Some Fixed Point Theorems in Menger Spaces and Applicationsshodhganga.inflibnet.ac.in/bitstream/10603/20254/1/some fixed point... · SOME FIXED POINT THEOREMS IN MENGER SPACES AND

SOME FIXED POINT THEOREMS IN

MENGER SPACES AND APPLICATIONS

A THESIS

SUBMITTED TO THE

KUMAUN UNIVERSITY, NAINITAL

FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

SUPERVISOR: SUBMITTED BY:

Dr. B.D. PANT SUNEEL KUMAR

DEPARTMENT OF MATHEMATICS MOH - GUJRATIYAN

R.H. GOVT. POSTGRADUATE COLLEGE JASPUR (U.S. NAGAR)

KASHIPUR (UTTARAKHAND) 244713 UTTARAKHAND 247712

ENROLMENT NO. KU 979220 MARCH 2008

IN THE LOVING MEMORY OF MY FATHER

Dr. B. D. PANT Reader in Mathematics

M. Sc., D.Phil. R.H. Govt. Postgraduate College

Kashipur (U.S.Nagar), Uttarakhand

Date: - -2008

CERTIFICATE

In forwarding the thesis with title ‘Some Fixed Point

Theorems In Menger Spaces And Applications’ submitted by

Mr. Suneel Kumar in fulfillment of the requirement for the award of

the Degree of Doctor of Philosophy in Mathematics of Kumaun

University, Nainital. I hereby certify that he has completed the

research work for the full period commencing from the date of his

application for registration prescribed under the ordinance 6 and

that the thesis embodies the result of his investigations conducted

during the period. He worked as a research scholar with me. It is

original piece of work and can be forwarded to experts for critical

examination.

Forwarded By:

Professor S. B. PANDEY Dr. B. D. PANT

Convener R. D. C. Mathematics Supervisor

Kumaun University, Nainital

PREFACE

‘Fixed Point Theory’ is a beautiful mixture of analysis (pure

and applied), topology and geometry. Topological ideas are present

in almost all the areas of today's mathematics. The subject of

topology itself consists of several different branches; such as point

set topology, algebraic topology and differential topology, which

have relatively little in common.

Fixed point theorems give the conditions under which

mappings (single or multivalued) have solutions. Over the last fifty

years or so, the theory of fixed points has been revealed as a very

powerful and important tool in the study of nonlinear phenomena. In

particular, fixed point techniques have been applied in such diverse

fields as Biology, Chemistry, Economics, Engineering, Game theory,

and Physics.

The 19th century had been an era of great advances in the art

of measurements. These advances stimulated simultaneously the

concern for the accompanying errors. However, it was believed that

Preface

ii

through careful design and ample data, the error in any

measurement could be made arbitrarily small. But with the advent of

quantum mechanics, this belief was shattered and it was accepted

that the “uncertainties” of measurements were inherent in the

process of measurement.

French mathematician Professor Maurice René Fréchet wrote

an outstanding doctoral dissertation ‘Sur Quelques Points Du Calcul

Fonctionnel’ submitted on 2 April 1906. In it, he introduced the

concept of a metric space and is a natural setting for many problems

in which notion of ‘distance’ appears. An essential feature is the fact

that, for any two points in the space, there is defined a positive

number called the distance between the two points. However, in

practice we find very often that this association of a single number

for each pair is, strictly speaking, an over-idealization. Therefore,

Professor Karl Menger introduced a probabilistic metric space

(briefly PM-space) as a generalization of metric space.

Originally, Menger had planned to collaborate with his former

student Abraham Wald on this subject, but Wald was killed in a

plane crash in India in 1950. However, others, such as Berthold

Preface

iii

Schweizer (a former student of Menger) and Abe Sklar (a colleague

of Menger) took up the work and developed what is now called the

theory of probabilistic metric spaces (Menger Spaces).

One of the important aspects to study in fixed point theory in

probabilistic metric spaces is to obtain the necessary and sufficient

conditions implying the existence of a fixed point; and if that is not

possible then to obtain minimal type condition to ensure the fixed

point. Hence, there is a good scope for the study on fixed point

theorems in Menger spaces.

Our aim of study is:

1. To improve and extend the known results considering new

contractive conditions.

2. To obtain common fixed point theorems for mappings having

reciprocal continuity and satisfying implicit relations.

3. To obtain common fixed point theorems for compatible and

weakly compatible expansion mappings.

4. To obtain coincidences and fixed points of probabilistic

densifying mappings.

5. To study related fixed point theorems.

6. To study the applications of fixed points theorems.

Preface

iv

The present dissertation contains six chapters. A chapter-wise

brief sketch is as follows:

The first chapter is introductory in nature. In this chapter, we

define some fundamental concepts and notations relevant to the

development of the fixed point theory in probabilistic metric spaces.

A brief survey of the development of the fixed point theory in

probabilistic metric spaces along with metric spaces has also been

presented.

The second chapter is devoted to the study of common fixed

points of contraction mappings. These results are established for

commuting, poitwise R -weakly commuting and compatible

mappings. The last theorem of this chapter is proved by taking the

reciprocal continuity of mappings.

The third chapter is intended to obtain some common fixed

point theorems for expansion mappings. The first result is proved for

surjective mappings and the remaining results for non-surjective

mappings.

Preface

v

In the fourth chapter, we obtain coincidences and fixed points

of probabilistic densifying mappings.

In the fifth chapter, we establish some related fixed point

theorems for two pairs of mappings in two and for three mappings in

three different Menger spaces respectively.

In the last chapter, we give a survey work on some

applications of fixed point theorems in PM-spaces to various fields.

The list of literature consulted has been placed at the end of

the dissertation as “REFERENCES”.

Our work is contained in chapter- II, III, IV, V and VI. A

part of the research work contained in this dissertation has already

been either published or accepted for publication in various

research journals.

ACKNOWLEDGEMENTS

The present dissertation entitled “Some Fixed Point

Theorems In Menger Spaces and Applications ” is a result of

study of the author under the supervision and meticulous guidance

of Dr. B.D. Pant . Very First, I express my deep sense of gratitude to

him for his deep interest, invaluable suggestions and constant

inspirations during the course of my study.

I also express my sincere thanks to Professor S. B. Pandey,

Department of Mathematics, S.S.J. Campus Almora, Kumaun

University, Professor (Retd.) S.L. Singh, Department of

Mathematics, Gurukul Kangri Vishwavidyalaya, Hardwar and

Professor R.C. Dimri, Department of Mathematics, H.N.B. Garhwal

University, Srinagar (Garhwal) for their intrinsic help and blessings.

During this study, several well-known professors of

Mathematics (especially fixed point theorists) from different parts of

world such as Prof. S.M. Kang, Gyeongsang National University,

Korea; Prof. B. Fisher, University of Leister, UK; Prof. M. Stojaković,

University of Novi Sad, Yugoslavia; Prof. D. Mihet, West University

Acknowledgements

vii

of Timisoara, Romania; Prof. A. Razani, Imam Khomeini

International University, Iran; Prof. S. Kutukcu, Ondokuz Mayis

University, Turkey; Prof. Z. Liu, Liaoning Normal University, China

and Prof. M.S. Khan, Sultan Qaboos University, Oman, helped me

by providing their research papers. I am extremely thankful for their

valuable cooperation and moral support. Also I wish to express my

gratefulness to all those mathematicians whose research papers or

books were used by me frequently during the present study.

I am highly obliged to Mrs. Veenapani Pant for her

affectionate and inspirational behaviour. Also, I am very thankful to

Dr. (Ms.) Anjali Pant (Govt. Polytechnic, Nainital) and Ms. Indu Pant

for their well wishes.

Also, I am very thankful to Mr. Santosh Kumar Chauhan for

his affection and blessings and my cousin Mr. Ashok Kumar

(Research Scholar, M.J.P. Rohilkhand University, Bareilly).

I express my indebtedness to my friends Mr. Suraj Singh

Saini (Research Scholar, IIT Delhi), Mr. Irshad Aalam, Mr. Naresh

Kumar and Mr. Chitresh Kumar Mittal.

Acknowledgements

viii

I dedicate this dissertation to my father Late Shri Ramdhari

Singh Chauhan who had been the main source of inspiration

behind this difficult task. During the course of this study, due to his

untimely death, I was totally broken down with sorrow. However, the

emotional support of my mother Mrs. Kamla Devi, my younger

brother Er. Ajay Kumar and my sister Ms. Sangeeta went a long way

in inspiring me to accomplish this task. It is really difficult for me to

find words to express my feelings and gratitude towards all my

family members.

Last but not the least; I extend my thanks to all the teachers,

my colleagues and friends who helped me directly or indirectly

during the progress of this task.

Date: SUNEEL KUMAR

E-mail: [email protected]/

[email protected]

CONTENTS

PREFACE i

ACKNOWLEDGEMENTS vi

CHAPTER I Introduction 1

CHAPTER II Common Fixed Points of Contraction 20

Mappings

CHAPTER III Common Fixed Points of Expansion 49

Mappings

CHAPTER IV Coincidences and Fixed Points of 69

Probabilistic Densifying Mappings

CHAPTER V Related Fixed Point Theorems 81

CHAPTER VI Applications 102

REFERENCES 113

LIST OF PUBLICATIONS 135

CHAPTER I

INTRODUCTION

In the foundation of this chapter, we give some fundamental

concepts and notions needed for the development of fixed point

theory in probabilistic metric spaces (briefly, PM-spaces) via metric

spaces. We begin with the notion of metric spaces and then define

notions of distribution functions and probabilistic metric spaces.

After it, we define convergence, Cauchy sequence, completeness,

continuity and probabilistic diameter, which are frequently used

terms. This is followed by defining the concepts of commutativity,

weak commutativity, R -weak commutativity, compatibility, weak

compatibility and reciprocal continuity of mappings in PM-spaces.

We shall also present a brief survey of the development of fixed

point theory in PM-spaces.

We know that in real measurements, assigning a fixed

number to the distance between two points is an over idealized way

of thinking. Indeed, the distance between two points is an average

of several measurements. Therefore, Karl Menger introduced the

Introduction

2

notion of a PM-space as a generalization of metric space. Fréchet

[39] wrote an outstanding doctoral dissertation ‘Sur Quelques Points

Du Calcul Fonctionnel’. In it, he introduced the concept of a metric

space, although he did not invent the name 'metric space' which is

due to Hausdorff [52]. In his dissertation, Fréchet [39] started a

whole new area with his investigations of functionals on a metric

space and formulated the abstract notion of compactness.

In general terms, a metric space is one (which could be a

plane, the surface of a sphere, three-dimensional space etc.) in

which we have the notion of ‘distance’, which fits in, with our

geometrical intuition.

DEFINITION 1.1 Let M be a nonempty set and +→× RMMd : ,

associating a nonnegative real number ),( vud -called the distance

between u and v with every pair ),( vu of elements of M . The

ordered pair ),( dM is called a metric space if d satisfies the

following axioms:

(M1) 0),( =vud if and only if vu = ; (identity of indiscernibles)

(M2) 0),( >vud ; (non-negativity)

(M3) ),(),( uvdvud = ; (symmetry)

(M4) );,(),(),( wvdvudwud +≤ for Mwvu ∈,, (Triangle inequality).

Introduction

3

The idea of introducing probabilistic notions into geometry

was one of the great thoughts of Menger. His motivation came from

the idea that positions, distances, areas, volumes etc. all are subject

to variations in measurement in practice. And as e.g., quantum

mechanics implies, even in theory, some measurements are

necessarily inexact. In 1942, Menger [83] published a note entitled

Statistical Metrics (also see [84]). The idea of Menger was to use

distribution functions instead of nonnegative real numbers as values

of the metric. The notion of a probabilistic metric space corresponds

to the situations when we do not know exactly the distance between

two points; we know only probabilities of possible values of this

distance. In this note he explained how to replace the numerical

distance between two points u and v by a function vuF , whose

value )(, xF vu at the real number x is interpreted as the probability

that the distance between u and v is less than x . B. Schweizer and

A. Sklar [124] took up the work initiated by Menger [83] and

developed what is now called the theory of probabilistic metric

spaces (see [125]).

Introduction

4

DEFINITION 1.2 A mapping +→ RRF : is called a distribution

function if it is non-decreasing and left continuous with 0)( =∈

xFinfRx

and .1)( =∈

xFsupRx

We shall denote by ℑ the set of all distribution functions while

H will always denote the specific distribution function defined by

>

≤=

.0,1

;0,0)(

x

xxH

DEFINITION 1.3 A PM-space is an ordered pair ),( FX , where X

is a nonempty set of elements and F is a mapping from XX × to

ℑ , the collection of all distribution functions. The value of F at

XXvu ×∈),( is represented by vuF , . The functions vuF , are

assumed to satisfy the following conditions:

(PM1) 1)(, =xF vu for all 0>x iff ;vu =

(PM2) ;0)0(, =vuF

(PM3) ;)()( ,, xFxF uvvu =

(PM4) if 1)(, =xF vu and 1)(, =yF wv

then 1)(, =+ yxF wu

Introduction

5

for all Xwvu ∈,, and ., +∈ Ryx

DEFINITION 1.4 A mapping ]1,0[]1,0[]1,0[: →×t is called a

triangular norm (abbreviated, t -norm) if the following conditions are

satisfied:

(T1) aat =)1,( for all ;]1,0[∈a

(T2) );,(),( abtbat =

(T3) ),(),( batdct ≥ for ;, bdac ≥≥

(T4) ));,(,()),,(( cbtatcbatt =

for all ].1,0[,,, ∈dcba

EXAMPLE 1.1 The following are the four basic t -norms:

(i) The minimum t -norm, Mt , is defined by

),,(),( yxminyxtM =

(ii) The product t -norm, Pt , is defined by

,.),( yxyxtP =

(iii) The Lukasiewicz t -norm, Lt , is defined by

),0,1(),( −+= yxmaxyxtL

Introduction

6

(iv) The weakest t -norm, the drastic product, Dt , is defined

by

=

=otherwise.

yxmaxifyxminyxtD

0

,1),(),(),(

As regards the pointwise ordering, we have the inequalities

.MPLD tttt <<<

DEFINITION 1.5 A Menger space is a triplet ),,( tFX , where

),( FX is a PM-space and t -norm t is such that the inequality

)}(),({)( ,,, yFxFtyxF wvvuwu ≥+

holds for all Xwvu ∈,, and all 0, ≥yx .

Every metric space ),( dX can be realized as a PM-space by

taking ℑ→× XXF : defined by )),(()(, vudxHxF vu −= for all vu,

in .X

Schweizer, Sklar and Thorp [126] proved that if ),,( tFX is a

Menger space with ,1),(10

=<<

xxtsupx

then ),,( tFX is a Hausdroff

Introduction

7

topological space in the topology induced by the family of ),( λε -

neighbourhoods

},0,0,:),({ >λ>ε∈λε XuUu

where

}.1)(:{),( , λ−>ε∈=λε vuu FXvU

The following definition is due to Schweizer and Sklar [125]:

DEFINITION 1.6 Let ),,( tFX be a Menger space with

1),(10

=<<

xxtsupx

.

A sequence }{ nu in X is said to be convergent to Xz ∈ if

for any given 0>ε and ,0>λ there exists a positive integer

),( λε= NN such that λ−>ε 1)(, zunF whenever Nn ≥ .

A sequence }{ nu in X is called a Cauchy sequence if for any

0>ε and ,0>λ there exists a positive integer ),( λε= NN such

that λ−>ε 1)(, mn uuF whenever Nmn ≥, .

A Menger space ),,( tFX is said to be complete if each

Cauchy sequence in X is convergent to some point z in X .

Introduction

8

The following definition is given in [115]:

DEFINITION 1.7 Let ),,( tFX be a Menger space. The mapping

XXA →: is continuous at Xu ∈ , if for each 10 <λ< , there exists

a real number 10 <δ< satisfying the following condition:

δ−≥1)(, xF vu implies λ−≥1)()(),( xF vAuA

for each 0>x and Xv∈ .

The notion of the probabilistic diameter of a nonempty set in a

PM- space was introduced by Egbert [28] as follows:

DEFINITION 1.8 Let A be a non-empty subset of X . A function

(.)AD defined by

=∈<

)()( ,,

yFinfsupxD vuAvuxy

A

is called probabilistic diameter of A . A is said to be bounded if

1)( =∈

xDsup ARx

.

DEFINITION 1.9 Two self-mappings A and S of a PM-space

),( FX are said to be commuting if SAzASz = for each z in .X

In 1982, Sessa [129] initiated the tradition of relaxing

commutativity in fixed point theorems by introducing the notion of

Introduction

9

weakly commuting mappings in metric spaces. The counterpart of

weakly commuting mappings in PM-spaces was studied by Singh

and Pant [140] as follows:


),( FX are said to be weakly commuting if )()( ,, xFxF SzAzSAzASz ≥

for each z in X and .0>x

REMARK 1.1 Every pair of commuting self-maps is weakly

commuting but the reverse is not true.

The following example is due to Singh, Pant and Talwar [144].

It shows that weakly commuting mappings need not to be

commuting.

EXAMPLE 1.2 Let },,{ wvuX = and let F be defined via

>ε

≤ε<

≤ε

=ε=ε=ε=ε

.2,1

,20,2/1

,0,0

)()()()( ,,,,

if

if

if

FFFF wvvwuwwu

and

>ε

≤ε<

≤ε

=ε=ε

.2/3,1

,2/30,2/1

,0,0

)()( ,,

if

if

if

FF uvvu

Introduction

10

Then ),,( MtFX is a Menger space. Let XXBA →:, be such that

vwAuvAuA === )(,)()( and uwBvBuB === )()()( . Then, it can

be verified that A and B are weakly commuting but not commuting.

In 1986, Jungck [65] introduced a new class of mappings,

known as compatible mappings and proved some fixed point

theorems in metric spaces. He showed that each pair of weakly

commuting self-maps is compatible but the reverse is not true. The

following example is due to Jungck [65]:

EXMAPLE 1.3 Let ),0[ ∞=M be endowed with usual metric and

MMSA →:, such that 3uAu = and 32uSu = . Then SAuASu ≠ .

So, A and S are not commuting on M and SuAuSAuASu −>− .

Therefore, A and S are not weakly commuting as well on M .

However, MSuAulimu ∈=−→ 00 and it implies

00 =−→ SAuASulimu . Therefore, A and S are compatible.

The counterpart of compatibility in a PM-space was

introduced in 1991, by Mishra [88] as follows:

Introduction

11


),( FX , will be called compatible if and only if 1)(, →xFnn SAuASu

for all 0>x , whenever }{ nu is a sequence in X such that

zSuAu nn →, for some z in X .

In 1994, R.P. Pant [100] introduced the concept of R -weakly

commuting mappings in metric space while in 2007, Kohli and

Vasishtha [72] extended the concept of R -weakly commuting

mappings in PM-spaces as follows:


),( FX are said to be R -weakly commuting if there exists a positive

real number R such that )/()( ,, RxFxF SzAzSAzASz ≥ for each z in

X and .0>x

REMARK 1.2 Clearly, every pair of weakly commuting mappings is

R -weakly commuting with .1=R

REMARK 1.3 Each pair of weakly commuting self-maps is R -

weakly commuting but the reverse is not true.

