequence {xn } is called convergent and converges to x if, for each ε ∈ (0, 1) and each t > 0, there exists n0 ∈ N such that M(xn , x,t) > 1 − ε, for all n ≥ n0 . For more details, one can see [8]. Definition 6.3. Let (X , M , ∗) be a fuzzy metric space. The mapping T : X → X is called fuzzy quasi-nonexpansive if M (T u, p, τ) ≥ M (u, p, τ) ,

(6.12)

for each u ∈ X and p ∈ Fix(T ). Theorem 6.6. Let (X , M , ∗) be a fuzzy metric space and T : X → X be a fuzzy quasi-nonexpansive self-map. Then the fixed point set Fix(T ) of T is closed. Proof. Assume that T is a fuzzy quasi-nonexpansive self-map. Let (un ) be a sequence in the set Fix(T ) converging to u. Then for 0 < r < 1 and each τ > 0 there exists n0 ∈ N such that un ∈ Br,τ (u0 ) for all n ≥ n0 . It follows that M (un , u, τ) > 1−r and hence 1 − M (un , u, τ) < r. By the definition of a fuzzy quasi-nonexpansive self map, we have M (u, un , τ) ≤ M (T u, un , τ) and using this, we find 1 − M (T u, un , τ) ≤ 1 − M (u, un , τ) < r and so 1 − M (T u, un , τ) < r.

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This implies M (T u, un , τ) → 1 as n → ∞. Then from (GV4), we can write   τ τ ∗ M un , T u, . M (u, T u, τ) ≥ M u, un , 2 2 Letting n → ∞, we get M (u, T u, τ) ≥ 1 and so, M (u, T u, τ) = 1, by (GV2). This last equality implies T u = u. Consequently, u ∈ Fix(T ). Example 6.16. Take X = (0, ∞), u∗v = uv for all u, v ∈ [0, 1] and consider the fuzzy metric defined in Equation (6.3). Define the self map T as  u; u ≥ u0 T u= . (6.13) u0 ; 0 < u < u0 Then, it is easy to check that T is a fuzzy quasi-nonexpansive self-map. Clearly, we have Fix(T ) = [u0 , ∞), and this is a closed set. This closed set can contain several geometric figures. For example, it is easy to see that the set Fix(T ) = [u0 , ∞) contains all of the circles C 1 ,τ (u0 + t) = 2u0 + 2t + τ, u0 +t−τ (where t is chosen such 2 2 that t − τ ≥ u0 ) and Cassini curves C 1 ,τ (u0 , u0 + t) = [u0 , u0 + t] (where t is chosen 2 such that t = u0 + τ). Example 6.17. Take X = R, u ∗ v = uv for all u, v ∈ [0, 1] and consider the fuzzy metric defined in Equation (6.1) where d (u, v) = |u − v|. Consider the mapping T : X → X defined by  u; u ≥ 0 T u= , (6.14) αu; u < 0 where 0 < α < 1. Then, T is a fuzzy quasi-nonexpansive self-map and we have Fix(T ) = [0, ∞). On the other hand, consider the mapping S : X → X defined by  5u; u > 0 Su= . u; u ≤ 0 For any p ∈ Fix(S ) = (−∞, 0] and u ∈ (0, ∞), we have M (T u, p, τ) = M (5u, p, τ) =

τ τ < M (u, p, τ) = . τ + |5u − p| τ + |u − p|

That is, S is not a fuzzy quasi-nonexpansive map. However, the fixed point set Fix(S ) is closed. This example shows that the converse statement of Theorem 6.6 does not hold in general.

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Example 6.18. Let X = {1, 2, 3} and define the fuzzy set M on X × X by M (u, v, τ) = 1 if u = v and M (u, v, τ) = 21 if u ̸= v, then (X , M , ∗m ) is a strong GV fuzzy metric space. Now consider the self-mapping T as T 1 = 1, T 2 = 3, T 3 = 3. Then T is a fuzzy quasi-nonexpansive and we have Fix(T ) = {1, 3} which is closed. However, the same mapping T fails to be fuzzy quasi-nonexpansive (at p = 1 τ . and u = 2) if we consider the standard fuzzy metric i.e. M(u, v, τ) = τ+d(u,v) Now, we investigate the case in which the fixed point set of a fuzzy quasinonexpansive map contains the closed ball Bρ,τ [p] for a given fixed point p and a chosen parameter τ > 0. Theorem 6.7. Let (X , M , ∗) be a fuzzy metric space, T : X → X be a fuzzy quasi-nonexpansive self-map and p ∈ Fix(T ). Suppose that there exists a parameter τ > 0 such that D1) M (u, T u, τ) ∗ M (T u, p, τ) < (1 − ρ) ∗ ρ, for each u ∈ X \ Fix(T ) where the number ρ is defined as ρ := inf {M (T u, u, τ) : u ∈ X \ Fix(T )} .

(6.15)

Then we have Bρ,τ [p] ⊂ Fix(T ). Proof. Let v ∈ Bρ,τ [p] be an arbitrary point. Conversely, assume that v ∈ / Fix(T ). Then by the definition of the number ρ, we can write ρ ≤ M (T v, v, τ).

(6.16)

Since T is a fuzzy quasi-nonexpansive self-map and p ∈ Fix(T ), we obtain M (T v, p, τ) ≥ M (v, p, τ) ≥ 1 − ρ.

(6.17)

This means that T v ∈ Bρ,τ [p]. Then, by (6.16), (6.17) and the monotonicity of ∗, we get M (T v, p, τ) ∗ M (v, T v, τ) ≥ M (v, p, τ) ∗ M (v, T v, τ) ≥ (1 − ρ) ∗ ρ, a contradiction with the condition (D1). Hence, we have T v = v for each v ∈ Bρ,τ [p]. That is, the closed ball Bρ,τ [p] is contained in the set Fix(T ). Example 6.19. Consider the fuzzy metric space used in Example 6.16 and the selfmap T defined in (6.13) for u0 = 2. For a number τ > 0, we have ρ

= inf {M (T u, u, τ) : 0 < u < 2} = inf {M (2, u, τ) : 0 < u < 2}   u+τ = inf :0 0 such that M (u, T u, τ) ∗ (M (T u, p1 , τ) ∗ M (T u, p2 , τ)) < ρ ∗ (1 − ρ) ,

(6.18)

for each u ∈ X \ Fix(T ) where the number ρ is defined as in (6.15). Then we have Cρ,τ [p1 , p2 ] ⊂ Fix(T ). Proof. Let τ > 0 be chosen such that the condition (6.18) is satisfied and v ∈ Cρ,τ [p1 , p2 ] be an arbitrary point. Conversely, assume that v ∈ / Fix(T ). Then by the definition of the number ρ, we can write ρ ≤ M (T v, v, τ).

(6.19)

Since T is a fuzzy quasi-nonexpansive self-map and p1 , p2 ∈ Fix(T ), we obtain M (T v, p1 , τ) ≥ M (v, p1 , τ), and

M (T v, p2 , τ) ≥ M (v, p2 , τ).

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By the monotonicity of ∗, we obtain M (T v, p1 , τ) ∗ M (T v, p2 , τ) ≥ M (v, p1 , τ) ∗ M (v, p2 , τ) ≥ (1 − ρ) .