Introduction

12

The following example due to Kohli and Vashistha [72], shows

that R -weakly commuting mappings need not to be weakly

commuting.

EXAMPLE 1.4 Let RX = , the set of all real numbers and let

=ε

>ε

ε−−

=ε

0,0

0,)(,

if

ifvu

expF vu

for all vu, in X .

Then ),( FX is a probabilistic metric space.

Let XXSA →:, be defined by 12)( −= uuA and 2)( uuS = . Then

ε−−

=ε2

,12

)(u

expF SAuASu and

ε−−

=ε2

,1

)/(uR

expRF SuAu

Therefore, for 2=R , A and S are R -weakly commuting mappings.

However, A and S are not weakly commuting mappings since

exponential function is strictly increasing.

In 1998, Jungck and Rhoades [67] termed a pair of self-

mappings to be weakly compatible if they commute at their

coincidence points and proved some fixed point theorems in metric

spaces. Note that each pair of compatible self-mappings is weakly

compatible but the reverse is not true. This fact can be easily

Introduction

13

understood from the following example given by Singh and Mishra

[135].

EXAMPLE 1.5 Let ),0[ ∞=M be a metric space with the usual

metric and MMSA →:, be defined by

∞∈

∈=

),1[,1

)1,0[,

uif

uifuAu and

)1( u

uSu

+= , if Mu ∈ ,

then A and S are not compatible on ,M but commute at their

coincidence point 0=u . Indeed A and S are weakly compatible.

In 2005, Singh and Jain [134] extended the notion of weakly

compatible mappings in PM-spaces as follows:


),( FX are said to be weakly compatible if they commute at their

coincidence points, i.e. if SzAz = for some ,Xz ∈ then .SAzASz =

Most of the common fixed point theorems for contraction

mappings invariably require a compatibility condition besides

assuming continuity of at least one of the mappings. In 1999, R.P.

Pant [103] noticed these criteria for fixed points of contraction

mappings and introduced a new continuity condition, known as

Introduction

14

reciprocal continuity and obtained a common fixed point theorem by

using the compatibility in metric spaces. He also showed that in the

setting of common fixed point theorems for compatible mappings

satisfying contraction conditions, the notion of reciprocal continuity is

weaker than the continuity of one of the mappings.

Note that if A and S are both continuous self-mappings, then

they are obviously reciprocally continuous but the converse is not

true. This fact can be easily understood from the following example

given by Kumar and Chugh [75]:

EXAMPLE 1.6 Let ]20,2[=M and d be usual metric on M . Define

mappings MMSA →:, by

>

==

2,3

2,2

uif

uifAu and

>

==

.2,6

2,2

uif

uifSu

It is to be noted that A and S are reciprocally continuous mappings

but they are not continuous.

Introduction

15

The counterpart of reciprocal continuity of mappings in PM-

spaces is as follows:


),( FX will be called reciprocally continuous if AzASun → and

SzSAun → , whenever }{ nu is a sequence in X such that


For several weaker forms of commuting maps, Singh and

Tomar [145] is a good reference.

The Banach fixed point theorem (also known as the

contraction mapping theorem or contraction mapping principle) is an

important tool in the theory of metric spaces. Perhaps, this is the

most frequently cited and famous theorem, which guarantees that a

contraction mapping defined on a complete metric space, has a

unique fixed point. The theorem is named after Stefan Banach [3]

and was first stated by him in 1922. A similar theorem does not hold

in a complete probabilistic metric space. The problem is that the

triangle function in such spaces is often not strong enough to

guarantee that the sequence of iterates of a point under a

contraction map is a Cauchy sequence [131]. Two different

Introduction

16

approaches have been pursued. One is to identify those triangle

functions which are strong enough to guarantee that the sequence

of iterates of a point is a Cauchy sequence [47]. The other is to

modify the original definition of a contraction map. The latter was

done by Hicks [53]. However, as shown by Schweizer, Sherwood

and Tardiff [123], a contraction map in Hicks’ sense is a contraction

map in a related metric spaces.

In [119], Rhoades summarized various types of contraction

mappings conditions for the existence of their fixed points. There are

a number of generalizations of Banach contraction principle in

different settings such as metric space ([18]), 2-metric space ([41]),

PM-space ([127], [150]), 2-PM-space ([14], [160]), fuzzy metric

space ([45], [85]), D -metric space ([23], [24]) and probabilistic D -

metric space ([17]).

Probabilistic contractions were first defined and studied by

V.M. Sehgal [127] in his doctoral dissertation at Wayne State

University.

DEFINITION 1.15 Let X be a nonempty set and let F be a

probabilistic distance on X . A mapping XXA →: is called a

probabilistic contraction if there exists )1,0(∈h such that

Introduction

17

.0,,),()()1.15.1( ,, >∀∈∀≥ xXvuxFhxF vuAvAu

The interpretation of (1.15.1) is as follows: The probability that the

distance between the image points AvAu, is less than hx is at

least as large as the probability that the distance between vu, is

less than x .

In [128], it is shown that any contraction map on a complete

Menger space in which the triangle inequality is formulated under

the strongest triangular norm Mt has a unique fixed point. In [131],

Sherwood showed that one can construct a complete Menger space

under Lt and a fixed point-free contraction map on that space.

Hicks [53] observed that fixed point theorems for certain

contraction mappings on a Menger space endowed with a ‘triangular

norm 'Mt may be obtained from corresponding results in metric

spaces. After it, there has been a vigorous study of fixed points of

the contraction mappings in PM-spaces; see ([7], [11], [16], [19],

[29], [44], [50], [51], [54], [55], [68], [91], [92], [105], [113], [114],

[141], [143], [148], [149], [153]).

Introduction

18

Banach contraction principle also yields a fixed point theorem

for a diametrically opposite class of mappings, viz. expansion

mappings. The study of fixed point of single expansion mapping in a

metric space is initiated by Wang, Li, Gao and Iseki [159].

Subsequently, a large number of results for such mappings have

been proved for 1-4 mappings in metric spaces; see ([106], [107],

[108], [110], [120], [121], [130], [132]). On the other hand, in 1987,

Pant, Dimri and Singh [95] introduced the notion of expansion

mappings on PM-spaces and proved some fixed and common fixed

point theorems for 1-2 mappings. Later, Vasuki [156], Kumar, Chugh

and Vats [76] and Kumar [74] also established some fixed point

theorems for expansion mappings in Menger spaces.

Furi and Vignoli [40] studied the fundamental properties of

Kuratowski’s measure of non-compactness of a bounded set in a

metric space and introduced the notion of densifying mappings. The

counterpart of the study of Kuratowski’s measure of non-

compactness in PM-spaces was initiated by Bocşan and Constantin

[7]. The concept of probabilistic densifying mapping was introduced

by Bocşan [5]. Later, numerous results have been proved for such

mappings; see ([12], [25], [94], [99], [137], [151]).

Introduction

19

Fisher [36] investigated the conditions ensuring the existence

of a relation between fixed points of two contraction mappings on

two metric spaces. Afterwards, a number of studies of the relation

between the fixed points of contraction mappings in two and three

different metric spaces have been done in [37], [38], [62], [63], [90].

In 2002, Pant [93] initiated the study of the relation between the

fixed points of two contraction mappings in two different Menger

spaces by generalizing the results of Fisher [36, 37].

The theory of PM-spaces is also of fundamental importance in

probabilistic functional analysis, nonlinear analysis and its

applications; see ([4], [13], [48]).

CHAPTER II

COMMON FIXED POINTS OF

CONTRACTION MAPPINGS

The study of fixed point theorems in probabilistic metric spaces

is of nascent interest and is an active area of research. The first effort

in this direction was made by Sehgal [127], who in his doctoral

dissertation initiated the study of contraction mapping theorems in

probabilistic metric spaces. Since then, the subject has been further

investigated by a host of authors including Sehgal and Bharucha-

Reid [128], Schweizer and Sklar [124, 125, 126], Schweizer,

Sherwoad and Tardiff [123], Sherwood [131], Bocşan [6], Cain and

Kasriel [9], Istrăteşcu [58, 59], Istrăteşcu and Săcuiu [60], Hicks [53,

54], Alimohammady, Esmaeli and Saadati [1], Hosseini and Saadati

[56] and others. Studies of these authors have culminated in an

elegant theory of fixed points in probabilistic metric spaces, which has

far reaching consequences and are useful in the study of existence of

Common fixed points of contraction mappings

21

solutions of operator equations in probabilistic metric spaces and

probabilistic functional analysis.

In 1975, Ćirić [19] introduced the notion of ‘generalized

contraction’ on a PM-space. Some fixed point theorems for a pair of

commuting mappings in PM-spaces have been proved in [20], [87],

[118], [154]. Singh and Pant [138] introduced the notion of

‘generalized contraction triplet’ in PM-spaces in which one of the

mappings commutes with the other two; and proved some common

fixed point theorems; also see [139]. Later, Singh, Mishra and Pant

[136] introduced the notion of ‘generalized contraction quadruplet’ in

PM-spaces. Chamola [10], Vasuki [155] and Kutukcu [80] studied the

fixed points of mappings satisfying new contraction conditions.

Subsequently, several contraction mapping theorems for commuting

mappings in PM-spaces have been proved in [21], [81], [117], [122],

[142], [143], [146], [147], [153].

There are a number of results on fixed points of mappings

satisfying implicit relations in metric spaces; see for instance [109],

[111], [112]. In 2005, MiheŃ [86] established a fixed point theorem

concerning probabilistic contractions satisfying an implicit relation.


22

In this chapter, we obtain some common fixed point theorems

for a triplet and quadruplet of mappings satisfying new contraction

conditions in Menger spaces. In the end of this chapter, we prove a

common fixed point theorem for pointwise R -weakly commuting

mappings having reciprocal continuity and satisfying an implicit

relation.

Singh and Pant [138] introduced the following:

DEFINITION 2.1 Three mappings BA, and T on a PM-space ),( FX

will be called a generalized contraction triplet );,( TBA if there exists

a constant )1,0(∈h such that for every Xvu ∈,

{

)}2(

),2(),(),(),()(

,

,,,,,

xF

xFxFxFxFminhxF

TuBv

TvAuTvBvTuAuTvTuBvAu ≥

holds for all .0>x

DEFINITION 2.2 Let BA, and T be mappings from X to itself. If

there exists a point 0u in X and a sequence }{ nu in X such that

1222212 , +++ == nnnn BuTuAuTu for ...2,1,0=n


23

then the space X will be called );,( TBA -orbitally complete with

respect to 0u or simply ))(;,( 0uTBA -orbitally complete if the closure

of ...}2,1:{ =nTun is complete.

DEFINITION 2.3 T will be called ))(;,( 0uTBA -orbitally continuous if

the restriction of T on the closure of ...}2,1:{ =nTun is continuous.

The following lemma is given in [139]:

LEMMA 2.1 Let }{ nu be a sequence in a Menger space ( )tFX ,, ,

where t is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . If

there exists a constant )1,0(∈h such that

...3,2,1),()( ,, 11=≥

−+nxFhxF

nnnn uuuu

then }{ nu is a Cauchy sequence in X .

LEMMA 2.2 Let ( )tFX ,, be a Menger space, if there exists )1,0(∈h

such that for all Xvu ∈, , )()( ,, xFhxF vuvu ≥ . Then vu = .

The foregoing theorems 2.1 and 2.2 (appearing in [96]) have

been published in

“ Journal of Natural & Physical Sciences 19 (1) (2005), 29-37”.


24

THEOREM 2.1 Let ( )tFX ,, be a Menger spaces, where t is

continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let

XXTBA →:,, satisfy the following condition:

)}2()2(),()(

),()(),()({))(()1.1.2(

,,,,

,,,,2

,

xFxFxFxF

xFxFxFxFminhxF

AuTvBvTuBvTvAuTu

BvTvTvTuAuTuTvTuBvAu ≥

for all vu, in X and )1,0(∈h . Further, assume that either

TAAT = or TBBT = . If there exists a point 0u in X such that X is

))(;,( 0uTBA –orbitally complete and T is ))(;,( 0uTBA –orbitally

continuous, then BA, and T have a unique common fixed point and

}{ nTu converges to the common fixed point.

PROOF. Let .0 Xu ∈ Define }{ nu as follows:

1222212 , +++ == nnnn BuTuAuTu for ...2,1,0=n

By (2.1.1),

)}()(

),()(),()(

),()({

))(())((

122122

122122122122

122122

1222212

,,

,,,,

,,

2,

2,

xFxF

xFxFxFxF

xFxFmin

hxFhxF

nnnn

nnnnnnnn

nnnn

nnnn

TuTuTuTu

TuTuTuTuTuTuTuTu

TuTuTuTu

BuAuTuTu

++

++++

++

+++

≥

=

giving


25

)}.()(

,))({())((

2212122

1222212

,,

2,

2,

xFxF

xFminhxF

nnnn

nnnn

TuTuTuTu

TuTuTuTu

+++

+++≥

since )}.(),({)2(221222222 ,,, xFxFminxF

nnnnnn TuTuTuTuTuTu +++≥

Now suppose that

2,

2, ))(())((

1222212xFhxF

nnnn TuTuTuTu +++≥

then

)()(1222212 ,, xFhxF

nnnn TuTuTuTu +++≥ .

Again suppose that

)}()({))((22121222212 ,,

2, xFxFminhxF

nnnnnn TuTuTuTuTuTu +++++≥

then

).()(1222212 ,, xFhxF


So in both cases, we have

).()(1222212 ,, xFhxF


Similarly,

).()(32123222 ,, xFhxF

nnnn TuTuTuTu ++++≥

Thus, in general,

).()(121 ,, xFhxF



26

By Lemma 2.1, }{ nTu is a Cauchy sequence in X . Since X is

))(;,( 0uTBA –orbitally complete, }{ nTu converges to a point z in

X .

Now we prove that .TzBz =

Let ),( λεBzU be a neighbourhood of Bz . By the continuity of ,T

TzTTu n →2 and TzTTu n →+12 . So there exists an integer

( )λε= ,NN such that for ,0, >λε

( ) λ−>ε− 1)2.1.2(2

1, h

hTzTun

F and ( ) λ−>ε−+

12

1,12 h

hTzTu n

F .

First, suppose that TAAT = , then by (2.1.1),

)}/2()/2(

),/()/(),/()/(

),/()/({

))(())(())((

122

1222

1222

2212

,,

,,,,

,,

2,

2,

2,

hFhF

hFhFhFhF

hFhFmin

FFF

nn

nnn

nnn

nnn

TTuTzBzTTu

BzTzTTuTTuBzTzTzTTu

TTuTTuTzTTu

BzATuBzTAuBzTTu

εε

εεεε

εε≥

ε=ε=ε

+

+

+

+

{ ( ) ( ){ }( ) ( ){ }

( ) ( ){ } ( ) ( ){ }{ }{ }}}/()/()/()/(

,,,

,,)/(

,,)/(

12222

22122

222

1222

,,,,

2

1,2

1,2

1,2

1,

2

1,2

1,,

2

1,2

1,,

hFhFhFhF

FFFF

FFhF

FFhFmin

nnnn

nnnn

nnn

nnn

TTuTTuTTuTzBzTzTzTTu

h

hBzTTuh

hTTuTzh

hTTuTzh

hTzTTu

h

hBzTTuh

hTTuTzTzTTu

h

hTTuTzh

hTzTTuTzTTu

εεεε

εεεε

εεε

εεε≥

+

+

+

+−+−

+−

+−


27

{ ( ) ( )( )( ) ( )

})/()/(

),/()/(),/()/(

,)/(,

,)/(,)/(

122

21222

22

2222

,,

,,,,

2,

2

2

1,

2

1,,2

1,,

hFhF

hFhFhFhF

hFF

FhFFhFmin

nn

nnnn

nn

nnnn

TTuTTuBzTz

TTuTzBzTzTTuTTuTzTTu

TzTTuh

hTzTTu

h

hTTuTzTzTTuh

hTzTTuTzTTu

εε

εεεε

εε

εεεε≥

+

+

−

−−

2)1( λ−> by (2.1.2)

giving λ−>ε 1)(,2 BzTTu nF for all .Nn ≥

Consequently, .BzTz =

To prove zTz = , let ),( λεTzU be a neighbourhood of Tz . Since

}{ nTu is a Cauchy sequence, there exists an integer ( )λε= ,NN

such that

( ) λ−>ε−+

1)3.1.2(2

1, 122 h

hTuTu nn

F for all .Nn ≥

By (2.1.1), we have

( ) ( ){ }{),/(),/(

),/(,

)}/2()/2(

),/()/(),/()/(

),/()/({

))(())((

1222

12212122

122

1222

1222

212

,,

,2

1,2

1,

,,

,,,,

,,

2,

2,

hFhF

hFFFmin

hFhF

hFhFhFhF

hFhFmin

FF

nnn

nnnnn

nn

nnn

nnn

nn

TuTuTzTu

TuTuh

hTzTuh

hTuTu

TuTzTzTu

TzTzTuTuTzTzTzTu

TuTuTzTu

BzAuTzTu

εε

εεε≥

εε

εεεε

εε≥

ε=ε

+

+++

+

+

+

+

+−


28

{ }{ }})/()/()/(),/(122212122 ,,,, hFhFhFhF

nnnnnn TuTuTuTzTzTuTuTu εεεε+++

( ){

})/()/(

),/()/(,))/((

),/()/(),/(

),/(),/(

12212

1212122

21222

122122122

,,

,,2

,

,,,

,,2

1,

hFhF

hFhFhF

hFhFhF

hFhFFmin

nnn

nnnn

nnnn

nnnnnn

TuTuTzTu

TuTzTzTuTuTu

TuTzTuTuTzTu

TuTuTuTuh

hTuTu

εε

εεε

εεε

εεε≥

++

+++

+

+++−

( )( )22

1, 122

ε≥ −+ h

hTuTu nn

F

2)1( λ−> by (2.1.3),

giving

λ−>ε+

1)(,12 TzTu nF for all .Nn ≥

So, Tzz = since .12 TzTu n →+

So far we have proved that .zTzBz ==

Now we claim that z is also a fixed point of A . For this, let .zAz ≠

By (2.1.1),

)}2()2(),()(

),()(),()({

))(())((

,,,,

,,,,

2,

2,

xFxFxFxF

xFxFxFxFmin

hxFhxF

AzTzBzTzBzTzAzTz

BzTzTzTzAzTzTzTz

BzAzzAz

≥

=

giving

2,

2, ))(())(( xFhxF AzzzAz ≥ , a contradiction.


29

Therefore, .zAz =

Thus, .zTzBzAz ===

Same result holds good if .TBBT =

To prove the uniqueness of z as a common fixed point of BA, and

,T let )( zw ≠ be another fixed point.