(6.20)

Then, by (6.19), (6.20) and the monotonicity of ∗, we get M (u, T u, τ) ∗ (M (T v, p1 , τ) ∗ M (T v, p2 , τ)) ≥ ρ ∗ (1 − ρ) , a contradiction with the condition (6.18). Hence, we have T v = v for each v ∈ Cρ,τ [p1 , p2 ]. That is, the set Cρ,τ [p1 , p2 ] is contained in the set Fix(T ). Example 6.20. Let the fuzzy metric space and the self-map T are the same as in Example 6.19. Let us choose the fixed points p1 = 5 and p2 = 6. Then, for τ = 1 and u ∈ (0, 2), we have ρ = 31 and M (u, T u, τ) ∗ (M (T u, p1 , τ) ∗ M (T u, p2 , τ)) = M (u, 2, 1) ∗ (M (2, 5, 1) ∗ M (2, 6, 1)) u+1 2+1 2+1 u+1 = = 2+1 5+1 6+1 14 2+1 3 < = 14 14 1 1 2 = . < 1− 3 3 9 Hence, the condition (6.18) is satisfied by T . Clearly, the set C 1 ,1 [5, 6] = 3 h√ i √ 28 − 1, 63 − 1 is contained in the fixed point set Fix(T ) = [2, ∞) . Remark 6.8. The converse statement of Theorem 6.8 does not hold in general. For an instance, consider Example 6.20. If we choose τ = 2, then we have ρ = 21 , and
it is easy to check that the condition (6.18) is satisfied only for each u ∈ 0, 23 . h√ i √ However, the set C 1 ,2 [5, 6] = 28 − 2, 112 − 2 is contained in the fixed point 2 set Fix(T ) = [2, ∞) .

6.5

CONCLUSION AND FUTURE SCOPE

The fuzzy metric fixed point theory has various advantages over the regular metric fixed point theory. This is due to the pliancy which the fuzzy concepts inherently possess. Even then it is not easy to translate the classical metric contractions and corresponding fixed point theorems in fuzzy setting. Such issues are discussed in Refs. [9,10,13,26]. Also, it is well known that Caristi’s [6] fixed point theorem is considered as one of the most beautiful extensions of Banach contraction theorem which also characterizes metric completeness. The Caristi’s fixed point theorem is extended

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by Abbasi et al. in the setting of fuzzy metric spaces. Most recently, an interesting ¨ ur and Tas¸ [20,21] which also have apidea of fixed circle was introduced by Ozg¨ plications in the area of neural network in terms of activation functions. Pursuing this approach of research, we introduce the notion of fixed circle in fuzzy setting and then utilize the ideas of Archimedean t-norm and Mh -triangular fuzzy metric to obtain fixed circle theorems. Moreover, we present a result which prescribed that the fixed point set of fuzzy quasi-nonexpansive mapping is always closed. Our results could be considered as an extension of fixed circle theory and quasi-nonexpansive mapping in the setting of fuzzy metric spaces. It is known that theoretical fixed point results are important for the study of artificial neural networks. Some fixed point theorems (for instance, Brouwer’s fixed point theorem and Banach fixed point theorem) have been intensively used (for example, see [18,29]). On the other hand, activation functions play a significant role to design a neural network. In Ref. [18], globally Lipschitzian activation functions were used in the study of global exponential stability of delayed cellular neural networks. In Ref. [29], the following activation function was used in the numerical example: T u=

1 (|u + 1| − |u − 1|) . 2

Let X = R, a ∗ b = ab for all a, b ∈ [0, 1] and consider the fuzzy metric defined in Equation (6.1) with the usual metric d (x, y) = |x − y|. Then, it is easy to see that T is a fuzzy quasi-nonexpansive self-map, and we have Fix(T ) = [−1, 1]. In addition, we note that the fuzzy quasi-nonexpansive self-map T , defined in Equation (6.14), is one of the most popular activation functions used in the neural networks. This function known as the Parametrized ReLU function. For the case α = 0.01, the corresponding function T is known as the Leaky ReLU function (see [5,25] for more details about the frequently used activation functions). These examples show the effectiveness of the obtained results. Before closing this chapter, we pose the following question:

✻ ✻

Question 2: Is it possible to generalize Theorem 2.1 of [7] in frame work of fuzzy metric?

■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ ■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ REFERENCES ❋✐❣✉#❡ -#♦❜❧❡♠1 ✐♥ ❋✉③③② 1. Abbasi N., Golshan H. M., Caristi’s fixed point theorem and its equivalences in fuzzy ❋✐❣✉#❡ ✐♥ ❋✉③③② metric spaces. Kybernetika, 2016;52:929–942. ▼❡"#✐❝ -#♦❜❧❡♠1 ❙♣❛❝❡1 2. Abbas M., Ali B., Romaguera S., Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat, 2015;29(6):1217– 1222. ¨ ur N. Y., Mlaiki N., Fixed-discs in rectangular metric spaces. Sym3. Aydi H., Tas¸ N., Ozg¨ metry, 2019;11(2):294. ¨ ur N. Y., Mlaiki N., Fixed discs in Quasi-metric spaces. Fixed Point 4. Aydi H., Tas¸ N., Ozg¨ Theory, 2021;22(1):59–74. 5. Calin O., Activation functions. In: Deep Learning Architectures. Springer Series in the Data Sciences. Springer, Cham, 2020.

▼❡"#✐❝ ❙♣❛❝❡1

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6. Caristi J., Fixed point theorem for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society, 1976;215:241–251. 7. Chaoha P., Phon-On A., A note on fixed point sets in CAT(0) spaces. Journal of Mathematical Analysis and Applications, 2006;320(2):983–987. 8. George A., Veeramani P., On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 1994;64:395–399. 9. Grabiec M., Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 1988;27:385– 389. 10. Gregori V., Sapena A., On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 2002;125:245–252. 11. Gregori V., Morillas S., Sapena A., Examples of fuzzy metrics and applications. Fuzzy Sets and Systems, 2011;170(1):95–111. 12. Gregori V., Minana J.-J., Morillas S., Some questions in fuzzy metric spaces. Fuzzy Sets and System, 2012;204:71–85. 13. Gregori V., Minana J.-J., On fuzzy ψ- contractive mappings. Fuzzy Sets and Systems, 2016;300:93–101. 14. Gopal D., Abbas M., Imdad M., ψ-weak contractions in fuzzy metric spaces. Iranian Journal of Fuzzy Systems, 2011;8(5):141–148. 15. Gopal D., Vetro C., Some new fixed point theorems in fuzzy metric spaces. Iranian Journal of Fuzzy Systems, 2014;11(3):95–107. 16. Gopal D., Mart´ınez-Moreno J., Suzuki type fuzzy Z -contractive mappings and fixed points in fuzzy metric spaces. Kybernetika, 2021;5(7):908–921. 17. Kramosil I., Michalek J., Fuzzy metric and statistical metric spaces. Kybernetica, 1975;11(5):336–344. 18. Mohamad S., Global exponential stability in DCNNs with distributed delays and unbounded activations. Journal of Computational and Applied Mathematics, 2007;205(1):161–173. ¨ ur N. Y., Tas¸ N., Celik U., New fixed-circle results on S-metric spaces. Bulletin of 19. Ozg¨ Mathematical Analysis and Applications, 2017;9(2):10–23. ¨ ur N. Y., Tas¸ N., Some fixed-circle theorems and discontinuity at fixed circle. AIP 20. Ozg¨ Conference Proceedings, 2018;1926:020048. ¨ ur N. Y., Tas¸ N., Some fixed-circle theorems on metric spaces. Bulletin of Mathemat21. Ozg¨ ical Analysis and Applications, 2019;42(4):1433–1449. ¨ ur N., Tas¸ N., Geometric properties of fixed points and simulation functions. 22. Ozg¨ arXiv:2102.05417. ¨ ¨ ur N. Y., Complex valued neural network with M¨obius 23. Ozdemir N., ˙Iskender B. B., Ozg¨ activation function. Communications in Nonlinear Science and Numerical Simulation, 2011;16(12):4698–4703. 24. Pomdee K., Sunyeekhan G., Hirunmasuwan P., The product of virtually nonexpansive maps and their fixed points. In Journal of Physics: Conference Series (vol. 1132, no. 1, p. 012025). IOP Publishing, Malaysia, 2018. 25. Sharma S., Sharma S., Athaiya A., Activation functions in neural networks. Towards Data Science, 2017;6(12):310–316. 26. Shukla S., Gopal D., Sintunavarat W., A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems, 2018;350:85–95. 27. Tomar A., Joshi M., Fixed point to fixed circle and activation function in partial metric space. Journal of Applied Analysis, 2022;1:15.