By (2.1.1),

}))((,1),(),({

)}2()2(),()(

),()(),()({

))(())((

2,,,

,,,,

,,,,

2,

2,

xFxFxFmin

xFxFxFxF

xFxFxFxFmin

hxFhxF

zwzwzw

AwTzBzTwBzTzAwTw

BzTzTzTwAwTwTzTw

BzAwzw

≥

≥

=

giving

).())(( ,2

, xFhxF zwzw ≥

This is possible only when zw= . Hence, z is a unique common

fixed point of BA, and T .

This completes the proof of the theorem.

REMARK 2.1 The result in [138] for a contraction triplet has been

proved by taking one of the mappings commuting with both of the

other two mappings. But in Theorem 2.1, one of the mappings


30

commutes with either of the two. Hence, Theorem 2.1 improves the

result in [138] and a number of other results as well.

COROLLARY 2.1 Let BAX ,, and T be as in Theorem 2.1. If the

mappings ,A B and T satisfy the following condition

)}2()2(),()2(

),2()(),()({))(().1.2(

,,,,

,,,,2

,

xFxFxFxF

xFxFxFxFminhxFa

BvTuAuTuBvTvAuTv

BvTuAuTuBvTvAuTuBvAu ≥

for all vu, in X and )1,0(∈h . Then the conclusion of Theorem 2.1

holds.

PROOF. The proof may be completed on the lines of the preceding

theorem.

THEOREM 2.2 Let ( )tFX ,, be a complete Menger spaces, where t

is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let

XXTBA →:,, satisfy the condition (2.1.1) for all vu, in X and

)1,0(∈h . Further, assume that either TAAT = or TBBT = and

).()()( XTXBXA ⊆∪ If T is continuous, then BA, and T have a

unique common fixed point.


31

PROOF. Let .0 Xu ∈ Define the sequence }{ nu in X given by the

rule 1222212 , +++ == nnnn BuTuAuTu for ...2,1,0=n

This can be done by virtue of ).()()( XTXBXA ⊆∪ Now the proof of

Theorem 2.1 works.

COROLLARY 2.2 Let BAX ,, and T be as in Theorem 2.2. If the

mappings ,A B and T satisfy the condition ).1.2( a for all vu, in X

and )1,0(∈h . Then the conclusion of Theorem 2.2 holds.

In 1987, Singh, Mishra and Pant [136] extended the result for a

triplet of self-mappings given by Singh and Pant [138] to a quadruplet

of self-mappings by introducing the notion of ‘generalized contraction

quadruplet );,( STBA ’ for self-mappings TSBA ,,, in PM-spaces as

follows:

DEFINITION 2.4 Four mappings SBA ,, and T on a PM-space

),( FX will be called a generalized contraction quadruplet );,( STBA

if there exists a constant )1,0(∈h such that for every ,, Xvu ∈

{

)}2(

),2(),(),(),()(

,

,,,,,

xF

xFxFxFxFminhxF

TvAu

BvSuBvTvSuAuTvSuBvAu ≥

holds for all .0>x


32

DEFINITION 2.5 Let SBA ,, and T be mappings from X to itself. If

there exists a point 0u in X and a sequence }{ nu in X such that

2212122 , +++ == nnnn TSuBSuTSuATu for ...2,1,0=n

then the space X will be called );,( STBA -orbitally complete with

respect to 0u or simply ))(;,( 0uSTBA -orbitally complete if the

closure of ...}2,1:{ =nSTun is complete.

DEFINITION 2.6 A mapping on X will be called ))(;,( 0uSTBA -

orbitally continuous if the restriction of the mapping on the closure of

...}2,1:{ =nSTun is continuous.

Next, we present some common fixed point theorems for

quadruplet of self-mappings satisfying the new contraction conditions

in PM-spaces. These results (see [78]) are published in

“ Varāhmihir Journal of Mathematical Sciences 5(2005), 227-234”.

THOEREM 2.3 Let ( )tFX ,, be a Menger spaces, where t is

continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let ,,, SBA

XXT →: satisfy the following condition:


33

)}2()2(),()(

),()(),()({))(()1.3.2(

,,,,

,,,,2

,

xFxFxFxF

xFxFxFxFminhxF

AuTvBvSuBvTvAuSu

BvTvTvSuAuSuTvSuBvAu ≥

for all vu, in X and )1,0(∈h . Further, assume that TSST = and

either (i) TAATSAAS == , or (ii) TBBTSBBS == , . If there exists a

point 0u in X such that X is ))(;,( 0uSTBA –orbitally complete and

T is ))(;,( 0uSTBA –orbitally continuous, then SBA ,, and T have a

unique common fixed point and }{ nSTu converges to the common

fixed point.

PROOF. The proof may be completed on the lines of Theorem 2.1.

REMARK 2.2 The result in [136] for a contraction quadruplet has

been proved for mappings satisfying ,,, TAATSAASTSST ===

SBBS = and TBBT = . But in Theorem 2.3 we have taken the

mappings satisfying the conditions TSST = and either (i)

TAATSAAS == , or (ii) TBBTSBBS == , . Being the present

condition weaker than those in [136], the above result improves the

result in [136] and a number of other results as well.


34

REMARK 2.3 By setting TS = in (2.3.1), we obtain Theorem 2.1 for

a contraction triplet.

COROLLARY 2.3 Let SBAX ,,, and T be as in Theorem 2.3. If

the mappings SBA ,, and T satisfy

)}2()2(,)2()(

),2()(),()({))(().3.2(

,,,,

,,,,2

,

xFxFxFxF

xFxFxFxFminhxFa

AuTvBvSuAuTvBvTv

BvSuAuSuBvTvAuSuBvAu ≥

for all vu, in X and )1,0(∈h . Then the conclusion of Theorem

2.3 holds.

PROOF. The proof may be completed on the lines of the preceding

theorem.

THEOREM 2.4 Let ( )tFX ,, be a complete Menger spaces, where t

is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let

XTSBA :,,, X→ satisfy the condition (2.3.1) for all vu, in X and

)1,0(∈h . Further, assume that TSST = and either (i)

TAATSAAS == , or (ii) TBBTSBBS == , and


35

).()()( XTSXBXA ⊆∪ If T and S are continuous, then SBA ,, and

T have a unique common fixed point.

COROLLARY 2.4 Let SBAX ,,, and T be as in Theorem 2.4. If

the mappings SBA ,, and T satisfy the condition ).3.2( a for all vu,

in X and )1,0(∈h . Then the conclusion of Theorem 2.4 holds.

As stated in first chapter, there is a tradition of relaxing

commutativity in fixed point theory. Sessa [129] formulated the notion

of weak commutativity. Subsequently, Jungck [65] gave the notion of

compatibility and Pant [100] defined R -weak commutativity and

obtained common fixed point theorems concerning them in metric

spaces.

These notions of improving commutativity have been extended

to PM-spaces by various authors. For example, Singh and Pant [140]

extended the notion of weak commutativity, Mishra [88] extended the

notion of compatibility and Kohli and Vashistha [72] extended the

notion of R -weak commutativity and proved common fixed point

theorems for contraction mappings by applying them in PM-spaces.


36

Subsequently, many authors obtained common fixed point theorems

for such weaker forms of commutativity in PM-spaces; see for

instance [11], [16], [54], [55], [57], [69], [116], [117], [144].

In 1994, R.P. Pant [100] introduced the concept of pointwise

R -weakly commuting mapping and proved two common fixed point

theorems in metric spaces. Introduction of the notion of pointwise

R -weakly commuting mappings made the scope of the study of

common fixed point theorems from the class of compatible mappings

to the wider class of pointwise R -weakly commuting mappings. Later,

Pant [101, 102] and Kumar and Chugh [75] also proved some

common fixed point theorems for such mappings in metric spaces. In

2007, Kohli and Vasishtha [72] extended the concept of pointwise

R -weakly commuting mappings in PM-spaces as follows:

DEFINITION 2.7 Two self-mappings A and S of a PM-space ),( FX

are said to be pointwise R -weakly commuting if given z in X there

exist 0>R such that )/()( ,, RxFxF SzAzSAzASz ≥ for .0>x

REMARK 2.4 It is obvious that A and S can fail to be pointwise

R -weakly commuting only if there is some z in X such that SzAz =


37

but ,SAzASz ≠ that is, only if they posses a coincidence point at

which they do not commute. This means that a contractive type

mapping pair cannot posses a common fixed point without being

pointwise R -weakly commuting since a common fixed point is also a

coincidence point at which the mappings commute, and contractive

conditions exclude the possibility of two types of coincidence points.

Also, compatible mappings are necessarily pointwise R -weakly

commuting since compatible mappings commute at their coincidence

points. However, pointwise R -weakly commuting need not to be

compatible as shown in the following example:

EXAMPLE 2.1 Let ]20,2[=X and let F be defined by

=ε

>ε−+ε

ε

=ε

0,0

0,)(,

if

ifvuF vu

Then ),( FX is a probabilistic metric space. Let A and S be self-

mappings of X defined as

≤<

>==

52,8

;52,2

u

uoruAu and

>−

≤<+

=

=

5,3

52,12

;2,2

uu

uu

u

Su


38

It can be verified that A and S are pointwise R -weakly commuting

mappings but not compatible. Also, neither A nor S is continuous,

not even at their coincidence points.

In 2005, MiheŃ [86] established a fixed point theorem

concerning probabilistic contractions satisfying an implicit relation.

This implicit relation is similar to that in [110]. In [110], Popa used the

family 4F of implicit real functions to find the fixed points of two pairs

of semi-compatible mappings in a d -compatible topological space.

Here 4F denotes the family of all real continuous functions satisfying

the following properties:

)( hF There exists 1≥h such that for every 0,0 ≥≥ vu

with 0),,,( ≥vuvuF or 0),,,( ≥uvvuF we have hvu ≥ .

)( uF 0)0,0,,( <uuF for all 0>u .

In the following we deal with the class Φ of all real continuous

functions RR →ϕ + 4)(: , non-decreasing in the first argument and

satisfying the following conditions:

)(I For 0, ≥vu , 0),,,( ≥ϕ vuvu or 0),,,( ≥ϕ uvvu implies that

vu ≥ .


39

)(II 0)1,1,,( ≥ϕ uu for all 1≥u .

EXAMPLE 2.2 Define 43214321 ),,,( dxcxbxaxxxxx +++=ϕ , where

Rdcba ∈,,, with 0=+++ dcba , 0,0,0 >+>+> bacaa and

.0>+ da Then .Φ∈ϕ

For 0, ≥vu and ,0),,,( ≥ϕ uvvu we have

,0)()( ≥+++ vcbuda

that is, vdauda )()( +≥+ . Hence vu ≥ , since .0>+ da

Again,

0),,,( ≥ϕ uvvu gives ,0)()( ≥+++ vdbuca

that is, .0)()( ≥+−+ vcauca Hence vu ≥ as .0>+ ca

Also, 0)1,1,,( ≥ϕ uu gives ,0)()( ≥+++ dcuba

that is, ),()( dcuba +−≥+ that is, ),()( bauba +≥+ as

.0=+++ dcba Hence, .1≥u .

As ϕ> ,0a is non-decreasing in the first argument and the result

follows.

EXAMPLE 2.3 Define 43214321 861214),,,( xxxxxxxx −+−=ϕ .

Then .Φ∈ϕ


40

R.P. Pant [103] introduced the notion of reciprocal continuity of

mappings and proved a common fixed point theorem in metric space.

The counterpart of reciprocal continuity of mappings in PM-spaces is

as follows:


),( FX , will be called reciprocally continuous if AzASun → and

SzSAun → , whenever }{ nu is a sequence in X such that


REMARK 2.5 If A and S are both continuous, then they are

obviously reciprocally continuous but converse is not true. Moreover,

in the setting of common fixed point theorems for compatible pair of

mappings satisfying contractive conditions, continuity of one of the

mappings A and S implies their reciprocal continuity but not

conversely.

LEMMA 2.3 Let ( )tFX ,, be a Menger space, where t is continuous

and satisfies xxxt ≥),( for all ]1,0[∈x . Let ),( SA and ),( TB be

pointwise R -weakly commuting pairs of self-mappings of X

satisfying:


41

(2.3.i) );()(),()( XSXBXTXA ⊆⊆

and for some Φ∈ϕ , there exists )1,0(∈h such that for all Xvu ∈,

and 0>x ;

(2.3.ii) 0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF TvBvSuAuTvSuBvAu and

(2.3.iii) .0))(),(),(),(( ,,,, ≥ϕ xFhxFxFhxF TvBvSuAuTvSuBvAu

Then the continuity of one of the mappings in compatible pair

),( SA or ),( TB on ( )tFX ,, implies their reciprocal continuity.

PROOF. First, assume that A and S are compatible and S is

continuous. We show that A and S are reciprocally continuous. Let

}{ nu be a sequence such that zAun → and zSun → for some

Xz ∈ as ∞→n . Since S is continuous, we have SzSAun → and

SzSSun → as ∞→n and since ),( SA is compatible, we have

.1)(, →xFnn SAuASu This implies that ,1)(, →xF SzASun

that is,

SzASun → as ∞→n . By (2.3.i), for each n , there exists nv in X

such that nn TvASu = . Thus, we have ,SzSSun → ,SzSAun →

SzASun → and SzTvn → as ∞→n whenever nn TvASu = .

Now we claim that SzBvn → as ∞→n . Suppose not, then by

(2.3.ii),


42

.0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxFnnnnnnnn TvBvSSuASuTvSSuBvASu

Letting ∞→n ,

,0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF SzBvSzSzSzSzBvSz nn

that is, .0))(,1,1),(( ,, ≥ϕ hxFhxF SzBvSzBv nn

Using )(I , we get 1)(, ≥hxF SzBvn for all .0>x Hence,

.1)(, =hxF SzBvn Thus, .SzBvn →

Again by (2.3.ii),

.0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxFnnnn TvBvSzAzTvSzBvAz

Letting ∞→n ,

.0)1),(,1),(( ,, ≥ϕ xFhxF SzAzSzAz

As ϕ is non-decreasing in the first argument, we have

.0)1),(,1),(( ,, ≥ϕ xFxF SzAzSzAz

Using )(I , we get 1)(, ≥xF SzAz for all 0>x , which gives

1)(, =xF SzAz implying .SzAz =

Thus, SzSAun → and AzSzASun =→ as ∞→n .

Therefore, A and S are reciprocally continuous on .X If the pair

),( TB is assumed to be compatible and T is continuous, the proof is

similar.


43

Now we are in a position to present the following result.

THEOREM 2.5 Let ( )tFX ,, be a complete Menger space, where t

is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let ),( SA and

),( TB be pointwise R -weakly commuting pairs of self-mappings of

X satisfying (2.3.i), (2.3.ii) and (2.3.iii). If one of the mappings in

compatible pair ),( SA or ),( TB is continuous, then SBA ,, and T

have a unique common fixed point.

PROOF. Let Xu ∈0 . By (2.3.i), we define the sequences }{ nu and

}{ nv in X such that for all ...2,1,0=n

12212 ++ == nnn TuAuv , .221222 +++ == nnn SuBuv

Now by (2.3.ii),

0))(),(),(),((12122212122 ,,,, ≥ϕ

++++hxFxFxFhxF

nnnnnnnn TuBuSuAuTuSuBuAu

that is,

.0))(),(),(),((12222121222212 ,,,, ≥ϕ

++++++hxFxFxFhxF

nnnnnnnn vvvvvvvv

Using )(I , we get

).()(1222212 ,, xFhxF

nnnn vvvv +++≥

Similarly, by (2.3.iii) and then using )(I , we get


44

).()(22123222 ,, xFhxF

nnnn vvvv ++++≥

Thus, for any n and ,x we have

).()( ,, 11xFhxF

nnnn vvvv −+≥

By Lemma 2.1, }{ nv is a Cauchy sequence in .X Since X is

complete, }{ nv converges to z . Its subsequences }{ 2nAu , }{ 12 +nBu ,

}{ 2nSu and }{ 12 +nTu also converge to z .

Now suppose that ),( SA is a compatible pair and S is

continuous. Then by Lemma 2.3, A and S are reciprocally

continuous, then AzASu n →2 and SzSAu n →2 . Compatibility of A

and S gives 1)(22 , →xF

nn SAuASu i.e. 1)(, →xF SzAz as ∞→n .

Hence, SzAz = .

Since ),()( XTXA ⊆ there exist a point p in X such that TpAz = .

By (2.3.ii),

,0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF TpBpSzAzTpSzBpAz

that is, .0))(,1,1),(( ,, ≥ϕ hxFhxF AzBpBpAz

Using )(I , we get 1)(, ≥hxF BpAz for all ,0>x which gives

.1)(, =hxF BpAz Hence, BpAz = .


45

Thus, TpBpSzAz === . Since A and S are pointwise R -weakly

commuting mappings, there exists 0>R such that

.1)/()( ,, =≥ RxFxF BpAzSAzASz

That is, SAzASz = and SSzSAzASzAAz === .

Similarly, since B and T are pointwise R -weakly commuting

mappings, we have

.TTpTBpBTpBBp ===

Again by (2.3.ii),

,0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF TpBpSAzAAzTpSAzBpAAz

that is, .0)1,1),(),(( ,, ≥ϕ xFhxF AzAAzAzAAz


.0)1,1),(),(( ,, ≥ϕ xFxF AzAAzAzAAz

Using )(II , we have 1)(, ≥xF AzAAz for all .0>x This gives

1)(, =xF AzAAz implying AzAAz = and .SAzAAzAz ==

Thus, Az is

a common fixed point of A and .S Similarly by (2.5.2), we have that

)( AzBp = is a common fixed point of B

and T . Thus, Az

is a

common fixed point of SBA ,, and T .

Finally, suppose that )( AzAp ≠ is another common fixed point

of SBA ,, and T . Then by (2.3.ii),


46

,0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF TApBApSAzAAzTApSAzBApAAz

that is, .0)1,1),(),(( ,, ≥ϕ xFhxF ApAzApAz


.0)1,1),(),(( ,, ≥ϕ xFxF ApAzApAz

Using )(II , we have 1)(, ≥xF ApAz for all ,0>x which gives

1)(, =xF ApAz implying .ApAz =

Thus, Az is a unique common fixed point of SBA ,, and T .


REMARK 2.6 Theorem 2.5 is an improved extension of the result of

Kumar and Chugh [75, Theorem 3.2] to PM-spaces.

Now taking XITS == (identity mapping) in Theorem 2.5, we have

the following result:

COROLLARY 2.5 Let A and B be self-mappings of X a complete

Menger space ( )tFX ,, such that for some Φ∈ϕ , there exists

)1,0(∈h such that for all Xvu ∈, and 0>x ,

).5.2( a 0))(),(),(),(( ,,,, ≥ϕ hxFxFxFhxF vBvuAuvuBvAu and

).5.2( b .0))(),(),(),(( ,,,, ≥ϕ xFhxFxFhxF vBvuAuvuBvAu


47

If A and B are reciprocally continuous mappings, then A and B


The following example illustrates Theorem 2.5.

EXAMPLE 2.4 Let += RX and let F be defined by

=ε

>ε−+ε

ε

=ε

0,0

0,)(,

if

ifvuF vu

Then ),( FX is a probabilistic metric space. Let SBA ,, and T be self-

mappings of X defined as

1,00 == AuA if 0>u

0=Bu

if 0=u

or 6>u , 2=Bu

if 60 ≤< u

2,00 == SuS

if 0>u

4,00 == TuT

if 6,60 −=≤< uTuu if .6>u

Then SBA ,, and T satisfy all the conditions of Theorem 2.5 with

)1,0(∈h and have a unique common fixed point 0=u . Clearly, A

and S are reciprocally continuous compatible mappings. However, A

and S are not continuous, not even at the common fixed point. The

mappings B and T are non-compatible because if we suppose that


48

there is a sequence }{ nu defined as ,1,1

6 ≥+= nn

un then ,0=nBu

0,0 =→ nn TBuTu and .2=nBTu Hence B and T are non-

compatible but pointwise R -weakly commuting since they commute

at their coincidence points.

CHAPTER III

COMMON FIXED POINTS OF

EXPANSION MAPPINGS

In 1977, Rhoades [119] summarized various types of

contractive mappings conditions for the existence of their fixed

points. In 1984, Wang, Li, Gao and Iseki [159] presented the

interesting work on expansion mappings in metric spaces

corresponding to some contractive mappings in [119] and proved

some fixed point theorems. Rhoades [120] and Taniguchi [152]

generalized the results of [159] for a pair of mappings. In 1986,

Jungck [65] introduced the notion of compatible mappings in metric

spaces. Later, Rhoades [121] and Kang and Rhoades [70]

established some common fixed point theorems for compatible pairs

using expansion type conditions in metric spaces. In 1995,

Jachymski [61] extended the main result of [70] for the mappings

),( SA and ),( TB commuting at their coincidence points, which

is essentially a weaker condition than compatibility (see Jungck [65],

Prop 2.2 and Jungck [66], Ex. 2.5). In 1998, Jungck and Rhoades

Common fixed points of expansion mappings

50

[67] introduced the notion of weakly compatible mappings on metric

spaces, which is in fact the commutativity of the mappings at their

coincidence points. The results by Kang and Rhoades [70] and

Jachymski [61] have been proved for surjective mappings. Pathak,

Kang and Ryu [107] and Sharma, Shau, Bounis and Bonaly [130]

proved some fixed point theorems for non-surjective expansion

mappings in metric spaces and d -complete topological spaces

respectively. In 2006, Pathak and Tiwari [108] established some

fixed point theorems for expansion mappings on metric spaces

using implicit relations.

In 1987, Pant, Dimri and Singh [95] introduced the notion of

expansion mappings on probabilistic metric spaces and proved

some fixed point theorems. In 1991, Vasuki [156], in 2006, Kumar,

Chugh and Vats [76] and in 2007, Kumar [74] also proved some

fixed point theorems for expansion mappings in Menger spaces.

The purpose of this chapter is twofold. First, we prove a result

for four expansion mappings, two of them being surjective, using

compatibility of mappings in Menger spaces. Secondly, we establish

some results for non-surjective expansion mappings. Theorem 3.3


51

and the corollary thereof are obtained by using the condition of

weak compatibility of mappings. Our results generalize and extend

several previously well-known results in Menger spaces.

Pant, Dimri and Singh [95] introduced the following:

DEFINITION 3.1 Let ),,( tFX be a Menger space. A mapping

XXT →: will be called an expansion mapping iff for a constant

1>h

)()()1.3( ,, xFhxF vuTvTu ≤ for all vu, in X and 0≥x .

The interpretation of (3.1) is as follows: The probability that the

distance between the image points TvTu, is less than hx is never

greater than the probability that the distance between vu, is less

than x .

The following lemma is given in [88].

LEMMA 3.1 If A and S are compatible self-mapping of a Menger

space ( )tFX ,, , where t is continuous and satisfies xxxt ≥),( for all

]1,0[∈x and zSuAu nn →, for some z in X ( }{ nu being a

sequence in X ), then AzSAun → provided A is continuous.


52

Now we prove a common fixed point theorem for four

expansion mappings using compatibility condition in complete

Menger spaces. This result (see [77]) has been published in

“Ganita 57 (1) (2006), 89-95”.


is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Further, let

SBA ,, and T be self-mappings of X satisfying the following

conditions:

(3.1.1) A and B are surjective;

(3.1.2) One of SBA ,, and T is continuous;

(3.1.3) ),( SA and ),( TB are compatible pairs;

(3.1.4) )()( ,, xFhxF TvSuBvAu ≤ for all vu, in X and .1>h

Then SBA ,, and T have a unique common fixed point in X .

PROOF. Let .0 Xu ∈ Since A and B are surjective, we choose

Xu ∈1 such that 001 vTuAu == and for this point ,1u there exists

a point 2u in X such that .112 vSuBu == Continuing in this

manner we obtain a sequence }{ nv in X as follows:


53

nnn vTuAu 2212)5.1.3( ==+ and .121222 +++ == nnn vSuBu

By (3.1.4),

).()(

)()(

22122212

2212122

,,

,,

xFxF

hxFhxF

nnnn

nnnn

vvTuSu

BuAuvv

++++

+++

=≤

=

By Lemma 2.1, }{ nv is a Cauchy sequence. Since X is complete,

}{ nv converges to some point z in X . Consequently, the

subsequences },{ 12 +nAu },{ 2nBu }{ 12 +nSu and }{ 2nTu also

converge to z .

Now suppose that A is continuous. Since A and S are

compatible, Lemma 3.1 implies

122

+nuA and AzSAu n →+12 as ∞→n .

By (3.1.4), we get

).()(212212

2 ,,xFhxF

nnnnTuSAuBuuA ++

≤

Letting ∞→n yields

)()( ,, xFhxF zAzzAz ≤ , which implies that Azz = .

Again by (3.1.4),

)()(22 ,, xFhxF

nn TuSzBuAz ≤ implying .Szz =

Let Spz = for some .Xp∈ Then we have

).()( ,, 12122 xFhxF TpSAuBpuA nn ++

≤


54

Letting ∞→n gives

),()( ,, xFhxF TpAzBpAz ≤ which implies that .Tpz =

Since B and T are compatible and ,zTpBp == 1)(, →xF TBpBTp

and hence .TzTBpBTpBz ===

Moreover, by (3.1.4), we have

).()( ,, 1212xFhxF TzSuBzAu nn ++

≤

Letting ∞→n yields

),()( ,, xFhxF TzzBzz ≤ which implies that .Tzz =

Therefore, z is a common fixed point of SBA ,, and T . In the

case of the continuity of B , the proof is similar.

Next, suppose that S is continuous. Since A and S are

compatible, Lemma 3.1 implies

122

+nuS and SzASu n →+12 as .∞→n

By (3.1.4), we have

).()(212

2212 ,, xFhxFnnnn TuuSBuASu

++≤

Letting ∞→n , we obtain .Szz =

Let Aqz = and Bwz = for some q and w in X respectively. Then

)()(,,

12212

xFhxFTwuSBwASu

nn ++≤ , which implies that .Twz = Since


55

B and T are compatible and ,zTwBw == 1)(, →xF TBwBTw and

hence, .TzTBwBTwBz === Moreover by (3.1.4), we have

)()( ,, 1212xFhxF TzSuBzAu nn ++

≤

which implies .Tzz = Further, we have

)()( ,, xFhxF TzSqBzAq ≤

so that .Sqz = Since A and S are compatible and ,zSqAq ==

1)(, →xF ASqSAq and hence .SzSAqASqAz === Therefore, z is

a common fixed point of SBA ,, and T . The proof for the case

in which T is continuous, is similar.

Finally, the uniqueness of z as a common fixed point of SBA ,,

and T is obvious from (3.1.4).

This completes the proof of theorem.

REMARK 3.1 Theorem 3.1 is an interesting extension of the result

of Kang and Rhoades [70] to PM-spaces.

By setting AB = and ST = in Theorem 3.1, the following

result is obtained:


56

COROLLARY 3.1 Let X be as in Theorem 3.1. Further, let A

and S be self-mappings of X satisfying the conditions

).1.3( a A is surjective;

).1.3( b One of A and S is continuous;

).1.3( c A and S are compatible;

).1.3( d )()( ,, xFhxF SvSuAvAu ≤ for all vu, in X and .1>h

Then A and S have a unique common fixed point in X .

PROOF. By setting AB = and ST = in (3.1.5), the sequence }{ nu

is as follows:

nn SuAu =+1

By ),.1.3( a

).()()(21211 ,,, xFhxFhxF

nnnnnn SuSuAuAuSuSu +++++≤=

So by Lemma 2.1, }{ nSu is Cauchy sequence and converges to

some point z in X . Consequently, the subsequence }{ nAu also

converges to z . Now, as in Theorem 3.1, it can be proved that z is

a unique common fixed point of A and .S


57

REMARK 3.2 Vasuki has proved a common fixed point theorem

[156, Theorem 2.3] for a pair of commuting mappings. Corollary 3.1

is, therefore, an improvement over the result of Vasuki [156] in the

sense that we have weakened the commutativity condition.

REMARK 3.3 In above Corollary, if the following conditions

).1.3( e )()( XAXS ⊆ ;

).1.3( f A is continuous;

are taken in place of conditions ).1.3( a and ).1.3( b , then

Corollary 3.1 is the generalization of result of Rhoades [121] to

PM-spaces.

In 2007, Kumar [74, Theorem 3.2] proved the following

common fixed point theorem in Menger spaces:


is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let A and S

be weakly compatible self mappings of X satisfying the following

conditions:

)()()1.2.3( ,, xFhxF SvSuAvAu ≤ for each Xvu ∈, , where 1>h and

for all 0>x ;


58

)()()2.2.3( XAXS ⊆ .

If one of the subspaces )(XA or )(XS is complete, then A and S


REMARK 3.4 The above theorem is clearly an improvement of Corollary 3.1 with

condition ).1.3( c replaced by weak compatibility of mappings. However, in Theorem

3.2, Kumar [74] have taken the space X complete and one of the subspaces (XA

or )(XS also complete. Our point of view about the above theorem is that

completeness of one of the subspaces )(XA or )(XS is sufficient and the theorem

holds without completeness of the space X .

Next, we prove some common fixed point theorems for a pair

of non-surjective expansion mappings in Menger spaces. Theorem

3.3 and corollary 3.2 have been proved by using the condition of

weak compatibility of mappings. These results (Theorem 3.3-3.4),

see [26], have been accepted for publication in

“Stud. Cerc. St. Ser. Matematica Universitatea Baca u 18(2008)”

THEOREM 3.3 Let ( )tFX ,, be a Menger space, where t is

continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Let A and S


59

be weakly compatible self mappings of X satisfying the following

conditions:

)()()1.3.3( XAXS ⊆ ;

)()())(()2.3.3( ,,2

, xFxFhxF SvAvSuAuAvAu ≤

for each Xvu ∈, , where 1>h and for all 0>x .



PROOF. Let Xu ∈0 . Since ),()( XAXS ⊆ choose Xu ∈1 such that

01 TuSu = . In general, choose 1+nu such that nn SuAu =+1 .

By ),2.3.3(

)()(

)()())((

211

111

,,

,,2

,

xFxF

xFxFhxF

nnnn

nnnnnn

AuAuAuAu

SuAuSuAuAuAu

+++

+++

≤

≤

giving

).()(211 ,, xFhxF

nnnn AuAuAuAu +++≤

Similarly,

).()(3221 ,, xFhxF

nnnn AuAuAuAu ++++≤

So in view of Lemma 2.1, }{ nAu is a Cauchy sequence. Since

)(XA is complete, }{ nAu has a limit in )(XA . Call it z . Hence


60

there exist a point p in X such that zAp = . Consequently, the

subsequence }{ nSu also converges to z .

By (3.3.2),

)()())(( ,,2

, xFxFhxFnnn SuAuSpApSuAp ≤

Letting ∞→n , we have

)()())(( ,,2

, xFxFhxF zzSpzzz ≤ implying zSp = .

Therefore,

zSpAp == .

Since A and S are weakly compatible, therefore,

.AzSzi.e.,ASpSAp ==

Now we show that z is a fixed point of A and S .

By (3.3.2),

).()())(( ,,2

, xFxFhxFnnn SuAuSzAzAuAz ≤


)()())(( ,,2

, xFxFhxF zzSzAzzAz ≤

which implies that zAz = . Hence, z is a common fixed point of A

and S .

To prove the uniqueness of z as a common fixed point of A and S ,

let )( zy ≠ be another fixed point. By (3.3.2),


61

).()(

)()())(())((

,,

,,2

,2

,

xFxF

xFxFhxFhxF

yyzz

SyAySyAzAyAzyz

≤

≤=

This implies that zy = and hence, z is a unique common fixed

point of A and S .

Now, we give corollary of the above theorem:

COROLLARY 3.2 Let ( )dM , be a metric space. Further, let A and

S be weakly compatible self mappings of M satisfying the following

conditions:

);()().2.3( MAMSa ⊆

( ) ),(),(),().2.3( 2 SvAvdSuAudhAvAudb ≥

for each Mvu ∈, , where 1>h . If one of the subspaces )(MA or

)(MS is complete, then A and S have a unique common fixed

point.

PROOF. Let .0 Mu ∈ Since );()( MAMS ⊆ choose Mu ∈1 such

that 01 SuAu = . In general, choose 1+nu such that nn SuAu =+1 .

By (3.2. b),


62

( )),(),(

),(),(),(

211

112

1

+++

+++

≥

≥

nnnn

nnnnnn

AuAudAuAudh

SuAudSuAudhAuAud

giving

),(),( 211 +++ ≥ nnnn AuAudhAuAud

which implies

).,(1

),( 121 +++ ≤ nnnn AuAudh

AuAud

Since 1>h , by Lemma of Jungck [64], }{ nAu is a Cauchy

sequence. Since )(MA is complete, }{ nAu has a limit in )(MA .

Call it z . Hence, there exist a point p in M such that zAp = .

Consequently, the subsequence }{ nSu also converges to z . Now,

applying the same technique as in Theorem 3.3, the conclusion of

Corollary holds.

REMARK 3.5 Corollary 3.2 extends the results of Pathak and Dubey

[106, Theorem 1] and Singh, Rajput and Saluja [132, Theorem 3.2]

to non-surjective mappings.

THEOREM 3.4 Let ( )tFX ,, be a Menger space where t is

continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Further, let A


63

and S be continuous self-mappings of X satisfying the following

conditions:

),()(),()()1.4.3( 2 XSAXAXAXA ⊆⊆

)}(,)(),({)()2.4.3( ,,,, 22 xFxFxFminhxF AvAuAvSAvuAAuSAvuA≤

for each Xvu ∈, , where 1>h and for all 0>x .

If the subspace )(XA is complete, then A or S has a fixed point or

A and S have a common fixed point.

PROOF. Let Xu ∈0 . Since )()( 2 XAXA ⊆ and )()( XSAXA ⊆ ,

choose a point Xu ∈1 such that 0012 vAuuA == , say, and for this

point ,1u there exists a point Xu ∈2 such that 112 vSuSAu == , say.

Continuing in this manner we obtain a sequence }{ nv in )(XA as

follows:

nnn vAuuA 22122 ==+ and 121222 +++ == nnn vAuSAu .

Now, if 122 += nn vv for any n , one has that nv2 is a fixed point of A

from the definition of }{ nv . It then follows that 2212 ++ = nn vv , which

implies that nv2 is also a fixed point of S .

Suppose that 122 +≠ nn vv , then by (3.4.2),


64

).(

)}(,)(,)({

)}(

),(),({

)()(

2212

22122212212

2212

2222122

12

22122122

,

,,,

,

,,

,,

xF

xFxFxFmin

xF

xFxFmin

hxFhxF

nn

nnnnnn

nn

nnnn

nnnn

vv

vvvvvv

AuAu

AuSAuuAAu

SAuuAvv

++

+++++

++

++++

+++

≤

≤

≤

=

Similarly, we have

)()(22321222 ,, xFhxF

nnnn vvvv ++++≤ .

In general, we have

)()(211 ,, xFhxF

nnnn vvvv +++≤ .

By Lemma 2.1, }{ nv is a Cauchy sequence and it converges to

some point z in )(XA . Consequently, the subsequences }{ 2nv ,

}{ 12 +nv and }{ 22 +nv also converge to z . By continuity of A and ,S

AzvAuuA nnn →==+ 22122 and SzvAuSAu nnn →== +++ 121222

as .∞→n

Thus, A and S have a common fixed point.


REMARK 3.6 Theorem 3.4 is an interesting extension of a result of

Pathak, Kang and Ryu [107, Corollary 2.3(2)] to PM-spaces.


65

Finally, we extend Theorem 3.2 of Kumar [74] to four self-

mappings.

THEOREM 3.5 Let ( )tFX ,, be a Menger space, where t is

continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Further, let

SBA ,, and T be self-mappings of X satisfying the following

conditions:

(3.5.1) )()( XAXT ⊆ and )()( XBXS ⊆ ;

(3.5.2) ),( SA and ),( TB are weakly compatible pairs;

(3.5.3) )()( ,, xFhxF TvSuBvAu ≤ for all vu, in X and .1>h

If )(XA is a complete subspace of ,X then SBA ,, and T have a

unique common fixed point in X .

PROOF. Let Xu ∈0 . By (3.5.1), we define the sequence }{ nv in X

such that for all ...2,1,0=n

nnn vTuAu 2212)4.5.3( ==+ and .121222 +++ == nnn vSuBu

By (3.5.3) and (3.5.4), we have

).()(

)()(

22122212

2212122

,,

,,

xFxF

hxFhxF

nnnn

nnnn

vvTuSu

BuAuvv

++++

+++

=≤

=


66

By Lemma 2.1, }{ nv is a Cauchy sequence. Since )(XA is

complete, }{ nv has a limit in )(XA . Call it, z . Hence there exists a

point p in zAp = . Consequently, the subsequences },{ 12 +nAu

},{ 2nBu }{ 12 +nSu and }{ 2nTu also converge to z .

By (3.5.3),

).()(22 ,, xFhxF

nn SuSpBuAp ≤


)()( ,, xFhxF zSpzz ≤ implying zSp = .

Therefore, zSpAp == .

Since A and S are weakly compatible, therefore, ASpSAp = that

is, AzSz = . But )()( XBXS ⊆ , so there exists Xq ∈ such that

.ApSpBq == Again by (3.5.3),

)()( ,, xFhxF TqSpBqAp ≤ implying .TqSp =

Hence, we have .TqBqSpApz ==== Since B and T are weakly

compatible, therefore, TBqBTq = that is, .TzBz =

Now we claim that zAz = . By (3.5.3), we get

)()()()( ,,,, xFxFhxFhxF zAzTqSzBqAzzAz ≤≤= , which is a

contradiction. Therefore, zAz = . Thus, we have zSzAz == .

Similarly, it can be proved that zTzBz == .


67

And hence, zTzBzSzAz ==== . Finally, the uniqueness of z as a

common fixed point of SBA ,, and T is obvious from (3.5.3).