✻ ✻ ✻ ✻

■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ ■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ ❋✐❣✉#❡ -#♦❜❧❡♠1 ✐♥ ❋✉③③② ■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ ❋✐❣✉#❡ -#♦❜❧❡♠1 ✐♥ ❋✉③③② ▼❡"#✐❝ ❙♣❛❝❡1 ■♥"#♦❞✉❝"✐♦♥ "♦ ❋✐①❡❞ ❋✐❣✉#❡ ✐♥ ❋✉③③② ▼❡"#✐❝ -#♦❜❧❡♠1 ❙♣❛❝❡1 ❋✐❣✉#❡ ✐♥ ❋✉③③② ▼❡"#✐❝ -#♦❜❧❡♠1 ❙♣❛❝❡1 ▼❡"#✐❝ ❙♣❛❝❡1

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28. Wang Z., Guo Z., Huang L., Liu X., Dynamical behaviour of complex-valued Hopfield neural networks with discontinuous activation functions. Neural Processing Letters, 2017;45(3):1039–1061. 29. Zhang Y., Wang Q. G., Stationary oscillation for high-order Hopfield neural networks with time delays and impulses. Journal of Computational and Applied Mathematics, 2009;231(1):473–477. ¨ ur N., On fixed figure problems in fuzzy metric 30. Gopal D., Mart´ınez-Moreno J., Ozg¨ spaces. Kybernetika, 2023;59(1):110—129.

of Fuzzy 7 Applications Metrics and Fixed-Point Theorems 7.1

INTRODUCTION

The applicability of fuzzy metrics is mainly due to the following two main advantages with respect to classical metrics: First, values given by fuzzy metrics are in the interval [0,1] regardless of the nature of the distance concept being measured. This implies that it is easy to combine different distance criteria that may originally be in quite different ranges but fuzzy metrics take to a common range. In this way, the combination of several distance criteria may be done in a straightforward way. Second, fuzzy metrics match perfectly with the employment of other fuzzy techniques since the value given by a fuzzy metric can be directly employed or interpreted as a fuzzy certainty degree. This allows to straightforwardly include fuzzy metrics as part of other complex fuzzy systems. In recent years, fuzzy metrics have been applied to color image filtering improving some filters when replacing classical metrics [6,7]. The fuzzy metric fixed point theory has various advantages over the regular metric fixed point theory. This is due to the pliancy which the fuzzy concepts inherently possess. Due to this reason, fuzzy metric fixed point theorems are widely used in solving various complex fuzzy problems including fuzzy operator equations.

7.2

IMAGE FILTERING USING FUZZY METRICS

According to the introductory section, fuzzy metrics provide a series of advantages with respect to classical metrics, which motivate the study of their application in practical problems. These advantages can be incorporated to deal with problems of different nature in distinct branches of Science and Engineering. In particular, several successful applications of fuzzy metrics in color image processing have already been published [5–10]. This suggests that the application of fuzzy metrics in other engineering problems is an interesting issue of research. In the following, to illustrate the benefits of fuzzy metrics, we give an application of fuzzy metrics to color image filtering where we combine two different fuzzy metrics into a unique fuzzy metric. Let us denote by F the digital image to be processed. We process the image by employing a sliding window approach. To process each image pixel, we take the pixels within a squared neighborhood W of size n × n centered on it, where n > 1 is an odd number. We denote by Fq the n2 color vectors associated with the pixels in W , where Fc denotes the color vector associated to the pixel under processing. Note that a color image pixel is characterized by its location in the image q = {q1 , q2 }, DOI: 10.1201/9781003427797-7

183

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given by two location coordinates, and by its three RGB color components, that is, Fq = {FqR , FqG , FqB }. We have used the RGB color space for being the space commonly used in image processing; however, other color spaces could be also employed, and it would be interesting to study the behavior of the filtering process in other cases. In RGB images, we may measure the distance between two color vectors attending to the similarity of their color components and also attending to the spatial closeness between the pixels. In this chapter, we propose to use a fuzzy metric that simultaneously represents both distance criteria. For this, on the one hand, we use the fuzzy metric R given as 3

R(Fi , Fj ) = ∏

l=1

min{Fil , Fjl } + K max{Fil , Fjl } + K

(7.1)

to measure the similarity between the color vectors Fi and Fj . This fuzzy metric is inspired in Equation (7.3), and it can be proven that it is in fact a fuzzy metric since the product of fuzzy metrics is also a fuzzy metric [11]. Notice that this fuzzy metric behaves, so that for two different pairs of consecutive values (or vectors), the fuzzy distance given by the metric is different. This effect can be alleviated by increasing the value of the K parameter in Equation (7.22). However, previous research concluded [6] that very high values of K should not be used to avoid decreasing the sensitivity of the fuzzy metric. The optimal value for K depends on both image and noise density, but, for RGB color images, it was found that values in [512, 2, 048] were appropriate. So, we have chosen K = 1, 024 since this hexa-decimal operation is computationally simple. On the other hand, the spatial closeness between two image pixels Fi and Fj is given by the fuzzy metric S defined as S(Fi , Fj ) =

t t + ||i − j||

(7.2)

where || · || denotes the Euclidean norm, and which is a special case of Equation (7.5) when n = 1 and where t is a parameter able to tune the sensitivity of the fuzzy metric. Now, we construct the product of the above fuzzy metrics R and S into a single fuzzy metric C, so that C(Fi , Fj ,t) = R(Fi , Fj )S(Fi , Fj ,t). In this way, the fuzzy metric C measures the closeness between two image pixels taking simultaneously into account the similarity between the color components and the spatial closeness of the pixels. The problem of filtering a digital image implies to remove the incorrect noisy pixels and replace them with noise-free ones. Within a sliding window approach, this can be performed by identifying in W a noise-free pixel Fˆc to replace the original one Fc . In a vector context, this identification can be made by employing the reduced ordering principle [1], where the noise-free vector Fˆc is identified as the vector in W maximizing the accumulation of an appropriate closeness measure with respect to all the other pixels in W . In our application, we aim to employ the fuzzy metric C as the measure to maximize. Since this fuzzy metric takes into account both the similarity between the color components and the spatial closeness of the pixels in the image,