REMARK 3.7 The above theorem is an improvement of Theorem

3.1 in the sense that it has been proved for non-surjective mappings

and the pairs ),( SA and ),( TB are taken only weakly compatible.

Let Φ

be the family of mappings such that for each

),0[),0[:, ∞→∞φ∈φ Φ is upper semi-continuous from the right

and non-decreasing in each coordinate variable with tt <φ )( for

each 0>t (see [8]). Using this, we now give the following result:

COROLLARY 3.3 Let SBA ,, and T be self-mappings of a metric

space ),( dM such that the following conditions hold:

(3.3. a) )()( MAMT ⊆ and )()( MBMS ⊆ ;

(3.5. b) ),( SA and ),( TB are weakly compatible pairs;

(3.5. c) ),(),( TvSudBvAud ≥φ for all vu, in M and .Φ∈φ

If )(MA is a complete subspace of ,M then SBA ,, and T have a

unique common fixed point in .M


68

PROOF. By virtue of (3.3.1), the sequence }{ nv in M is as defined

in (3.5.4). Then by (3.5.c), (3.5.4) and Lemma of Kang and

Rhoades [70, Lemma 2.2], }{ nv is a Cauchy sequence. Now the

proof of Corollary involves the same technique as in Theorem 3.5.

THEOREM 3.6 Let X be as in Theorem 3.5. Further, let SBA ,,

and T be self-mappings of X satisfying the conditions (3.5.1),

(3.5.2) and

(3.6.1) )}(),(),({)( ,,,, xFxFxFminhxF TvBvAuSuTvSuBvAu ≤ for all

vu, in X and .1>h

If )(XA is a complete subspace of ,X then SBA ,, and T have a

unique common fixed point in X .

PROOF. The proof may be completed on the lines of Theorem 3.5.

CHAPTER IV

COINCIDENCES AND FIXED POINTS

OF PROBABILISTIC DENSIFYING MAPPINGS

The study of Kuratowski measure of non-compactness in a

probabilistic metric space was initiated by Bocşan and Constantin,

[7]. The concept of probabilistic densifying mappings was introduced

by Bocşan [5]. Later, Hadžić [46], Tan [151], Chamola, Pant and

Singh [12], Dimri and Pant [25], Pant, Dimri and Chandola [94], Pant

Tiwari, Singh [99], and Singh and Pant [137] proved some results for

such mappings. In [42], Ganguly, Rajput and Tuteja introduced the

notion of probabilistic nearly densifying mappings.

The purpose of this chapter is to establish coincidence and

common fixed point theorems for certain classes of nearly

densifying mappings in complete Menger spaces. First, we give

definitions and terminology, which play an important role in this

chapter.

Coincidences and fixed points of probabilistic densifying mappings

70

DEFINITION 4.1 A semi group G is said to be left reversible if for

any Gsr ∈, there exist Gba ∈, such that .sbra =

It is easy to see that the notion of left reversibility is equivalent

to the statement that any two right ideals of G have nonempty

intersection.

DEFINITION 4.2 Let G be a family of self-mappings in .X A subset

Y of X is called G -invariant if YgY ⊆ for all .Gg ∈

DEFINITION 4.3 Let *G be the semi group generated by G under

composition ∗ . Clearly, }0:{* ≥⊇ ngG n for any Gg ∈ and

}:{}{)( ** GgguuuG ∈∪= for .Xu ∈

DEFINITION 4.4 Let .:,, XXrgf → Also, let XG ∈ and

},,{ rgfG = , then }0,,:{)( 00* ≥= kjiurgfuG kji for .0 Xu ∈

We restate the notion of probabilistic diameter for the sake of

quick reference:


71

DEFINITION 4.5 Let A be a non-empty subset of X . A function

(.)AD defined by

=∈<

)()( ,,

yFinfsupxD vuAvuxy

A

is called probabilistic diameter of A . A is said to be bounded if

1)( =∈

xDsup ARx

.

The following definition is due to Bocşan and Constantin [7].

DEFINITION 4.6 For a probabilistic bounded subset A of X ,

)(xAα defined by ∃≥ε=α :0{)( supxA a finite cover A of A such

that ε≥)(xDS for all ∈S A } is called Kuratowski’s function.

The following properties of Kuratowski’s functions are proved in [7].

(a) ℑ∈α A , the set of distribution functions;

(b) );()( xDx AA ≥α

(c) If XBA ⊂⊂≠φ , then );()( xx BA α≥α

(d) )};(),({)( xxminx BABA αα=α ∪

(e) Let A be the closure of A in the −λε ),( topology on X . Then

)()( xx AA α=α ;

(f) A is probabilistic precompact (totally bounded) iff ,HA =α


72

where H denotes the specific distribution function defined by

>

≤=

.0,1

0,0)(

x

xxH

The notion of probabilistic densifying mapping was given in

[5].

DEFINITION 4.7 Let ( )FX , be a PM-space. A continuous mapping

f of X into X is called a probabilistic densifying mapping iff for

every subset A of ,X HA <α implies AAf α>α )( .

The notion of probabilistic nearly densifying mapping was

introduced in [42].

DEFINITION 4.8 A self-mapping XXf →: is probabilistic nearly

densifying if ,)( AAf α>α whenever ,HA <α ,HA ⊂ and A is

−f invariant.

First, we prove some results on coincidence and fixed points

of probabilistic nearly densifying mappings. These results (Theorem

4.1-4.3), see [98], have been published in

“ Ganita 58 (2007), 9-15”


73

THEOREM 4.1 Let gf , and r be three continuous and nearly

densifying self-mappings on a complete Menger space ( )tFX ,,

such that 1),(1

=<

xxtsupx

and r commutes with f and g . If for all

Xvu ∈, , the following conditions are satisfied -

)},,(),,(),,({),()1.1.4( 1221 gvrvfururvrumingvfu φφφ>φ for rvru ≠

and ;gvfu ≠

)},,(),,(),,({),()2.1.4( 1112 fvrvgururvruminfvgu φφφ>φ for rvru ≠

and ,fvgu ≠

where 1φ and 2φ are real valued mappings from XX x to ζ , the

collection of all distribution functions, with either 1φ or 2φ upper

semi-continuous (u.s.c.) and 1),(),( 21 =φ=φ uuuu for all Xu ∈ .

Further, if for some )(, 0*

0 uGXu ∈ is bounded, then f and r or g

and r have a coincidence point.

PROOF. For ,0 Xu ∈ let ).( 0uGA =

Then ).()()(}{ 0 ArAgAfuA ∪∪∪=

If ,HA >α then

))()()(({ }0 ArAgAfuA ∪∪∪α>α


74

)}()()({ ArAgAfmin ∪∪=

,Aα> a contradiction

This implies that A is compact.

Let )()( AfgrB n

Nn∈∩= .

Then it is easy to see that BBfgr =)( and B is non-empty compact

subset of A . By the continuity of gf , and r , it follows that

,AAf ⊂ ,AAg ⊂ AAr ⊂ . Further, it is clear that ,)( BBf ⊂

BBg ⊂)( and .)( BBr ⊂

Note that BArfgrAfgrrBr n

Nn

n

Nn⊂∩⊂∩=

∈∈)()()()()( and

);()()()( BrBrfBrfgBfgrB ⊂⊂== this implies that ,)( BBr = so

.)(2 BBr =

Now, assume that 1φ is upper semi-continuous. Then the function

,: ℑ→BT defined by ),()( 1 guruuT φ= is u.s.c. So T assumes its

maximal value at some point p in B . Clearly, ),(2 Brp∈ so there is

a Bw∈ such that )(2 wrp = . Suppose that neither f and r nor g

and r have a coincidence point. Then

))(),(())(( 1 wgfgwrfgwfgT φ=


75

))(),((1 wgfgwfrgφ= by (4.1.1),

))}(),((

)),(),(()),(),(({

2

22

22

wgfgwrfg

wfrgwgrwrfgwgrmin

φ

φφ>

)),(),(( 22 wfrgwgrφ= by (4.1.2)

),(),(

))(),((

))}(),((

)),(),(()),(),(({

221

22

221

221

pTgprp

wgrwrr

wfrgwgr

wgrwrrwgrwrrmin

==

φ=

φ

φφ>

a contradiction to the selection of p . Hence f and r or g and r

must have a coincidence point.

Same result holds good if 2φ is upper semi-continuous.


REMARK 4.1 The above theorem extends the results of Khan and

Liu [71, Theorem 3.1 and Corollary 3.3] to PM-spaces.

COROLLARY 4.1 Let fX , , g and r be as in Theorem 4.1.

Further, let f , g and r satisfy the following conditions:

),,(),().1.4( 21 rvrugvfua φ>φ for rvru ≠ and ;gvfu ≠


76

),,(),().1.4( 12 rvrufvgub φ>φ for rvru ≠ and ,fvgu ≠

where 1φ and 2φ are as stated in Theorem 4.1 for all Xvu ∈, . If for

some )(, 0*

0 uGXu ∈ is bounded, then the conclusion of Theorem

4.1 holds.

THEOREM 4.2 Let fX , , g and r be as in Theorem 4.1. Further,

let f , g and r satisfying (4.1.1) and (4.1.2), have a common

coincidence point z , then rz is a unique common fixed point of ,f

g and r .

PROOF. We have

rzgzfz == .

Commutativity of r with f and g gives

).()()()()( rzggzrrzrfzrrzf ====

Now let rzzr ≠2, then by (4.1.1), we have

)},(),,(),,({

),(),(

12

22

2

12

1

gzrzfrzzrrzzrmin

gzfrzrzzr

φφφ>

φ=φ,

),,( 22 frzzrφ= by (4.1.2)


77

)},(),,(),,({ 22

12

1 gzrzgrzzrrzzrmin φφφ>

),,( 21 rzzrφ= a contradiction.

Hence, .2 rzzr = Thus, rz is a common fixed point of f , g and r .

The uniqueness of rz as a common fixed point of f , g and r

follows from (4.1.1) and (4.1.2).

THEOREM 4.3 Let f , g and r be three continuous and nearly

densifying self-mappings on a complete Menger space ( )tFX ,,

such that 1),(1

=<

xxtsupx

and r commutes with f and g . If for all

Xvu ∈, , the following conditions are satisfied -

),,(),,(),,({),()1.3.4( gvrvfururvrumingvfu φφφ>φ

})],(),([2

1 furvgvru φ+φ for rvru ≠ and ;gvfu ≠

where φ is real valued u.s.c. function from XX x to ζ and

1),( =φ uu for all Xu ∈ . Further, if for some )(, 0*

0 uGXu ∈ is

bounded, then f and r or g and r have a coincidence point z .

Further, rz is the unique common fixed point of f , g and r .

PROOF. Proof is obvious from Theorem 4.1 and Theorem 4.2.


78

REMARK 4.2 Theorem 4.3 is an extension of the result of Ganguly,

Rajput and Tuteja [42, Theorem 3] for triplet of mappings.

DEFINITION 4.9 Let .:,,, XXsrgf → Also, let XG ∈ and

},,,,{ srgfG = then }0,,,:{)( 00* ≥= lkjiusrgfuG lkji for

.0 Xu ∈

Now we extend Theorem 4.1 to two pairs of mappings.

THEOREM 4.4 Let the continuous and nearly densifying self-

mappings rgf ,, and s on a complete Menger space ( )tFX ,,

such that 1),(1

=<

xxtsupx

satisfy the following

sfgfsgfgs ==)1.4.4( and ;rsgrgsgrs ==

)},,(),,(),,({),()2.4.4( 1221 gvsvfurusvrumingvfu φφφ>φ for svru ≠

and ;gvfu ≠

)},,(),,(),,({),()3.4.4( 1112 fvrvgusurvsuminfvgu φφφ>φ for rvsu ≠

and ,fvgu ≠

for all Xvu ∈, and 1φ and 2φ being real valued mappings from

XX x to ,ℑ with either 1φ or 2φ is u.s.c. and 1),(),( 21 =φ=φ uuuu

for all Xu ∈ . Further, if for some )(, 0*

0 uGXu ∈ is bounded and


79

*G is left reversible, then f and r or g and s have a coincidence

point.

PROOF. Applying the same technique as in Theorem 4.1, the proof

is obvious.

COROLLARY 4.2 Let rgfX ,,, and s be as in Theorem 4.4.

Further, let rgf ,, and s satisfy (4.4.1) and the following

conditions:

),,(),().2.4( 21 svrugvfua φ>φ for svru ≠ and ;gvfu ≠

),,(),().2.4( 12 rvsufvgub φ>φ for rvsu ≠ and ,fvgu ≠

where 1φ and 2φ are as stated in Theorem 4.4 and Xvu ∈, . If for

some )(, 0*

0 uGXu ∈ is bounded and *G is left reversible, then the

conclusion of Theorem 4.4 holds.

THEOREM 4.5 Let rgfX ,,, and s be as in Theorem 4.4. Further,

let rgf ,, and s satisfying (4.4.1), (4.4.2) and (4.4.3), have a

common coincidence point z , then sz is a unique common fixed

point of rgf ,, and s .


80

THEOREM 4.6 Let rgfX ,,, and s be as in Theorem 4.4. Further,

let rgf ,, and s satisfy (4.4.1) and the following conditions

),,(),,(),,({),()1.6.4( gvsvfurusvrumingvfu φφφ>φ

})],(),([2

1 fusvgvru φ+φ for rvru ≠ and ;gvfu ≠

for all Xvu ∈, and φ is real valued u.s.c. function from XX x to ζ

and 1),( =φ uu for all Xu ∈ . Further, If for some )(, 0*

0 uGXu ∈ is

bounded and *G is left reversible, then f and r or g and s have a

coincidence point z . Further, sz is a unique common fixed point of

rgf ,, and s .

CHAPTER V

RELATED FIXED POINT THEOREMS

As stated in the first Chapter, contraction mapping theorems

have great importance in fixed point theory. Fisher [36, 37]

investigated the conditions ensuring the existence of a relation

between fixed points of two contraction mappings on two metric

spaces. In 2002, Pant [93] initiated the study of the relation between

the fixed points of two contraction mappings in two different Menger

spaces by generalizing the results of Fisher [36, 37].

In this chapter, we extend the results of Pant [93] to two pairs

of mappings in two, and to three mappings in three different Menger

spaces respectively.

In [93], Pant proved the following

Related fixed point theorems

82

THEOREM 5.1 Let ( )tFX ,, and ( )tGY ,, be complete Menger

spaces, where t is continuous and satisfies xxxt ≥),( for all

]1,0[∈x . If T is a continuous mapping from X to Y and S is a

mapping from Y to X satisfying

{

)}(),2(

),2(),(),(),()(

,,

,,,,,

xGxF

xFxFxFxFminhxF

uTTuSTuu

uSTuuSTuSTuuuuuSTSTu

′′

′′′′′ ≥

{

)}(),2(

),2(),(),(),()(

,,

,,,,,

xFxG

xGxGxGxGminhxG

vSSvTSvv

vTSvvTSvTSvvvvvTSTSv

′′

′′′′′ ≥

for all uu ′, in X , vv ′, in Y and some )1,0(∈h , then ST has a

unique fixed point z in ,X TS has a unique fixed point p in Y .

Further, pTz = and .zSp =

Throughout this chapter, YX , and Z stand for complete

Menger spaces ( )tFX ,, , ( )tGY ,, and ( )tHZ ,, respectively.

First, we extend Theorem 5.1 to two pairs of mappings in two

different Menger spaces. This result (see [97]) has been published in

“Var āhmihir Journal of Mathematical Sciences 6(2006), 471-476”.


83

THEOREM 5.2 Let ( )tFX ,, and ( )tGY ,, be complete Menger

spaces, where t is continuous and satisfies xxxt ≥),( for all

]1,0[∈x . Further, let BA, be mappings from X to Y and TS, be

mappings from Y to X satisfying

{

)}(),2(

),2(),(),(),()()1.2.5(

,,

,,,,,

xGxF

xFxFxFxFminhxF

uBAuSAuu

uTBuuTBuSAuuuuuTBSAu

′′

′′′′′ ≥

( ) {

)}(),2(

),2(),(),(),()(2.2.5

,,

,,,,,

xFxG

xGxGxGxGminhxG

vTSvBSvv

vATvvATvBSvvvvvATBSv

′′

′′′′′ ≥

for all uu ′, in X , vv ′, in Y and some )1,0(∈h . If one of the

mappings SBA ,, and T is continuous, then SA and TB have a

unique common fixed point z in X and BS and AT have a unique

common fixed point p in Y . Further, pBzAz == and .zTpSp ==

PROOF. Let 0u be an arbitrary point in X . We define sequences

{ }nu and { }nv in X and Y respectively in the following manner

nnnnnn uTvvBuvAu 22212122 ,,)3.2.5( === −+ and

1212 −− = nn uSv for ...3,2,1=n

By (5.2.2),


84

)()(212122 ,, xGxG

nnnn ATvBSvvv −+=

)}./(),/({

)}/(),/2(),/2(

),/(),/(),/({

212212

212221212

122212212

,,

,,,

,,,

hxFhxGmin

hxFhxGhxG

hxGhxGhxGmin

nnnn

nnnnnn

nnnnnn

uuvv

uuvvvv

vvvvvv

−−

−+−

+−−

=

≥

since

)}./(),/({)/2(1222121212 ,,, hxGhxGminhxG

nnnnnn vvvvvv +−+−≥

Similarly by (5.2.1),

)}./(),/({)(212122212 ,,, hxGhxFminxF

nnnnnn vvuuuu +−+≥

In general, we have

)}/(),/({)()5.2.5(

)}/(),/({)()4.2.5(

,,,

,,,

111

111

hxFhxGminxG

hxGhxFminxF

nnnnnn

nnnnnn

uuvvvv

vvuuuu

−−+

+−+

≥

≥

Repeated use of (5.2.4) and (5.2.5) gives

)}/(),/({)(111 ,

1,,

nvv

nuuuu hxGhxFminxF

nnnnnn +−+−≥ and

)}/(),/({)( 1,

1,, 111

−−−−+

≥ nuu

nvvvv hxFhxGminxG

nnnnnn

for n= 1,2,3… Thus, as ( )xFnnn uu 1,,

+∞→ and ( )xG

nn vv 1, + both

tend to 1. Therefore, { }nu and { }nv are Cauchy sequences with

limits z and p in X and Y respectively. If A is continuous, then by

(5.2.3), we have


85

pAzAulimvlim nnnn ===+ 212)6.2.5( .

Now, let ),( λεSAzU be an −λε ),( neighbourhood of SAz . Since

zulim nn =2 and Azvlim nn =+12 , there exists an integer ( )λε= ,NN

such that for ,0, >λε

( ) ( ) λ−>ελ−>ε −−+−

1,1)7.2.5(2

1,2

1, 12212 h

huuh

huz nnn

FF and

( ) λ−>ε−+

12

1, 12 h

hvAz n

G .

Then by (5.2.1),

( ) ( )

( ){ ( ) ( )

( ) ( ) ( )}( ){ ( ) ( )

( ) ( )}( ){ ( ) ( )

( )}ε

εεε≥

εε

εεε≥

εεε

εεε≥

ε=ε

−

−−−

+−

−−−

−

−−−

−

−−

−

h

hvAz

h

huuh

huzh

huz

vAzuu

h

hSAzuh

huzuz

vAzSAzuuz

uuSAzzuz

TBuSAzuSAz

n

nnnn

nnn

nnn

nnn

nnn

nn

G

FFFmin

hGhF

FFhFmin

hGhFhF

hFhFhFmin

FF

2

1,

2

1,2

1,2

1,

,,

2

1,2

1,,

,,,

,,,

,,

2

2121212

2212

121212

2122

21212

122

,,,

/,/

,,,/

/,/2,/2

,/,/,/

since

( ) ( ) ( ){ }hFhFminhFnnnn uuuzuz /,//2

212122 ,,, εε≥ε−−

and

( ) ( ) ( ){ }hFhFminhF SAzzzuSAzu nn/,//2 ,,, 1212

εε≥ε−−

.

Therefore, from (5.2.7), we get


86

( ) ,12, λ−>ε

nuSAzF implying zSAz = .

Hence, with the help of (5.2.6), we have

.)8.2.5( zSpSAz ==

Again, let ),( λεBSpU be an −λε ),( neighbourhood of BSp . Since

pvlim nn =−12 and Spzulim nn ==2 , there exists an integer

( )λε= ,NN such that

( ) ( ) λ−>ελ−>ε −−+

1,1)9.2.5(2

1,2

1, 1222 h

hvvh

hvp nnn

GG

and ( ) λ−>ε− 12

1, 2 h

huSp n

F .

Then by (5.2.2),

( ) ( )

( ){ ( ) ( )

( ) ( ) ( )}( ){ ( ) ( )

( )}ε

εεε≥

εεε

εεε≥

ε=ε

−

−−−+

+

+

+

h

huSp

h

hvvh

hvph

hvp

uSpBSpvvp

vvBSppvp

ATvBSpvBSp

n

nnnn

nnn

nnn

nn

F

GGGmin

hFhGhG

hGhGhGmin

GG

2

1,

2

1,2

1,2

1,

,,,

,,,

,,

2

12222

2212

1222

212

,,

/,/2,/2

,/,/,/

since

)}/(),/({)/2(122212 ,,, hGhGminhG

nnnn vvvpvp εε≥ε++

and

)}/(),/({)/2( ,,, 22hGhGminhG BSpppvBSpv nn

εε≥ε .



87

( ) λ−>ε+

112, nvBSpG implying pBSp = .

So by (5.2.8) and (5.2.6), we get

pBzBSp ==)10.2.5( and pBzAz == .

Again using (5.2.1),

( ) ( )

{})/(),/2(

),/2(),/(),/(,)/(

,,

,,,,

,,

hxGhxF

hxFhxFhxFhxFmin

xFxF

BzAzzz

TBzzTBzzzzzz

TBzSAzTBzz

≥

=

)/(, hxF TBzz≥ , a contradiction.

Therefore, TBzz = .

Using (5.2.10) and (5.2.8), we get

zTpTBz ==)11.2.5( and zTpSp == .

So far, we have prove that

pBzAzpBSpzTBzSAz ===== ;; and zTpSp == .

Now, using (5.2.2). we can easily prove pATp = .

Same results hold well if one of the mappings SB, and T is

continuous.

Now to prove the uniqueness of z as a common fixed point of SA

and TB , we suppose that TB has another fixed point )( zz ≠′ . Then

by (5.2.1), we have


88

( ) ( )

{}

}{)/(

)/(),/2(),/2(,1,1),/(

)/(),/2(

),/2(),/(),/(,)/(

,

,,,,

,,

,,,,

,,

hxG

hxGhxFhxFhxFmin

hxGhxF

hxFhxFhxFhxFmin

xFxF

zBAz

zBAzzzzzzz

zBAzSAzz

zTBzzTBzSAzzzz

zTBSAzzz

′

′′′′

′′

′′′′

′′

=

≥

≥

=

)/(, hxG zATBBSz ′= , by (5.2.2)

{})/(),/2(),/2(

),/(),/(,)/(min

2,

2,

2,

2,

2,

2,

hxFhxGhxG

hxGhxGhxG

zTBSAzBSAzzBzATBAz

zATBzBBSAzAzzBAz

′′′

′′′≥

)/( 2, hxF zz ′≥ , a contradiction.

Therefore, z is the unique fixed point of TB .

Similarly, it can be proved that z is the unique fixed point of SA .

To prove the uniqueness of p as a common fixed point of BS

and AT , let us suppose that AT has another fixed point )( pp ≠′ .

Then by (5.2.2),

{})/(),/2(

),/2(),/(),/(,)/(

)()(

,,

,,,,

,,

hxFhxG

hxGhxGhxGhxGmin

xGxG

pTSpBSpp

pATppATpBSpppp

pATBSppp

′′

′′′′

′′

≥

=

giving

)/()()12.2.5( 2,, hxFxG pTzpp ′′ ≥


89

Let TByypT ==′ . Since TB has a unique fixed point z , therefore

zy = and hence by (5.2.12), we have pp ′= . Similarly, we can show

that p is the unique fixed point of BS .


REMARK 5.1 By setting QBA == and PST == in the above

theorem, we obtain Theorem 5.1 for two mappings P and Q .

REMARK 5.2 Theorem 5.2 is also an interesting extension of Fisher

and Murthy [38] to PM-spaces.

COROLLARY 5.1 Let ( )tFX ,, be a complete Menger space, where

t is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . Further, let

BA, TS, be self-mappings of X satisfying

{

)}(),2(

),2(),(),(),()().1.5(

,,

,,,,,

xFxF

xFxFxFxFminhxFa

BvAuSAuv

TBvuTBvvSAuuvuTBvSAu ≥

( ) {

)}(),2(

),2(),(),(),()(.1.5

,,

,,,,,

xFxF

xFxFxFxFminhxFb

TvSuBSuv

ATvuATvvBSuuvuATvBSu ≥


90

for all vu, in X and some )1,0(∈h . If one of the mappings SBA ,,

and T is continuous, then SA and TB have a unique common fixed

point z and BS and AT have a unique common fixed point p .

Further, pBzAz == and .zTpSp ==

Now, we extend Theorem 5.1 to three mappings in three

different Menger spaces.

THEOREM 5.3 Let ( ) ( )tGYtFX ,,,,, and ( )tHZ ,, be complete

Menger spaces, where t is continuous and satisfies xxxt ≥),( for all

]1,0[∈x . If T is a continuous mapping from X to Y , S is a

continuous mapping from Y to Z and R is a mapping from Z to X

satisfying

{

})(),(),2(),2(

),(),(),()()1.3.5(

,,,,

,,,,

xHpGxFxF

xFxFxFminhxF

uSTSTuuTTuRSTuuuRSTu

uRSTuRSTuuuuuRSTRSTu

′′′′

′′′′ ≥

{

})(),(),2(),2(

),(),(),()()2.3.5(

,,,,

,,,,

xFxHxGxG

xGxGxGminhxG

vRSRSvvSSvTRSvvvTRSv

vTRSvvTRSvvvvTRSTRSv

′′′′

′′′′′ ≥

{

})(),(),2(),2(

),(),(),()()3.3.5(

,,,,

,,,,

xGxFxHxH

xHxHxHminhxH

wTRTRwwRRwSTRwwwSTRw

wSTRwwSTRwwwwSTRSTRw

′′′′

′′′′′ ≥


91

for all uu ′, in X , vv ′, in Y and zz ′′′′, in Z , where (((( )))).1,0∈∈∈∈h Then

RST has a unique fixed point z in X , TRS has a unique fixed point

p in Y and STR has a unique fixed q in Z . Further,

qSppTz == , and .zRq =

PROOF. Let 0u be an arbitrary point in X . We define sequences

{ } { }nn vu , and { }nw in YX , and Z respectively in the following

manner

( ) nnnnn

n SvwTuvuRSTu === − ,,)4.3.5( 10 for ...3,2,1====n

By (5.3.2),

{

)}/(),/(),/2(),/2(

),/(),/(),/(

)()(

,,,,

,,,

,,

1111

111

11

hxFhxHhxGhxG

hxGhxGhxGmin

xGxG

nnnnnnnn

nnnnnn

nnnn

uuwwvvvv

vvvvvv

TRSvTRSvvv

−−+−

+−−

−+

≥

=

giving

{

)}/(

),/(),/()()5.3.5(

,

,,,

1

111

hxF

hxHhxGminxG

nn

nnnnnn

uu

wwvvvv

−

−−+≥

since { )}/(),/()/2(1111 ,,, hxGhxGminhxG

nnnnnn vvvvvv +−+−≥ .

Similarly by (5.3.3),


92

}{{

})/(),/(

),/(),/(),/(

)/(),/(),/()(

2,

2

2,,,

,,,,

1,1

111

1111

hxFhxH

hxGhxFhxHmin

hxGhxFhxHminxH

nnnn

nnnnnn

nnnnnnnn

uuww

vvuxuww

vvuuwwww

−−

−−−

+−−+

≥

≥

on using inequality (5.3.5) giving

)}./(

),/(),/({)()6.3.5(

2,

2,,,

1

111

hxF

hpGhpHminpH

nn

nnnnnn

xx

yyzzzz

−

−−+≥

Similarly by (5.3.1) and then using inequality (5.3.5) and (5.3.6), we

have

)}./(

),/(),/({)()7.3.5(

2,

2,,,

1

111

hxH

hxGhxFminxF

nn

nnnnnn

ww

vvuuuu

−

−−+≥

Repeated use of (5.3.5), (5.3.6) and (5.3.7) gives

{ }{ }

}{ .)/(),/(),/()(

,)/(),/(),/()(

,)/(),/(),/()(

1010101

101,0101

1010101

,1

,1

,,

1,

11,,

,,1

,,

nvv

nuu

nwwww

nuu

nww

nvvvv

nww

nvv

nuuuu

hxGhxFhxHminxH

hxFhxHhxGminxG

hxHhxGhxFminxF

nn

nn

nn

−−

−−−

−

≥

≥

≥

+

+

+

Thus, as ( ) ( )xGxFnnnnn vvuu 11 ,, ,,

++∞→ and ( )xH

nn ww 1, + all tend to

1. Therefore, { } { }nn vu , and { }nw are Cauchy sequences with limits


93

pz, and q in YX , and Z respectively. Then the continuity

conditions on T and S together with (5.3.4) imply

,1 pTzTulimvlim nnnn === − qSpSvlimwlim nnnn === .

Now, let ( )λε,RSTzU be a neighbourhood of RSTz . Since zulim nn = ,

Tzvlim nn = and STzwlim nn = , there exists an integer ( )λε= ,NN

such that for ,0, >λε

( ) ( ) ,1,1)8.3.5(2

1,2

1, 1

λ−>ελ−>ε −−+ h

huuh

huz nnn

FF

( ) λ−>ε−+

12

1, 1 h

hvTz n

G and ( ) λ−>ε−+

12

1, 1 h

hwSTz n

H .

Then by (5.3.1),

( ) ( )

( ){ ( ) ( ) ( )

( ) ( ) ( )}( ){ ( ) ( )

( ) ( ) ( )}( ){ ( ) ( ),,,

/,/,/

,,/

/,/,/2

,/2,/,/,/

2

1,2

1,2

1,

,,,

2

1,2

1,,

,,,

,,,,

,,

11

111

11

11

11

1

εεε≥

εεε

εεε≥

εεε

εεεε≥

ε=ε

−−−

+−

++

+++

++

++

++

+

h

huuh

huzh

huz

wSTzvTzuu

h

hRSTzuh

huzuz

wSTzvTzRSTzu

uzuuRSTzzuz

RSTuRSTzuRSTz

nnnn

nnnn

nnn

nnn

nnnn

nn

FFFmin

hHhGhF

FFhFmin

hHhGhF

hFhFhFhFmin

FF

( ) ( )},,2

1,2

1, 11

εε −−++ h

hwSTzh

hvTz nn

HG

since

( ) ( ) ( ){ }hFhFminhFnnnn uuuzuz /,//2

11 ,,, εε≥ε++

and


94

( ) ( ) ( ){ }hFhFminhF RSTzuuuRSTzu nnnn/,//2 ,,, 11

εε≥ε++

.


( ) ,11, λ−>ε

+nuRSTzF

which implies that zRSTz = and so z is a fixed point of RST .

Now, we have

pTzTRSTzTRSp ===

and so qSpSTRSpSTRq === .

Hence, p and q are fixed points of TRS and STR respectively.

To prove the uniqueness of z as a fixed point of RST , let z′

be another fixed point of RST .

Then by (5.3.1), for any 0>x

( ) ( )

{}

{ })/(,)/(

)/(),/(),/2(

),/2(),/(),/(,)/(

,,

,,,

,,,,

,,

hxHhxGmin

hxHhxGhxF

hxFhxFhxFhxFmin

xFxF

zSTSTzzTTz

zSTSTzzTTzRSTzz

zRSTzzRSTzRSTzzzz

zRSTRSTzzz

′′

′′′

′′′′

′′

≥

≥

=

Again by (5.3.2),

{})/(),/(),/2(),/2(

),/(),/(,)/(

)/()/(

2,

2,

2,

2,

2,

2,

2,

,,

hxFhxHhxGhxG

hxGhxGhxGmin

hxGhxG

zzzSTSTzTzzTzTTz

zTzTTzTzzTTz

zTRSTTRSTzzTTz

′′′′

′′′

′′

≥

=


95

{ })/(,)/( 2,

2, hxFhxHmin zzzSTSTz ′′≥

Hence, we have

( ) { })/(

)/(),/(),/(

,

,2

,2

,,

hxH

hxHhxFhxHminxF

zSTSTz

zSTSTzzzzSTSTzzz

′

′′′′

≥

≥

),/(, hxH zSTRSTSTRSTz ′≥ by (5.3.3)

{})/(),/(),/2(),/2(

),/(),/(,)/(

2,

2,

2,

2,

2,

2,

2,

hxGhxFhxHhxH

hxHhxHhxHmin

zTTzzzSTzzSTzSTSTz

zSTzSTSTzSTzzSTSTz

′′′′

′′′≥

This gives

( ) )/( 2,, hxFxF zzzz ′′ ≥ , which is a contradiction.

Therefore, z is a unique fixed point of RST .

Similarly, it can be proved that p is a unique fixed point of TRS

and q is a unique fixed point of STR .

Finally, we have to prove that zRq = .

For this, we have

( ) ( )RqRSTSTRqRRq ==

and so Rq is a fixed point of RST . Since z is the unique fixed point

of RST , it follows that zRq = .


96

REMARK 5.3 Theorem 5.3 is an interesting extension of the results

of Jain [62] and Jain, Sahu and Fisher [63] to PM-spaces.

COROLLARY 5.2 Let ( )tFX ,, be a complete Menger space, where

t is continuous and satisfies xxxt ≥),( for all ]1,0[∈x . If

XXTSR →:,, , with T and S continuous, and satisfy

{

})(),(),2(),2(

),(),(),()().2.5(

,,,,

,,,,

xFxFxFxF

xFxFxFminhxFa

STvSTuTvTuRSTuvRSTvu

RSTvvRSTuuvuRSTvRSTu ≥

{

})(,)(),2(),2(

),(),(),()().2.5(

,,,,

,,,,

xFxFxFxF

xFxFxFminhxFb

RSwRSvSwSvTRSvwTRSwv

TRSwwTRSvvwvTRSwTRSv ≥

{

)}(),(),2(),2(

),(),(),()().2.5(

,,,,

,,,,

xFxFxFxF

xFxFxFminhxFc

TRuTRwRuRwSTRwuSTRuw

STRuuSTRwwuwSTRuSTRw ≥

for all wvu ,, in X and some ( ).1,0∈h Then RST has a unique

fixed point z , TRS has a unique fixed point p and STR has a unique

fixed point q . Further, qSppTz == , and zRq = and if qpz == ,

then z is a unique fixed point of SR, and T .


97

PROOF. The existence of pz, and q as the fixed points of

TRSRST , and STR respectively follows from Theorem 3.1. If

qpz == , then z is of course a common fixed point of SR, and T .

Now, suppose that T has another fixed point z′ . Then on using

),.2.5( a

{})/(),/(),/2(),/2(

),/(),/(,)/(

)()(

,,,,

,,,

,,

hxFhxFhxFhxF

hxFhxFhxFmin

xFxF

zSTSTzzTTzRSTzzzRSTz

zRSTzRSTzzzz

zRSTRSTzzz

′′′′

′′′

′′

≥

=

),/(, hxF zz ′≥ a contradiction.

This proves the uniqueness of z as a fixed point of T . Similarly, it

can be proved that z is the unique fixed point of S and R .

THEOREM 5.4 Let YX , and Z be as in Theorem 5.3. If T is a

continuous mapping from X to Y , S is a continuous mapping from

Y to Z and R is a continuous mapping from Z to X satisfying

{

})(

),(),(),()()1.4.5(

,

,,,,

xH

xGxFxFminhxF

STuSv

TuvRSTuuRSvuRSTuRSv ≥

{

)}(

),(),(),()()2.4.5(

,

,,,,

xF

xHxGxGminhxG

RSvRw

SvwTRSvvTRwvTRSvTRw ≥


98

{

)}(

),(),(),()()3.4.5(

,

,,,,

xG

xFxHxHminhxH

TRwTu

RwuSTRwwSTuwSTRvSTu ≥

for all u in X , v in Y and w in Z , where ( ).1,0∈h Then the


PROOF. By (5.4.2) and (5.3.4),

)}/(

),/(),/(),/({

)()(

,

,,,

,,

1

11

11

hxF

hxHhxGhxGmin

xGxG

nn

nnnnnn

nnnn

uu

wwvvvv

TRSvTRwvv

−

−+

−+

≥

=

giving

{ })/(),/()()4.4.5( ,,, 111hxFhxHminxG

nnnnnn uuwwvv −−+≥

Similarly by (5.4.3) and then using inequality (5.4.4), we have

{ })/(),/()()5.4.5( 2,,, 111

hxGhxFminxHnnnnnn vvuuww −−+

≥

Similarly by (5.4.1) and then using inequality (5.4.4) and (5.4.5), we

have

{ })/(),/()()6.4.5( 2,

2,, 111

hxHhxGminxFnnnnnn wwvvuu −−+

≥

Repeated use of (5.4.4), (5.4.5) and (5.4.6) give

{ })/(,)/()(10101 ,,,

nww

nvvuu hxHhxGminxF

nn≥

+


99

{ }{ })/(,)/()(

)/(,)/()(

10101

10101

,1

,,

1,

1,,

nvv

nuuww

nuu

nwwvv

hxGhxFminxH

hxFhxHminxG

nn

nn

−

−−

≥

≥

+

+

Thus as ( ) ( )xGxFnnnnn vvuu 11 ,, ,,

++∞→ and ( )xH

nn ww 1, + all tend to

1. Therefore, { } { }nn vu , and { }nw are Cauchy sequences with limits

pz, and q in YX , and Z respectively. Now applying the same

technique as in Theorem 5.3, the conclusion of theorem holds.