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Table 7.1 Suboptimal Values of the t Parameter in Terms of PSNR When Filtering the Test Images (Figure 7.1) Using a 5 × 5 Filter Window for Different Noise Percentages Image noise Range of t values for suboptimal performance

10 %

20 %

30 %

[0.05,0.75]

[0.25,1]

[0.5,1.5]

Fˆc will be identified as the vector in W which is simultaneously the most similar and spatially close to all the other vectors in W . In the following, we detail the procedure to determine Fˆc . For each vector Fk in the filter window, an accumulated measure Ak =

∑

C(Fk , Fj ,t)

j∈W, j̸=k

to all the other vectors in the window has to be calculated to perform the reduced ordering. The Ak values are ordered in a descending sequence(in which the value located in the r-th position is written as A(r) ) as follows: A(0) ≥ A(1) ≥ · · · ≥ A(n2 −1) This order implies the same ordering of the Fk vectors: F(0) ≥ F(1) ≥ · · · ≥ F(n2 −1) Then, the filter output Fˆc will be the vector F(0) occupying the lowest rank in the ordered sequence. Notice that since the filter output is always one of the vectors in the filtering window, no false colors can be introduced by the filter. Also, in the above procedure, the t parameter is able to tune the importance of the spatial criterion, as commented below. In an analogous context, the well-known vector median filter (VMF) [1] employs the Euclidean metric as the distance criteria between the vectors. This vector filter is known to be very robust, but the resulting images are frequently too smoothed, and consequently, edges and fine details are not properly preserved. On the other hand, in the approach we propose, we can see that the inclusion of the spatial criterion helps to improve the preservation of image details while the noise is also reduced. This is achieved because the output vector in each filtering window is determined as a vector which is spatially close to all the other vectors in the window. Therefore, we avoid the possibility of replacing a pixel with another located far from it, which is not appropriate to preserve edge sand details. The importance of the spatial criterion can be tuned by modifying the value of the t parameter. Higher values of t reduce the sensitivity of the combined fuzzy metric to

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Figure 7.1 PSNR value obtained when filtering the Pills (solid blue), Gold hill (solid green), and Baboon (solid red) images contaminated with 10% noise as a function of the t parameter. Blue, green, and red dotted lines represent the PSNR value of the corresponding images filtered with the VMF. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the spatial criterion and increase the noise reduction capability of the filter, whereas lower values imply more importance of the spatial criterion and so better preservation of the original image. This implies that higher values of t are appropriate for higher noise densities, since in these cases, the filter robustness is more important than the preservation of original details. This can be seen in Table 7.1, where, for different noise percentages, we show a range of possible values for t which can obtain suboptimal performance. We have found these optimal values through extensive experimental results using the test images in Figures 7.2–7.4 contaminated with different noise percentages where we have searched for the values that maximize the peak signal-to-noise ratio (PSNR) quality measure. However, this measure hardly agrees with the image visual quality so, to obtain more visually pleasing images, we found that it is better to take the higher values in the determined intervals. From Table 7.1, we may conclude that the optimal value of t increases as the noise percentage in the image increases. In Figure 7.1, we show the PSNR value for different output images as a function of t. We found that, with respect to the PSNR obtained by the VMF, there is a wide range of t values able to achieve significantly improved performance. Notice that high PSNR values are obtained also for very low t values (t < 0.1). Using such a low value increases the importance of the spatial criterion and so the imagepreserving capability of the filter, which leads to high PSNR performance. However,

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also some noise is preserved, which means that these results are not appropriate from the visual quality point of view. This not only happens for PSNR but also for other image quality measures and, in fact, the obtention of an image quality measureable to match the visual image quality is an open problem in this field. For this reason, we suggest to use a little higher values for t, which lead to better noise reduction and, so, more visually pleasing output images. Our experiments indicate that, for percentage of noise in the image in [5,30], the value of t can be safely set proportionally in the interval [0.25,1.5] to obtain suboptimal performance. Similar results that agree with Table 7.1 can be obtained for higher noise percentages. Figures 7.2–7.4 show three noisy images filtered with both the VMF and the proposed method, using an appropriate value of t in each case, along with their quantitative image quality in terms of the PSNR. We can see that both methods are able

Figure 7.2 Performance comparison: (a) Detail of Pills image where 10% of pixels are noisy, PSNR = 18.33 (b) output obtained using the VMF with a filter window of size 5 × 5, PSNR = 22.47 and (c) output obtained with the proposed method, PSNR = 25.63 (t = 0.5).

Figure 7.3 Performance comparison: (a) Detail of Goldhill image where 20% of pixels are noisy, PSNR = 15.99 (b) output obtained using the VMF [1] with a filter window of size 5 × 5, PSNR = 24.68 and (c) output obtained with the proposed method, PSNR = 27.87 (t = 1).

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Figure 7.4 Performance comparison: (a) Detail of Baboon image where 30% of pixels are noisy, PSNR = 14.20 (b) output obtained using the VMF [1] with a filter window of size 5 × 5, PSNR = 21.56 and (c) output obtained with the proposed method, PSNR = 23.07 (t = 1.5).

to suppress noise but, whereas the VMF generates too blurry output images, the proposed method is able to better preserve image edges and details, and so, to improve both the visual quality and the quantitative quality of the obtained images. Notice also that the proposed method is able to work well over images corrupted with different percentages of noise since its behavior can be adapted by varying the value of the parameter t.

7.3

APPLICATION OF THE FUZZY METRIC M0 TO MEASURE PERCEPTUAL COLOR DIFFERENCES

Apart from the interesting theoretical properties of the fuzzy metrics studied in previous sections, it is interesting as well to note that they have application in a variety of practical problems. Indeed, they have been previously used to filter color image sand to measure the degree of consistency of elements in a dataset [6,13,20,21]. Here we focus on a different application of the fuzzy metric M0 that takes advantage of the homotetique invariant property that this fuzzy metric satisfies (see [2,3]). Indeed, M0 fulfills that, for any λ ∈ R: (I) M0 (λ x, λ y) = M0 (x, y) Also, if z > 0 (II) M0 (x + z, y + z) > M0 (x, y) As we will see later on, there exist practical problems where these properties are pretty interesting. However, in practical applications, it is more appropriate to use the

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Table 7.2 Values of STRESS Obtained by Different Color Difference Formulas for the COM Dataset Color Difference Formula