REMARK 5.4 Theorem 5.4 is an extension of Pant [93, Theorem 2]

for three mappings in three different Menger spaces and is a

generalization of the result of Nung [90] to PM-spaces.

COROLLARY 5.3 Let X be as in Corollary 5.2. If XXTSR →:,,

be continuous mappings satisfying

{ )}(),(),(),()( ,,,,, xFxFxFxFminhxF STuSvTuvRSTuuRSvuRSTuRSv ≥

{ )}(),(),(),()( ,,,,, xFxFxFxFminhxF RSvRwSvwTRSvvTRwvTRSvTRw ≥

{ )}(),(),(),()( ,,,,, xFxFxFxFminhxF TRwTuRwuSTRwwSTuwSTRwSTu ≥

for all wvu ,, in X and some ( ).1,0∈h Then the conclusion of

Corollary 5.2 holds.


100

Now, we extend Theorem 5.3 and Theorem 5.4 by using new

contraction conditions.



Y to Z and R is a mapping from Z to X satisfying

{

})()(),()(

),()(

),()())(()1.5.5(

,,,,

,,

,,2

,

xFxHxHxF

xFxG

xGxFminhxF

RSTuuuSTSTuuSTSTuuRSTu

uRSTuuTTu

uTTuRSTuuuRSTRSTu

′′′′

′′′

′′ ≥

{

})()(),()(

),()(

),()())(()2.5.5(

,,,,

,,

,,2

,

xGxFxFxG

xGxH

xHxGminhxG

TRSvvvRSRSvvRSRSvvTRSv

vTRSvvSSv

vSSvTRSvvvTRSTRSv

′′′′

′′′

′′ ≥

{

})()(),()(

),()(

),()())(()3.5.5(

,,,,

,,

,,2

,

xHxGxGxH

xHxF

xFxHminhxH

STRwwwTRTRwwTRTRwwSTRw

wSTRwwRRw

wRRwSTRwwwSTRSTRw

′′′′

′′′

′′ ≥

for all uu ′, in X , vv ′, in Y and ww ′, in Z , where ).1,0(∈h Then

the conclusion of Theorem 5.3 holds.


101



Y to Z and R is a continuous mapping from Z to X satisfying

{

})()(),()(

),()(

),()())(()1.6.5(

,,,,

,,

,,2

,

xFxHxHxF

xFxG

xGxFminhxF

RSvuSTuSvSTuSvRSTuu

RSTuuTuv

TuvRSvuRSTuRSv ≥

{

})()(),()(

),()(

),()())(()2.6.5(

,,,,

,,

,,2

,

xGxFxFxG

xGxH

xHxGminhxG

TRwvRSvRwRSvRwTRSvv

TRSvvSvw

SvwTRwvTRSvTRw ≥

{

})()(),()(

),()(

),()())(()3.6.5(

,,,,

,,

,,2

,

xHxGxGxH

xHxF

xFxHminhxH

STuwTRwTuTRwTuSTRww

STRwwRwu

RwuSTuwSTRwSTu ≥

for all u in X , v in Y and w in Z , where ( ).1,0∈h Then the


CHAPTER VI

APPLICATIONS

In this chapter, we point out the applications of fixed point

theory (especially, Menger probabilistic metric spaces). Also, we

present applications of some of our results in fuzzy metric spaces.

Fixed point theory in probabilistic metric spaces can be

considered as a part of probabilistic analysis, which is a very

dynamic area of mathematical research. Karl Menger [83] made a

contribution to resolving the interpretative issue of Quantum

Mechanics. He proposed transferring the probabilistic notions of

quantum mechanics from the physics to the underlying geometry

(see [84]). As stated in earlier chapters, PM-space is the

probabilistic generalization of metric space. In fact, it is suitable to

look upon the distance concept as a statistical or probabilistic rather

than deterministic one because the advantage of a probabilistic

approach is that it permits from the initial formulation a greater

flexibility rather than that offered by a deterministic approach. The

Applications

103

concept of a probabilistic metric space corresponds to the situations

when we do not know exactly the distance between two points; we

know only probabilities of possible values of this distance. Such a

concept may have very important applications in quantum particle

physics particularly in connections with both string and ∞e theory,

which were introduced and studied by a well-known scientist

Mohamed Saladin El Naschie [30-33]. It is also of fundamental

importance in probabilistic functional analysis, nonlinear analysis

and applications; see ([4], [13], [48]).

Fixed point theory is one of the famous and traditional

theories in mathematics and has a broad set of applications. In this

theory, contraction is one of the main tools to prove the existence

and uniqueness of a fixed point. In addition, Sehgal and Bharucha-

Reid [128], Schweizer and Skalr [124, 125] studied contraction in

PM-spaces. In 1996, a group of mathematicians; Chang, Lee, Cho,

Chen, Kang and Jung [15] presented a research paper, in which

they obtained a generalized contraction mapping principle in PM-

spaces and applied it to prove the existence theorems of solutions

to differential equations in these spaces. Also, Hadžić, Pap and

Applications

104

Budinčević [50] presented the application of fixed point theorem in

random equations.

There has been always a tendency in mathematics to regard

the concept of Probability as one of the basic mathematical

concepts. In fact, the more general (i.e., not necessarily probabilistic

in nature) concept of “uncertainty” is considered a basic ingredient

of some basic mathematical structures. Consonant with this trend is

the Menger’s theory of probabilistic metric spaces (see [27]).

Fixed-point theory in fuzzy metric spaces for different

contractive-type mappings is closely related to that in probabilistic

metric spaces (refer [13, Chapters VIII, IX], [48, Chapters 3–5], [85],

[128]). Various mathematicians; for example, Hadžić and Pap [49],

Razani and Shirdaryazdi [116], Razani and Kouladgar [115] and Liu

and Li [82] have studied the applications of fixed point theorems in

PM-spaces to fuzzy metric spaces.

In 1968, the concept of fuzzy sets was introduced by Zadeh

[161]. It constitutes yet another example where the concept of

uncertainty was introduced in the theory of sets, in a non-

Applications

105

probabilistic manner. Since then, to use this concept in topology and

analysis many authors have extensively developed the theory of

fuzzy sets and applications. For example, Deng [22], Ereeg [34],

Fang [35], Kaleva and Seikkala [69], Kramosil and Michalek [73]

have introduced the concept of fuzzy metric spaces in different

ways. Grabiec [45] followed Kramosil and Michalek [73] and

obtained the fuzzy version of Banach contraction principle.

Moreover, it appears that the study of Kramosil and Michalek [73] of

fuzzy metric spaces paves the way for developing a smoothing

machinery in the field of fixed point theorems, in particular for the

study of contractive type maps. George and Veeramani [43]

modified the concept of fuzzy metric spaces introduced by Kramosil

and Michalek [73] and defined Hausdorff topology of metric spaces,

which was later proved to be metrizable. They also showed that

every metric induces a fuzzy metric. Consequently, the last three

decades were very productive for fuzzy mathematics and the recent

literature has observed the fuzzification in almost every direction of

mathematics such as arithmetic, topology, graph theory, probability

theory, logic etc. Fuzzy set theory has applications in applied

sciences such as neural network theory, stability theory,

mathematical programming, modeling theory, engineering sciences,

Applications

106

medical sciences (medical genetics, nervous system), image

processing, control theory, communication etc. No wonder that fuzzy

fixed point theory has become an area of interest for specialists in

fixed point theory, or fuzzy mathematics has offered new

possibilities for fixed point theorists.

First, in the following text, the preliminaries are collected.

The following definition is given in [161].

DEFINITION 6.1 A fuzzy set A in X is a function with domain X

and values in ]1,0[ .

The following definition is due to Kramosil and Michlek [73]:

DEFINITION 6.2 The triplet ),,( tMX is a fuzzy metric space if X

is an arbitrary set, t is a continuous t -norm, M is a fuzzy set in

),0[2 ∞×X satisfying the following conditions:

(FM1) 1),,( =xvuM for all 0>x

iff ;vu =

(FM2) ;0)0,,( =vuM

(FM3) ;),,(),,( xuvMxvuM =

(FM4) ( ) );,,(),,(),,,( yxwuMywvMxvuMt +≤

(FM5) ]1,0[),0[:),,( →∞xvuM is left continuous

for all Xwvu ∈,, and 0, >yx .

Applications

107

Note that ),,( xvuM can be thought of as the degree of

nearness between u and v with respect to x . We identify vu = with

1),,( =xvuM for all 0>x and 0),,( =xvuM with ∞ and we can

find some topological properties and examples of fuzzy metric

spaces in [43]. In the following example (see [43]), we know that

every metric induces a fuzzy metric:

EXAMPLE 6.1 Let ),( dX be a metric space. Define abbat =),( (or

),(),( baminbat = ) for all Xvu ∈, and 0>x ,

.),(

),,().1.6(vudx

xxvuMi

+=

Then ),,( tMX is a fuzzy metric space. We call this fuzzy metric M

induced by the metric d the standard fuzzy metric. On the other

hand, note that there exists no metric on X satisfying ).1.6( i .

The following lemma and definition is given in [45].

LEMMA 6.1 For all ),,(,, xvuMXvu ∈ is a non-decreasing

function.

Applications

108

DEFINITION 6.3 Let ),,( tMX be a fuzzy metric space.

A sequence }{ nu in X is said to be convergent to a point

Xu ∈ (denoted by uulim nn

=∞→

), if 1),,( =∞→

xuuMlim nn

for all 0>x .

A sequence }{ nu in X is called a Cauchy sequence if

1),,( =+∞→

xuuMlim nmnn

for all 0>x and 0>m .

A fuzzy metric space in which every Cauchy sequence is

convergent is said to be complete.

Throughout this chapter, ),,( tMX is considered to be the

fuzzy metric space with condition

(FM6) 1),,( =∞→

xvuMlimx for all vu, in .X

The following lemmas 6.2 and 6.3 are given in [89].

LEMMA 6.2 Let }{ nu be a sequence in a fuzzy metric space

),,( tMX with the condition (FM-6). If there exists a number

)1,0(∈h such that

),,(),,( 112 xuuMhxuuM nnnn +++ ≥

for all 0>x and ...3,2,1=n then }{ nu is a Cauchy sequence in X .

Applications

109

LEMMA 6.3 If for all 0,, >∈ xXvu and for a number )1,0(∈h ;

),,,(),,( xvuMhxvuM ≥

then vu = .

First, we give the application of Theorem 2.1 to fuzzy metric

spaces as follows:

THEOREM 6.1 Let BA, and T be three self-mappings of fuzzy

metric space ),,( tMX satisfying the following condition:

)}2,,()2,,(,),,(),,(

),,,(),,(

),,,(),,({),,()1.1.6( 2

xAuTvMxBvTuMxBvTvMxAuTuM

xBvTvMxTvTuM

xAuTuMxTvTuMminhxBvAuM ≥

for all vu, in X and )1,0(∈h . Further, assume that either

TAAT = or TBBT = . If there exists a point 0u in X such that X is

))(;,( 0uTBA –orbitally complete and T is ))(;,( 0uTBA –orbitally

continuous, then BA, and T have a unique common fixed point

and }{ nTu converges to the common fixed point.

Next, we give the application of Theorem 2.3 to fuzzy metric

spaces as follows:

Applications

110

THEOREM 6.2 Let SBA ,, and T be four self-mappings of fuzzy

metric space ),,( tMX satisfying the following condition:

)}2,,()2,,(,),,(),,(

),,,(),,(

),,,(),,({),,()1.2.6( 2

xAuTvMxBvSuMxBvTvMxAuSuM

xBvTvMxTvSuM

xAuSuMxTvSuMminhxBvAuM ≥

for all vu, in X and )1,0(∈h . Further, assume that TSST = and

either (i) TAATSAAS == , or (ii) TBBTSBBS == , . If there exists

a point 0u in X such that X is ))(;,( 0uSTBA –orbitally complete

and T is ))(;,( 0uSTBA –orbitally continuous, then SBA ,, and T

have a unique common fixed point and }{ nSTu converges to the

common fixed point.

In 1994, Mishra, Sharma and Singh [89] introduced the notion

of compatible mappings under the name of asymptotically

commuting mappings in fuzzy metric spaces. In 1999, Vasuki [157]

extended the notion of pointwise R -weak commuting mappings to

fuzzy metric spaces. In [133], Singh and Jain extended the notion of

weak compatibility to fuzzy metric spaces. In 2002,

Balasubramaniam, Murlisankar and Pant [2] extended the concept

of reciprocally continuous mappings to fuzzy metric spaces while in

Applications

111

[104], Pant and Jha made a connection between continuity and

reciprocal continuity in fuzzy metric spaces.

Now we give an application of Theorem 2.5 to fuzzy metric

spaces as follows:

THEOREM 6.3 Let ),( SA and ),( TB be pointwise R -weakly

commuting pairs of self-mappings of a complete fuzzy metric space

),,( tMX satisfying

(6.3.1) );()(),()( XSXBXTXA ⊆⊆

and some Φ∈ϕ , there exists )1,0(∈h such that for all Xvu ∈, and

0>x ,

(6.3.2)

( ) ;0),,(),,,(),,,(),,,( ≥ϕ hxTvBvMxSuAuMxTvSuMhxBvAuM

(6.3.3)

( ) .0),,(),,,(),,,(),,,( ≥ϕ xTvBvMhxSuAuMxTvSuMhxBvAuM

If one of the mappings in compatible pair ),( SA or ),( TB is

continuous, then SBA ,, and T have a unique common fixed

point.

REMARK 6.1 Theorem 6.3 is an extension of the result of Pant and

Jha [104] to implicit relation.

Applications

112

Next, we give an application of Theorem 3.3 to fuzzy metric

spaces.

THEOREM 6.4 Let ),,( tMX be a fuzzy metric space with

),(),( baminbat = for all ba, in ]1,0[ . Let A and S be weakly

compatible self mappings of X satisfying the following conditions:

)()()1.4.6( XAXS ⊆ ;

),,(),,(),,()2.4.6( 2 xSvAvMxSuAuMhxAvAuM ≤ ; for each

Xvu ∈, , where 1>h and for all 0>x .



REMARK 6.2 Theorem 6.4 is an improvement of Kumar [74,

Theorem 4.1] in the sense that we have taken completeness of one

of the subspaces, not the whole space.

For the proof of Theorems 6.1, 6.2, 6.3 and 6.4 given in this

chapter, let ),,,()(, xvuMxF vu = then ),,( tFX is a Menger

space. Then the results are corresponding Theorems obtained from

2.1, 2.3, 2.5 and 3.3 respectively.

REFERENCES

1. Alimohammady, M., Esmaeli, A. and Saadati, R., Completeness

results in probabilistic metric spaces, Chaos, Solitons & Fractals

(2007), doi:10.1016/j.chaos.2007.01.072.

2. Balasubramaniam, P., Murlisankar, S. and Pant, R.P., Common

fixed points of four mappings in a fuzzy metric space, J. Fuzzy

Math. 10 (2) (2002), 379-384.

3. Banach, S., Sur les operations dans les ensembles absraites et

leurs applications, Fund. Math. 3 (1922), 133-181.

4. Bharucha-Reid, A.T., Fixed point theorems in probabilistic

analysis, Bull. Amer. Math. Soc. 82 (5) (1976), 641-657.

5. Bocşan, Gh., On some fixed point theorems in probabilistic

metric spaces, Universitatea Din Timisoara, Facultatea De Ale

Naturii 24 (1974), 1-7.


metric spaces, Seminar on the Theory of Functions and Applied

Mathematics, A: Probabilistic Metric Spaces, No. 24, Univ. of

Timişoara (1974).

References

114

7. Bocşan, Gh. and Constantin, Gh., The Kuratowski function and

some application to probabilistic metric spaces, Atti Acad. Naz.

Lincei 55 (1973), 236-240.

8. Boyd, D.W. and Wong, J.S.W., On nonlinear contractions, Proc.

Amer. Math. Soc. 20 (1969), 458-464.

9. Cain Jr., G.L. and Kasriel, R.H., Fixed and periodic points of

local contraction mappings on probabilistic metric spaces, Math.

Systems Theory 9 (1976), 289-297.

10. Chamola, K.P., Fixed points of mappings satisfying a new

contraction condition in random normed spaces, Math. Japon.

33 (6) (1988), 821-825.

11. Chamola, K.P., Dimri, R.C. and Pant, B.D., On nonlinear

contractions on Menger spaces, Ganita 39 (1) (1988), 49-54.

12. Chamola, K.P., Pant, B.D. and Singh, S.L., Common fixed point

theorems for probabilistic densifying mappings, Math. Japon. 36

(1991), 769-775.

13. Chang, S.S., Cho, Y.J. and Kang, S.M., Probabilistic Metric

Spaces and Nonlinear Operator Theory, Sichuan Univ. Press

(Chengdu), 1994.

References

115

14. Chang, S.S. and Huang, N.J., On the generalized 2-metric

spaces and probabilistic 2-metric spaces with applications to

fixed point theory, Math. Japon. 34 (6) (1989), 885-900.

15. Chang, S.S., Lee, B.S., Cho, Y.J., Chen, Y.Q., Kang, S.M., and

Jung, J.S., Generalized contraction mapping principle and

differential equations in probabilistic metric spaces, Proc. Amer.

Math. Soc. 124 (8) (1996), 2367-2376.

16. Chang, T.H., Common fixed point theorems in Menger spaces,

Bull. Inst. Math. Acad. Sinica 22 (1994), 17-22.

17. Chugh, R., Kumar, S. and Vats, R.K., A generalization of

Banach contraction principle in probabilistic D-metric spaces,

Math. Sci. Res. J. 7 (2) (2003), 41-46.

18. Ćirić, Lj.B., A generalization of Banach contraction principle,

Proc. Amer. Math. Soc. 45 (2) (1974), 267-274.

19. Ćirić, Lj.B., On fixed points of generalized contractions on

probabilistic metric spaces, Publ. Inst. Math. (Beograd) (N.S.)

18 (32) (1975), 71-78.

20. Dedeic, R. and Sarapa, N., On common fixed point theorems

for commuting mappings on Menger spaces, Radovi Mat. 4

(1988), 269-278.

References

116

21. Dedeić, R., and Sarapa, N., A common fixed point theorem for

three mappings on Menger spaces, Math. Japon. 34 (1989),

919–923.

22. Deng Z.K., Fuzzy pseudo-metric spaces. J. Math. Anal. Appl.

86 (1982), 74–95.

23. Dhage, B.C., Generalized metric spaces and mappings with

fixed point, Bull. Cal. Math. Soc. 84 (1992), 329-336.

24. Dhage, B.C. and Rhoades, B.E., A comparison of two

contraction principles, Math. Sci. Res. Hot-Line 3 (8) (1999), 49-

53.

25. Dimri, R.C. and Pant, B.D., Fixed points of probabilistic

densifying mappings, J. Natur. Phy. Sci. 16 (1-2)(2002), 69-76.