Stress

CIELAB CIE94 CIEDE2000 ∆EM1∗ ∆EM1∗

0.428 0.335 0.292 0.347 0.348

M ∗ fuzzy metric(which also satisfies (II)), instead of the M0 , because the presence of the t parameter makes this fuzzy metric more adaptive to the particular problem. On the other hand, M0 is in fact M ∗ when t = 0. Notice that both M0 and M ∗ are suitable only for scalar values, and that for vector values, the combination of several fuzzy metrics needs to be considered (Table 7.2). In particular, one application that matches the behavior of these two fuzzy metrics regards the modeling of the perception of physical magnitudes such as colors, sounds, or weights. It is known that the perception threshold of changes in these magnitudes increases as the magnitudes themselves increase [14,15,22]. That is to say, the perceived difference between two magnitude values x, y is different from that for the values x + k, y + k, whenever k > 0. In particular, the perceived difference will be larger in the former case than in the latter, which agrees with (II). This situation can be observed in the case of perceptual color differences and, since the M ∗ fuzzy metric behaves accordingly to this situation, M ∗ can be used to appropriately devise color difference formulas as explained in the following. A color sample is usually represented as a tern in a particular color space. Among the different color spaces, a well-known one, especially in computer graphics, is the Hue-Chroma-Lightness (HCL) color space [16], where a color sample s is represented as a tern s = (Hs ,Cs , Ls ). In such a tern: Hue, Hs , is usually represented as an angle in [0◦ , 360◦ ] where 0◦ , 90◦ , 180◦ , and 270◦ correspond to approximately pure red, yellow, green and blue, respectively. Cs ∈ [0, 100] represents the Chroma of the color, where 0 is associated with neutral grey, black or white; and Ls ∈ [0, 100] represents the lightness of the sample, where 0 represents no lightness (absolute black color) and 100 represents the maximum lightness (absolute white color). A series of experimental datasets, BFD-P, Leeds, RIT-Dupont, and Witt, which are combined to form the COM dataset, have been obtained in order to characterize the perceptual difference between pairs of color samples [12,16–19,23,24]. In these datasets, each pair of color samples is associated with a value ∆V , which represents the experimental perceptual difference between them. On the other hand, color difference formulas are used to obtain, from two terns representing a pair of color samples,

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the computed perceptual difference between them, usually denoted by ∆E. Since the objective of color difference formulas is to model human perception, all formulas try to obtain ∆E values as close (or correlated) as possible to the ∆V values. One well-known color difference formula is the CIELAB formula [24], that corresponds with the Euclidean distance in the CIELAB color space. The performance of a color difference formula is assessed by measuring how close the ∆E values computed for the experimental datasets are to the ∆V values. A well-established figure of merit for this closeness is the STRESS coefficient, which provides values in the interval [0,1], where lower values indicate a higher closeness. In Table 7.1, we can see that the value of STRESS for the CIELAB formula over the COM dataset is 0.428. By analysing the experimental datasets, it has been observed that the sensitivity to differences in Chroma decreases as the value of Chroma increases. Notice that this fact is related to the Weber–Frechner and Stevens observations [14,15,22]. According to this, we propose to use the M ∗ fuzzy metric to model the similarity between two chroma values Cs ,Cr as M ∗ (Cs ,Cr ) =

min{(Cs ,Cr )} + kC max{(Cs ,Cr )} + kC

where kC is a parameter to adjust the behavior as desired. An analogous observation can be made with respect to lightness. So we propose to measure the similarity between two lightness values Ls , Lr as M ∗ (Ls , Lr ) =

min{(Cs ,Cr )} + kL max{(Cs ,Cr )} + kL

where kL is another adjusting parameter. Using these two expressions, we build a more complex expression to obtain a new color difference formula. We want also to take into account the CIELAB color ∗ , so, we employ the standard fuzzy metric deduced from ∆E ∗ . Given difference, ∆Eab ab that the product of these fuzzy metrics is also a fuzzy metric [11], we can use a productory to join these three criteria. Finally, to obtain a difference formula, we use the involutive negation as follows: ∆EM1∗ (s, r) = 1 − (M ∗ (Ls , Lr )M ∗ (Cs ,Cr )(

t ∗ )) t + ∆Eab

(7.3)

where kL , kC , and t are parameters able to tune the importance of each criterion. ∗ also includes lightness and chroma differences, alternatively However,since ∆Eab ∗ in Equation (7.3) with ∆H, which represents only Hue we propose to replace ∆Eab q ∗ and is given by ∆H = differences in ∆Eab obtaining

∗2 − |L − L |2 − |C −C |2 , and so ∆Eab s r s r

∆EM2∗ (s, r) = 1 − (M ∗ (Ls , Lr )M ∗ (Cs ,Cr )( where we have three adjusting parameters, as above.

t )) t + ∆H

(7.4)

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191

∗ It is interesting to point out that ∆EM1∗ can be seen as a modification of the ∆Eab using a correction term inspired in the Weber–Fechner and Stevens laws which are represented by an appropriate fuzzy metric. On the other hand, ∆EM2∗ is a color difference formula that corresponds with the representation of the Weber–Frechner and Stevens laws by means of fuzzy metrics. We have performed extensive experimental assessments varying the values of the adjusting parameters kL , kC and t in the range [0, 100] to obtain the optimal parameter setting for the formulas proposed in Equations (7.3) and (7.4). With optimal parameter setting, ∆EM1∗ is able to obtain a STRESS value for the COM dataset of 0.347 (with kL = 2, kC = 4, t = 11), whereas ∆EM2∗ obtained STRESS of 0.348 (with kL = 4, kC = 12,t = 40). Notice that, in both cases, a significant improvement with ∗ is obtained. This means that M ∗ has been successfully used to take respect to ∆Eab into account the facts related to the Weber–Fechner and Stevens laws. It should be ∗ does not incorporate these laws, they are considered in also noted that whereas ∆Eab more recent color difference formulas such as the CIE94 and CIEDE2000 formulas. We also compare the performance of the proposed formulas with these recent ones in Table 7.1, where we can see that the performance of our formulas is pretty close to the one of the CIE94.

7.4 7.4.1

APPLICATIONS TO FUZZY FIXED POINT THEOREMS APPLICATIONS TO DIFFERENTIAL EQUATIONS

In this section, following [4], we study the initial value problem for some classes of second-order differential equations. First, we consider the autonomous case, as follows. Let T > 0, I = [0, T ] and consider the problem: x′′ (t) = ξ (x(t), x(t), . . . , x(t)), t ∈ I,

x(0) = α, x′ (0) = β ,

(7.5)

(k)

where α, β ∈ R and ξ : Rk = R× · · · ×R → R is a continuous function. The Green’s function associated with (7.5) is given by G(t, τ) = t − τ for t > τ, and G(t, τ) = 0 for 0 ≤ t ≤ τ, in such a way that, as it can be easily seen, the solution to Equation (7.5) is given by the solution of the integral equation of the following form: Z T

x(t) =

0

Z t

=

0

G(t, τ)ξ (x(τ), x(τ), . . . , x(τ))dτ + ζ (t) (t − τ)ξ (x(τ), x(τ), . . . , x(τ))dτ + ζ (t), for t ∈ I,

(7.6)

where ζ (t) = α + βt. To define the concept of solution to Equation (7.5), we consider C2 (I, R), the space of all functions from I into R having continuous second order derivative on I. A solution to Equation (7.5) is a function x ∈ C2 (I, R) which satisfies the conditions in Equation (7.5). The procedure we follow to prove the existence of solutions to

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problem (7.5) is to establish a connection between them and the solutions to the integral equation (7.6). To study the existence of solutions to the integral equation (7.6), we consider C(I, R), the Banach space of all continuous functions from I = [0, T ] into R, endowed with the supremum norm, defined as: ∥x∥∞ := supt∈I | x(t) |, x ∈ C(I, R). Noticethat C(I, R) is also a Banach space with the Bielecki norm given by ∥x∥B := supt∈I | x(t) | e−bt , x ∈ C(I, R), where b > 0 is arbitrary but fixed. It is easy to see that the two norms ∥ · ∥∞ and ∥ · ∥B are equivalent on I = [0, T ]. For our purpose, we use the Bielecki norm instead of the supremum norm. The Bielecki  metric induced by the Bielecki norm is given by the expression dB (x, y) := supt∈I | x(t) − y(t) | e−bt , x, y ∈ C(I, R). The standard fuzzy metric MdB : [C(I, R)]2 × (0, ∞) → [0, 1] induced a , ∀ x, y ∈ C(I, R), ∀a > 0. by dB is defined as: MdB (x, y, a) = a + dB (x, y) Then, it is easy to see that the min-fuzzy metric space (C(I, R), MdB , ∗m ) is an M-complete fuzzy metric space. Define an operator Φ : [C(I, R)]k → C(I, R) by Rt [Φ(x1 , x2 , . . . , xk )](t) = 0 (t − τ)ξ (x1 (τ), x2 (τ), . . . , xk (τ))dτ + ζ (t), for t ∈ I and x1 , . . . , xk ∈ C(I, R). It is obvious that the solutions to the integral equation (7.6) coincide with the fixed points of the operator Φ, i.e., x ∈ C(I, R) such that [Φ(x, x, . . . , x)](t) = x(t), for every t ∈ I. Moreover, a solution to Equation (7.5) trivially satisfies the integral equation (7.6) and, if x ∈ C(I, R) is a solution to the integral equation (7.6), then we can prove that x ∈ C2 (I, R) and the conditions in (7.5) are fulfilled. Hence, the solutions to the initial value problem (7.5) are the fixed points of the operator Φ. All these considerations allow us to prove the existence and uniqueness of solution to the initial value problem (7.5), as established in the following theorem. Theorem 7.1. Let k be a positive integer and suppose that the following conditions are satisfied: 1. ξ : Rk → R is a continuous function; 2. there exists L > 0 such that, for all z1 , z2 , . . . , zk , zk+1 ∈ R, we have | ξ (z1 , . . . , zk ) − ξ (z2 , . . . , zk+1 ) | ≤ L max1≤i≤k | zi − zi+1 | . Then the initial value problem (7.5) has a unique solution. Proof. We consider (C(I, R), MdB , ∗m ) the min-fuzzy metric space induced by the Bielecki metric dB on C(I, R). Since b > 0 can be selected arbitrarily, we choose b = kLT > 0 then, for all x1 , . . . , xk , xk+1 ∈ C(I, R), we have dB (Φ(x1 , . . . , xk ), Φ(x2 , . . . , xk+1 )) Z t −bt = sup (t − τ)[ξ (x1 (τ), . . . , xk (τ)) − ξ (x2 (τ), . . . , xk+1 (τ))]e dτ 0 t∈I

≤ sup t∈I

≤ sup t∈I

Z t 0

Z t 0

(t − τ) | ξ (x1 (τ), . . . , xk (τ)) − ξ (x2 (τ), . . . , xk+1 (τ)) | e−bt dτ (t − τ) L max {| xi (τ) − xi+1 (τ) |} e−bτ eb(τ−t) dτ 1≤i≤k

Applications of Fuzzy Metrics and Fixed-Point Theorems

193

  Z t ≤ L max {dB (xi , xi+1 )} sup e−bt (t − τ)ebτ dτ 1≤i≤k 0 t∈I   Z t −bt bτ ≤ L max {dB (xi , xi+1 )} sup te e dτ 1≤i≤k

≤

t∈I

0

LT (1 − e−bT ) max {dB (xi , xi+1 )} . b 1≤i≤k

Since b = kLT , we have, for all x1 , . . . , xk , xk+1 ∈ C(I, R), dB (Φ(x1 , . . . , xk ), Φ(x2 , . . . , xk+1 )) ≤ λ · max {dB (xi , xi+1 )} , 1≤i≤k

LT 1 (1 − e−bT ) = (1 − e−bT ) < 1. Therefore, b k a MdB (Φ(x1 , . . . , xk ), Φ(x2 , . . . , xk+1 ), a) = a + dB (Φ(x1 , . . . , xk ), Φ(x2 , . . . , xk+1 ))    −1 a 1 ≥ = λ · max −1 +1 , a + λ · max {dB (xi , xi+1 )} 1≤i≤k MdB (xi , xi+1 , a)

where 0 < λ =

1≤i≤k

´ c operator. for all x1 , x2 , . . . , xk+1 ∈ C(I, R) and a > 0. Thus, Φ is a fuzzy-Preˇsi´c-Ciri´ Besides, it is clear that condition (H2*) holds for arbitrary choices of x1 , . . . , xk ∈ (n,m) 1 , for j = n, . . . , m − 1. Hence, by C(I, R), since λ ∈ (0, 1), taking s j = s j = j( j+1) Theorem 7.25, Φ has a fixed point in C(I, R), which gives a solution to Equation (7.5). Furthermore, for x, y ∈ C(I, R), we have, by (b), dB (Φ(x, . . . , x), Φ(y, . . . , y)) Z t = sup (t − τ)[ξ (x(τ), . . . , x(τ)) − ξ (y(τ), . . . , y(τ))]e−bt dτ 0 t∈I Z t h ≤ sup (t − τ) | ξ (x(τ), . . . , x(τ)) − ξ (x(τ), . . . , x(τ), y(τ)) | t∈I

0

i + · · · + | ξ (x(τ), y(τ), . . . , y(τ)) − ξ (y(τ), . . . , y(τ)) | e−bt dτ Z t

(t − τ) k L | x(τ) − y(τ) | e−bt dτ   Z t kLT −bt bτ ≤ kL dB (x, y) sup te e dτ ≤ (1 − e−bT ) dB (x, y). b 0 t∈I

≤ sup t∈I

0

Since b = kLT , we have dB (Φ(x, . . . , x), Φ(y, . . . , y)) ≤ (1 − e−bT )dB (x, y) < dB (x, y), for all x, y ∈ C(I, R) with x ̸= y. Therefore, by the definition of MdB , MdB (Φ(x, . . . , x), Φ(y, . . . , y), a) > MdB (x, y, a), ∀ a > 0.

(7.7)

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Thus, all the conditions of Theorem 7.25 are satisfied and, in consequence, Φ has a unique fixed point in C(I, R), which is the unique solution to the initial value problem (7.5). Remark 7.1. Note that, in the proof of Theorem 7.1, if k ≥ 2, an alternative way to check the validity of Equation (7.7) in order to achieve the uniqueness of solutions is to use the ideas in Corollary 4.7, since, with the selection of the minimum t-norm ∗m and, for example, ri = 21i , i = 1, . . . , k, condition (47) is reduced to for each u, v ∈ X fixed with u ̸= v, there exists t > 0 such that  i−1 h  1 > M(u, v,t), for every i = 1, . . . , k − 1. λ · M(u,v,t/2 i) − 1 + 1 To check its validity, we take u, v ∈ X fixed with u ̸= v. For M = MdB and a > 0, we have    −1    −1 a/2i + dB (u, v) 1 λ· −1 +1 = λ· −1 +1 MdB (u, v, a/2i ) a/2i  −1 a dB (u, v) a 2i = λ· + 1 , = = a/2i λ · dB (u, v) + 2ai λ · 2i · dB (u, v) + a so that, since dB (u, v) > 0, this expression is greater than MdB (u, v, a), for every i = 1, . . . , k − 1, if and only if λ · 2i < 1, for every i = 1, . . . , k − 1, that is, λ < 21−k . LT Note that λ is taken as λ = (1 − e−bT ) in the proof of Theorem 7.1. Thus, in b ´ c operator, we just have to choose b > 0 order to prove that Φ is a fuzzy-Preˇsi´c-Ciri´ such that b > LT . Now, for the validity of (47), it suffices to choose b > 0 with LT 2k−1 (1 − e−bT ) < 1, that is, b > LT 2k−1 . This provides uniqueness of solution b for k ≥ 2. Instead of problem (7.5), we could have considered a non-autonomous problem of the type x′′ (t) = ξ (t, x(t), x(t), . . . , x(t)), t ∈ I = [0, T ],

x(0) = α, x′ (0) = β ,

(7.8)

where α, β ∈ R and ξ : I × Rk → R is continuous. Taking ζ (t) = α + βt, t ∈ I, it is clear that the solutions to Equation (7.8) coincide with those of the integral equation Z T

x(t) =

0

Z t

=

0

G(t, τ)ξ (τ, x(τ), x(τ), . . . , x(τ))dτ + ζ (t) (t − τ)ξ (τ, x(τ), x(τ), . . . , x(τ))dτ + ζ (t), for t ∈ I,

(7.9)

e : [C(I, R)]k → C(I, R) defined as and also with the fixed points of the mapping Φ Rt e 1 , x2 , . . . , xk )](t) = (t − τ)ξ (τ, x1 (τ), x2 (τ), . . . , xk (τ))dτ + ζ (t), for t ∈ I and [Φ(x 0 x1 , . . . , xk ∈ C(I, R). Thus, the following extension of Theorem 7.25 is as follows.

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Applications of Fuzzy Metrics and Fixed-Point Theorems

Theorem 7.2. Let k be a positive integer and suppose that the following conditions are satisfied: 1. ξ : I × Rk → R is a continuous function; 2. there exists a nonnegative and integrable function L : I → R such that, for all t ∈ I and z1 , z2 , . . . , zk , zk+1 ∈ R, we have | ξ (t, z1 , . . . , zk ) − ξ (t, z2 , . . . , zk+1 ) | ≤ L(t) max | zi − zi+1 | 1≤i≤k

and there exists b > 0 such that   Z t k sup e−bt (t − τ)L(τ)ebτ dτ < 1. 0

t∈I

(7.10)

Then the initial value problem (7.8) has a unique solution. Proof. We consider again (C(I, R), MdB , ∗m ) the min-fuzzy metric space induced by the Bielecki metric dB on C(I, R), where b > 0 is given by the statement. Hence, similarly to the proof of Theorem 7.25, we have, for all x1 , . . . , xk , xk+1 ∈ C(I, R), e 1 , . . . , xk ), Φ(x e 2 , . . . , xk+1 )) dB (Φ(x ≤ sup

Z t 0

t∈I

(t − τ) L(τ) max {| xi (τ) − xi+1 (τ) |} e−bτ eb(τ−t) dτ 1≤i≤k

≤ λ max {dB (xi , xi+1 )} , 1≤i≤k

  Z t 1 where 0 < λ := sup e−bt (t − τ)L(τ)ebτ dτ < ≤ 1. k 0 t∈I Besides, if x, y ∈ C(I, R), by (b), e . . . , x), Φ(y, e . . . , y)) dB (Φ(x, Z t h i ≤ sup (t − τ) L(τ) | x(τ) − y(τ) | + · · · + L(τ) | x(τ) − y(τ) | e−bt dτ t∈I

0

= k sup t∈I

Z t 0

(t − τ) L(τ) | x(τ) − y(τ) | e−bτ eb(τ−t) dτ ≤ k dB (x, y)λ ,

e . . . , x), Φ(y, e . . . , y)) < dB (x, y), for all x, y ∈ C(I, R) with x ̸= y so that dB (Φ(x, (dB (x, y) > 0). Since (H2*) also holds, then Theorem 7.25 applies. Remark 7.2. Condition (7.10) trivially holds there exists b > 0  such that   if  Z t Z t 1 1 −bt bτ −bt bτ sup e L(τ)e dτ < , or if lim sup e L(τ)e dτ < , b→∞ kT kT 0 0 t∈I t∈I     Z t Z t since k sup e−bt (t − τ)L(τ)ebτ dτ ≤ kT sup e−bt L(τ)ebτ dτ . t∈I

0

t∈I

0

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Recent Advances and Applications of Fuzzy Metric Fixed Point Theory

In particular, if L is nonnegative, integrable, and bounded (there exists L > 0 such that L(t) ≤ L , t ∈ I), then     Z t Z t 1 − e−bT −bt bτ −bt bτ , k sup e (t − τ)L(τ)e dτ ≤ kT L sup e e dτ = kT L b 0 0 t∈I t∈I so that it is enough to choose b > kT L . Finally, we consider an impulsive problem of the type x′′ (t) = ξ (t, x(t), x(t), . . . , x(t)), t ∈ I = [0, T ], t ̸= t j , j = 1, . . . , l, j = 0, . . . , l, x′ (t + x(t + j ) = β j, j ) = α j,



(7.11)

where α j , β j ∈ R, j = 0, . . . , l, 0 = t0 < t1 < t2 < · · · < tl < tl+1 = T and ξ : I × Rk → R is such that, for every j = 0, . . . , l, its restriction to (t j ,t j+1 ] × Rk is continuous and admits a continuous extension to the set [t j ,t j+1 ] × Rk . To define the concept of solution to problem (7.11), we consider the spaces PC(I, R) = {x : I → R : x is continuous in I \ {t1 , . . . ,tl } − and ∃ x(t + j ), x(t j ) = x(t j ), j = 1, . . . , l} = {x : I → R : x ∈ C((t j ,t j+1 ), R), j = 0, . . . , l, and ∃ x(0+ ) = x(0), − x(T − ) = x(T ), x(t + j ), x(t j ) = x(t j ), j = 1, . . . , l}, E := {x ∈ PC(I, R) : x ∈ C2 (I \ {t1 , . . . ,tl }, R) ′′ − ′ − ′′ + and ∃ x′ (t + j ), x (t j ), x (t j ), x (t j ), j = 1, . . . , l} 2 = {x ∈ PC(I, R) : x ∈ C ((t j ,t j+1 ), R), j = 0, . . . , l, and ∃ x′ (0+ ) = x′ (0), x′′ (0+ ) = x′′ (0), x′ (T − ) = x′ (T ), x′′ (T − ) = x′′ (T ), ′′ − ′ − ′′ + x′ (t + j ), x (t j ), x (t j ), x (t j ), j = 1, . . . , l}. Hence, a solution to Equation (7.11) is a function x ∈ E satisfying the conditions in Equation (7.11). For this problem (7.11), the Green’s function G : I × I → R is given by  t − τ, t j < τ < t ≤ t j+1 , for some j = 0, . . . , l, G(t, τ) = 0, otherwise. We can also use the functions G j : [t j ,t j+1 ] × [t j ,t j+1 ] → R, j = 0, . . . , l, defined by G j (t, τ) = t − τ, if t j ≤ τ < t ≤ t j+1 , and G j (t, τ) = 0, if t j ≤ t ≤ τ ≤ t j+1 , in such a way that G(t, τ) = G j (t, τ), for (t, τ) ∈ (t j ,t j+1 ) × (t j ,t j+1 ). Therefore, taking ζ j (t) = α j + β j (t − t j ), for t ∈ (t j ,t j+1 ], and j = 0, . . . , l, the solutions to Equation (7.11) are the solutions x ∈ PC(I, R) to the family of integral equations: Z T

x(t) =

0

G(t, τ)ξ (τ, x(τ), x(τ), . . . , x(τ))dτ + ζ j (t)

Z t j+1

= tj

Z t

= tj

G j (t, τ)ξ (τ, x(τ), x(τ), . . . , x(τ))dτ + ζ j (t) (t − τ)ξ (τ, x(τ), x(τ), . . . , x(τ))dτ + ζ j (t), t ∈ (t j ,t j+1 ].

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Applications of Fuzzy Metrics and Fixed-Point Theorems

The space PC(I, R) is a Banach space with the supremum norm defined as ∥x∥PC := supt∈I | x(t) |= max0≤ j≤l supt∈(t j ,t j+1 ] | x(t) |, x ∈ PC(I, R) and also with n o the equivalent norm ∥x∥PCB := max sup | x(t) | e−b(t−t j ) , x ∈ PC(I, R), where 0≤ j≤l t∈(t ,t ] j j+1

b > 0 is arbitrary but n fixed. The distance oinduced by ∥ · ∥PCB is dPCB (x, y) := max0≤ j≤l supt∈(t j ,t j+1 ] | x(t) − y(t) | e−b(t−t j ) , for x, y ∈ PC(I, R). b : [PC(I, R)]k → PC(I, R), given by We consider the mapping Φ Z t

b 1 , x2 , . . . , xk )](t) = [Φ(x

tj

(t − τ)ξ (τ, x1 (τ), x2 (τ), . . . , xk (τ))dτ + ζ j (t),

for t ∈ (t j ,t j+1 ], j = 0, . . . , l, and x1 , . . . , xk ∈ PC(I, R), whose fixed points are the solutions sought. We prove the following existence and uniqueness result for problem (7.11). Theorem 7.3. Let k be a positive integer and suppose that the following conditions are satisfied: 1. ξ : I ×Rk → R is such that its restriction to (t j ,t j+1 ]×Rk is continuous and admits a continuous extension to the set [t j ,t j+1 ] × Rk , for j = 0, . . . , l. 2. there exists a nonnegative and integrable function L : I → R such that, for all t ∈ I and z1 , z2 , . . . , zk , zk+1 ∈ R, we have | ξ (t, z1 , . . . , zk ) − ξ (t, z2 , . . . , zk+1 ) | ≤ L(t) max | zi − zi+1 | 1≤i≤k

and there exists b > 0 such that k max



sup

0≤ j≤l t∈(t ,t ] j j+1

e−bt

Z t

(t − τ)L(τ)ebτ dτ



< 1.

(7.12)

tj

Then the initial value problem (7.11) has a unique solution. Proof. We take (PC(I, R), MdPCB , ∗m ) the min-fuzzy metric space induced by the metric dPCB on PC(I, R), where b > 0 is given in the statement. Analogously to the proof of Theorem 7.25, we have, for all x1 , . . . , xk , xk+1 ∈ PC(I, R), b 1 , . . . , xk ), Φ(x b 2 , . . . , xk+1 )) dPCB (Φ(x ≤ max

sup

Z t

0≤ j≤l t∈(t ,t ] t j j j+1

(t − τ) L(τ) max {| xi (τ) − xi+1 (τ) |} e−b(τ−t j ) eb(τ−t) dτ 1≤i≤k

≤ λ max {dPCB (xi , xi+1 )} , 1≤i≤k

where 0 < λ := max

sup

0≤ j≤l t∈(t ,t ] j j+1



−bt

Z t

e

tj

bτ

(t − τ)L(τ)e dτ


0). The proof is concluded by Theorem 7.25, due to the validity of (H2*). Remark 7.3. If L is nonnegative, integrable and bounded (with upper bound L > 0), then condition (7.12) trivially holds since   Z t −bt bτ k max sup e (t − τ)L(τ)e dτ 0≤ j≤l t∈(t ,t ] j j+1

tj

  Z t n o kT L −bt bτ ≤ kT L max sup e e dτ = max 1 − e−b(t j+1 −t j ) . b 0≤ j≤l 0≤ j≤l t∈(t ,t ] tj j j+1 and the same choice of b > kT L is useful.

7.5

CONCLUSION

In this chapter, we have identified some fuzzy metrics that are very much applicable in dealing with problems concerning image processing and image filtering. The applications of fuzzy fixed point theorems are also discussed in solving some nonlinear differential equations. This suggests that it could be interesting to study the application of fuzzy metrics in other engineering problems. In particular, within the image denoising field, there exist other types of noise (e.g. Gaussian noise) for which the filtering solutions make use of different metrics. So, it could be studied if some fuzzy metric can provide improvements with respect to metrics commonly used.

REFERENCES 1. Astola J., Haavisto P., Neuvo Y., Vector median filters. Proceedings of the IEEE, 1990;78:678–689. 2. Gregori V., Morillas S., Sapena A., Examples of fuzzy metrics and applications. Fuzzy Sets and Systems, 2011;170:95–111. 3. Gregori V., Minana J. J., Morillas S., Some questions in fuzzy metric sapces. Fuzzy Sets and Systems, 2012;204:71–85. ´ c operators and applica4. Shukla S., Gopal D., Rodr´ıguez-L´opez R., Fuzzy-Preˇsi´c-Ciri´ tions to certain nonlinear differential equations. Mathematical Modelling and Analysis, 2016;21(6):811–835.

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Index admissible 61–62, 64–68, 70, 73, 75–77

increasing generator 7–8 KM-fuzzy metric 10–12, 21, 58–59, 152–154 Mh-triangular 171–172, 180

Bielecki norm 192 characteristic function 1 compact 21–22, 27, 45, 52–53

nested sequence 32–33, 35–40 non-Archimedean 18, 21, 45, 60–61, 68, 70, 89–92, 165

decreasing generator 6 Edelstein 10, 20, 55, 80, 86

point wise convergent 22–23, 143 principal fuzzy metric 28, 31, 38, 40, 44

fuzzy complement 4–5 fuzzy diameter 32–36, 38–40 fuzzy intersection 5–7 fuzzy set 1–5, 8–10, 50, 54, 80, 83, 85, 92–93, 104, 115, 124, 130, 132, 165, 170, 175, 177, 179 fuzzy union 7

quasi-nonexpansive 164, 175–178, 180 strong fuzzy metric 18–20, 28, 76, 79, 171 Tirado contraction 57–58, 80, 84, 86

GV-fuzzy metric x, 11–12, 22–23, 45, 58, 79, 154–155, 165

upper semicontinuous 139 weakly compatible 119, 135–139, 145, 147–154

H-type t-norm 97, 112

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