26. Dimri, R.C., Pant, B.D. and Kumar, Suneel, Fixed point of a

pair of non-surjective expansion mappings in Menger spaces,

Stud. Cerc. St. Ser. Matematica Universitatea Bacau. 18 (2008)

(To Appear).

27. Drossos, C.A., Stochastic Menger spaces and convergence in

probability, Rev. Roum. Math. Pures Et Appl. 22 (8) (1977),

1069-1076.

28. Egbert, R.J., Products and quotients of probabilistic metric

spaces, Pacific J. Math. 24 (3) (1968), 437-455.

References

117

29. Elamrani, M., Mbarki A., and Mehdaoui, B., Nonlinear

contractions and semigroups in general complete probabilistic

metric spaces, Panam. Math. J. 4 (11) (2001), 79-87.

30. El Naschie M.S., On the uncertainty of Cantorian geometry and

two-slit experiment, Chaos, Solitons & Fractals 9 (3) (1998),

517–529.

31. El Naschie M.S., A review of E-infinity theory and the mass

spectrum of high energy particle physics, Chaos, Solitons &

Fractals 19 (2004), 209–236.

32. El Naschie M.S., On a fuzzy Kahler-like Manifold which is

consistent with two-slit experiment, Int. J. Nonlinear Sci. and

Numerical Simulation 6 (2005), 95–98.

33. El Naschie M.S., The idealized quantum two-slit gedanken

experiment revisited-Criticism and reinterpretation, Chaos,

Solitons & Fractals 27 (2006), 9–13.

34. Ereeg M.A., Metric spaces in fuzzy set theory. J. Math. Anal.

Appl. 69 (1979), 338–353.

35. Fang, J.X., On fixed point theorems in fuzzy metric spaces,

Fuzzy Sets and Systems 46 (1992) 107–113.

36. Fisher, B., Fixed point on two metric spaces, Glasnik Mat. 16

(36) (1981), 333-337.

References

118

37. Fisher, B., Related fixed point on two metric spaces, Math.

Sem. Notes Kobe Univ. 10 (1982), 17-26.

38. Fisher, B. and Murthy, P.P., Related fixed point theorems for

two pairs of mappings on two metric spaces, Kyungpook Math.

J. 37 (1997), 343-347.

39. Fréchet, M., Sur quelques points du calcul fonctionnel, Rendic.

Circ. Mat. Palermo 22 (1906), 1–74.

40. Furi, M. and Vignoli, A., A fixed point theorem in complete

metric spaces, Boll. Un. Mat. Ital. 2 (4) (1969), 505–509.

41. Gähler, S., 2-metric Räume and ihre topologische Struktur,

Math. Nachr. 26 (1963/64), 115-148.

42. Ganguly, A., Rajput, A.S. and Tuteja, B.S., Fixed points of

probabilistic densifying mappings, J. Indian Acad. Math. 13 (2)

(1991), 110-114.

43. George, A. and Veeramani, P., On some results in fuzzy metric

spaces, Fuzzy Sets and Systems 64 (1994) 395–399.

44. Ghaemi, M.B. and Razani, A., Fixed and periodic points in the

probabilistic normed and metric spaces. Chaos, Solitons &

Fractals 28 (2006), 1181-1187.

45. Grabiec, M., Fixed points in fuzzy metric space, Fuzzy Sets and

Systems 27 (1988), 385-389.

References

119

46. Hadžić O., Fixed point theorems in probabilistic metric and

random normed spaces, Math. Sem. Notes Kobe Univ. 7

(1979), 260-270.

47. Hadžić, O., A generalization of the contraction principle in PM-

spaces, Rev Res, ZB Rad (Kragujevac) 10 (1980), 13–21.

48. Hadžić, O. and Pap, E., Fixed point theory in probabilistic metric

spaces. Dordrecht: Kluwer Academic publishers 2001.

49. Hadžić, O. and Pap, E., A fixed point theorem for multivalued

mappings in probabilistic metric spaces and an application in

fuzzy metric spaces, Fuzzy Sets and Systems 127 (2002), 333–

344.

50. Hadžić, O., Pap, E. and Budinčević, M., A generalization of

Tardiff’s fixed point theorem in probabilistic metric spaces and

applications to random equations, Fuzzy Sets and Systems 156

(2005), 124–134.

51. Hadžić, O., Pap, E. and Radu, V., Generalized contraction

mapping principles in probabilistic metric spaces, Acta Math.

Hungar. 101 (1-2) (2003), 131-148.

52. Hausdorff, F., Grundzuge der Mengenlehre, Leipzig: Veit and

Comp. (1914).

References

120

53. Hicks, T.L., Fixed point theory in probabilistic metric spaces,

Review of Research, Faculty of Science, Univ. of Novi Sad 13

(1983), 63-72.

54. Hicks, T.L., Fixed point theory in probabilistic metric spaces II,

Math. Japon. 44 (3) (1996), 487-493.

55. Hicks, T.L., Rhoades, B.E. and Saliga, L.M., Fixed point theory

in probabilistic metric spaces III, Math. Japon. 50 (3) (1999),

385-389.

56. Hosseini, S.B. and Saadati, R., Completeness results in

probabilistic metric spaces, I. Commun. Appl. Anal. 9 (2005),

549-554.

57. Huang, Y.Y. and Hong, C.C., Fixed points of compatible

mappings in complete Menger spaces, Sci. Math. 1 (1) (1998),

69-83.

58. Istrăteşcu, V.I., Fixed point theorems for some classes of

contraction mappings on non-Archimedean probabilistic metric

space, Publ. Math. (Debrecen) 25 (1978), 29-34.

59. Istrăteşcu, V.I., Fixed Point Theory, D. Reidel Publishing Co.

Holland, 1981.

References

121

60. Istrăteşcu, V.I. and Săcuiu, I., Fixed point theorems for

contraction mappings on probabilistic metric spaces, Rev.

Roumaine Math. Pures. Appl. 18 (1973), 1375-1380.

61. Jachymski, J.R., Fixed point theorems for expansive mappings,

Math. Japon. 42 (1) (1995), 131-136.

62. Jain, R.K., Fixed points on three metric spaces, Bull. Cal. Math.

Soc. 87(1995), 463-466.

63. Jain, R.K., Sahu, H.K. and Fisher, B., Related fixed point

theorems for three metric spaces, Novi Sad J. Math. Soc. 26

(1996), 11-17.

64. Jungck, G., Commuting mappings and fixed points, Amer. Math.

Monthly 83 (1976), 261-263.

65. Jungck, G., Compatible mappings and common fixed points,

Internat. J. Math. Math. Sci., 9 (4) (1986), 771-779.

66. Jungck, G., Common fixed points for commuting and

compatible maps on compacta, Proc. Amer. Math. Soc., 103 (3)

(1988), 977-983.

67. Jungck, G. and Rhoades, B.E., Fixed points for set valued

functions without continuity, Indian J. Pure Appl. Math. 29 (3)

(1998), 227-238.

References

122

68. Kadian, T. and Chugh, R., Fixed point theorems for weakly

compatible mappings in ε -chainable probabilistic metric

spaces, Int. Math. F. 3 (3) (2008), 135 - 145

69. Kaleva, O. and Seikkala, S., On fuzzy metric spaces. Fuzzy Set

and Systems 12 (1984), 215–229.

70. Kang, S.M. and Rhoades, B.E., Fixed point for four mappings,

Math. Japon. 37 (6) (1992), 1053-1059.

71. Khan, M.S. and Liu, Z., On coincidence points of densifying

mappings, Turkish J. Math. 21 (1997), 269-276.

72. Kohli, J.K. and Vashistha, S., Common fixed point theorems in

probabilistic metric space, Acta Math. Hungar. 115 (1-2) (2007),

37-47.

73. Kramosil, I. and Michalek, J., Fuzzy metric and statistical metric

spaces, Kybernetika 11 (1975) 336–344.

74. Kumar, S., Common fixed point theorems for expansion

mappings in various spaces, Acta Math. Hungar. (2007), doi:

10.1007/s10474-007-6142-2.

75. Kumar, S. and Chugh, R., Common fixed points theorems using

minimal commutativity and reciprocal continuity conditions in

metric spaces, Sci. Math. Japon. 56 (2) (2002), 269-275.

References

123

76. Kumar, S., Chugh, R. and Vats, R.K., Fixed point theorems for

expansion mappings theorems in various spaces, Stud. Cerc.

St. Ser. Matematica Universitatea Bacau 16 (2006), 47-68.

77. Kumar, S. and Pant, B.D., A common fixed point theorem for

expansion mappings in probabilistic metric spaces. Ganita 57

(1) (2006), 89-95. MR2290267

78. Kumar, Suneel and Pant, B.D., Some common fixed point

theorems for mappings satisfying a new contraction condition in

Menger spaces, Varāhmihir J. Math. Sci. 5 (1) (2005), 227-234.

MR2204665.

79. Kutukcu, S., A fixed point theorem in Menger spaces, Int. Math.

F. 1 (32) (2006), 1543-1554.

80. Kutukcu, S., A fixed point theorem for contraction type

mappings in Menger spaces, Am. J. Appl. Sci. 4 (6) (2007),

371-373.

81. Kutukcu, S., Yildiz, C. and Tuna, A., On common fixed points in

Menger probabilistic metric spaces, Int. J. Contemp. Math. Sci.

2 (8) (2007), 383-391.

82. Liu, Y. and Li, Z., Coincidence point theorems in probabilistic

and fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007)

58– 70.

References

124

83. Menger, K., Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28

(1942), 535-537.

84. Menger, K., Probabilistic geometry, Proc. Nat. Acad. Sci. U.S.A.

37 (1951), 226-229.

85. Mihet, D., A Banach contraction theorem in fuzzy metric

spaces, Fuzzy Sets and Systems 144 (2004), 431-439.

86. Mihet, D., A generalization of a contraction principle in

probabilistic metric spaces. Part II, Internat. J. Math. Math. Sci.

2005: 5 (2005) 729–736.

87. Mihet, D., A note on a fixed point theorem in Menger

probabilistic quasi-metric spaces, Chaos, Solitons & Fractals

(2007), doi:10.1016/ j.chaos.2007.10.029.

88. Mishra, S.N., Common fixed points of compatible mappings in

PM-spaces, Math. Japon. 36 (2) (1991), 283-289.

89. Mishra, S.N., Sharma, N. and Singh, S.L., Common fixed points

of maps on fuzzy metric spaces, Internat. J. Math. Math. Sci. 17

(2) (1994), 253-258.

90. Nung, N.P., A fixed point theorem in three metric spaces, Math.


References

125

91. Ouahab, A., Lahrech, S., Rais, S., Mbarki, A. and Jaddar, A.,

Fixed point theorems in complete probabilistic metric spaces,

Applied Math. Sci. 1 (46) (2007), 2277-2286.

92. Pant, B.D., Fixed point theorems in probabilistic metric spaces,

D. Phil. Thesis, Garhwal Univ. Srinagar, 1984.

93. Pant, B.D., Relation between fixed points in Menger spaces, J.

Indian Acad. Math. 24 (1) (2002), 135-142.

94. Pant, B.D., Dimri, R.C. and Chandola, V.B., Some results on

fixed points of probabilistic densifying mappings, Bull. Cal.

Math. Soc. 96 (3) (2004), 189-194.

95. Pant, B.D., Dimri, R.C. and Singh, S.L., Fixed point theorems

for expansion mapping on probabilistic metric spaces, Honam

Math. J. 9 (1) (1987), 77-81.

96. Pant, B.D. and Kumar, Suneel, Some common fixed point

theorems for commuting mappings in Menger spaces, J. Natur.

Phy. Sci. 19 (1) (2005), 29-37.

97. Pant, B.D. and Kumar, Suneel, A related fixed point theorem for

two pairs of mappings in two Menger spaces, Varāhmihir J.

Math. Sci. 6 (2) (2006), 471-476.

References

126

98. Pant, B.D., Kumar, Suneel and Aalam, I., Coincidence and fixed

points of probabilistic densifying mappings, Ganita 58 (1)

(2007), 9-15.

99. Pant, B.D., Tiwari, B.M.L. and Singh, S.L., Common fixed point

theorem for densifying mappings in probabilistic metric spaces,

Honam Math. J. 5 (1983), 151-154.

100. Pant, R.P., Common fixed points of non-commuting mappings,

J. Math. Anal. Appl. 188 (1994), 436-440.

101. Pant, R.P., Common fixed point theorems for contractive maps,

J. Math. Anal. Appl. 226 (1998), 251-258.

102. Pant, R.P., Common fixed points of four mappings, Bull. Cal.

Math. Soc. 90 (4) (1998), 281-286.

103. Pant, R.P., A common fixed point theorem under a new

condition, Indian J. Pure Appl. Math. 30 (2) (1999), 147-152.

104. Pant, R.P. and Jha, K., A remark on common fixed points of

four mappings in a fuzzy metric space, J. Fuzzy Math. 12 (2)

(2004), 433-437.

105. Pap, E., Hadžić, O. and Mesiar, R., A fixed point theorem in

probabilistic metric spaces and an application, J. Math. Anal.

Appl. 202 (1996), 433-449.

References

127

106. Pathak, H.K. and Dubey, R.P., Some fixed point theorems for

certain expansion mappings, Acta Ciencia Indica 16 (1990), 99-

102.

107. Pathak, H.K., Kang, S.M. and Ryu, J.W., Some fixed points of

expansion mapping, Internat. J. Math. Math. Sci. 19 (1) (1996),

97-102.

108. Pathak, H.K. and Tiwari, R., Fixed point theorems for expansion

mappings satisfying implicit relations, Filomat 20 (1) (2006), 43-

57.

109. Popa, V., A general fixed point theorem for weakly compatible

mappings in compact metric spaces, Turkish J. Math. 25

(2001), 465-474.

110. Popa, V., Fixed points for non-surjective expansion mappings

satisfying an implicit relation, Bul. ŞtiinŃ. Univ. Baia Mare Ser. B

Fasc. Mat.-Inform. 18 (1) (2002), 105-108.

111. Popa, V., A general common fixed point theorem of Meir and

Keeler type for noncontinuous weak compatible mappings,

Filomat 18 (2004), 33-40.

112. Popa, V., Fixed point theorems for mappings satisfyng a new

type of implicit relation, Stud. Cerc. St. Ser. Matematica

Universitatea Bacau. 15 (2005), 123-128.

References

128

113. Radu, V., Ideas and methods in fixed point theory for

probabilistic contractions, Seminar on Fixed Point Theory Cluj-

Napoca 3 (2002), 73-98.

114. Razani, A., A fixed point theorem in the Menger probabilistic

metric space, New Zealand J. Math. 35 (2006), 109-114.

115. Razani, A. and Fouladgar, K., Extension of contractive maps in

the Menger probabilistic metric space, Chaos, Solitons &

Fractals 34 (2007), 1724–1731.

116. Razani, A. and Shirdaryazdi, M., A common fixed point theorem

of compatible maps in Menger space, Chaos, Solitons &

Fractals 32 (1), (2007), 26-34.

117. Regan, D.O. and Saadati, R., Nonlinear contraction theorems in

probabilistic spaces, Appl. Math. Comput. 195 (1) (2008), 86-

93.

118. Rezaiyan, R., Cho, Y.J. and Saadati, R., A common fixed point

theorem in Menger probabilistic quasi-metric spaces. Chaos,

Solitons & Fractals (2006), doi:10.1016/j.chaos. 2006.10.007.

119. Rhoades, B.E., A comparison of various definitions of

contractive mappings, Trans. Amer. Math. Soc. 226 (1977),

257-290.

References

129

120. Rhoades, B.E., Some fixed point theorems for pairs of

mappings, Jñānãbha 15 (1985), 151-156.

121. Rhoades, B.E., An expansion mapping theorem, Jñānãbha 23

(1993), 151-152.

122. Sarapa, N., On common fixed point theorems for commuting

mappings on Menger spaces, Functional Analysis (Dubronik) 4

(1993), 257-273.

123. Schweizer, B., Sherwoad, H. and Tardiff, R.M., Contractions on

probabilistic metric spaces, examples and counter examples,

Stochastica 12 (1988), 5-17.

124. Schweizer, B. and Sklar, A., Statistical metric spaces, Pacific J.

Math. 10 (1960), 313-334.

125. Schweizer. B. and Sklar, A., Probabilistic Metric Spaces, North

Holland Series 1983.

126. Schweizer, B., Sklar, A. and Thorp, E., The metrization of

statistical metric spaces, Pacific J. Math. 10 (1960), 673-675.

127. Sehgal, V.M., Some fixed point theorems in functional analysis

and probability, Ph. D. Dissertation, Wayne State Univ. 1966.

128. Sehgal, V.M. and Bharucha-Reid, A.T., Fixed points of

contraction mappings on probabilistic metric spaces, Math.

Systems Theory 6 (1972), 97-102.

References

130

129. Sessa, S., On a weak commutativity condition of mappings in

fixed point considerations, Publ. Inst. Math. (Beograd)(N.S.) 32

(46) (1982), 149-153.

130. Sharma, B.K., Sahu, D.R., Bounias, M. and Bonaly, A., Fixed

points for non-surjective expansion mappings, Internat. J. Math.

Math. Sci. 21 (2) (1998), 277-288.

131. Sherwood, H., Complete probabilistic metric spaces, Z. Wahr.

verw. Geb. 20 (1971), 117- 128.

132. Singh, A., Rajput, A. and Saluja, A.S., Some fixed point

theorems of expansion mappings in complete metric spaces,

Vikram Math. J. 20 (2000), 22-30.

133. Singh, B. and Jain, S., Weak compatibility and fixed point

theorems in fuzzy metric spaces, Ganita 56 (2) (2005), 167-176.

134. Singh, B. and Jain, S., A fixed point theorem in Menger space

through weak compatibility, J. Math. Anal. Appl. 301 (2005),

439-448.

135. Singh, S.L. and Mishra, S.N., Remarks on Jachymski’s fixed

point theorems for compatible maps, Indian J. Pure Appl. Math.

28 (5) (1997), 611-615.

References

131

136. Singh, S.L., Mishra, S.N. and Pant, B.D., General fixed point

theorems in probabilistic metric and uniform spaces,

Indian J. Math. 29 (1) (1987), 9-21.

137. Singh, S.L. and Pant, B.D., A fixed point theorem for

probabilistic densifying mappings Indian J. Phy. Natur. Sci. 3

(B) (1983), 21-24.

138. Singh, S.L. and Pant, B.D., Common fixed point theorems for

commuting mappings in probabilistic metric spaces, Honam

Math. J. 5 (1) (1983), 139-150.

139. Singh, S.L. and Pant, B.D., Common fixed point theorems in

probabilistic metric spaces and extension to uniform spaces,

Honam Math. J. 6 (1) (1984), 1-12.

140. Singh, S.L. and Pant, B.D., Common fixed points of weakly

commuting mappings on non-Archimedean Menger spaces,

Vikram Math. J. 6 (1986), 27-31.

141. Singh, S.L. and Pant, B.D., Coincidence theorems, Math.

Japon. 31 (5) (1986), 783-789.

142. Singh, S.L. and Pant, B.D., Common fixed points of family of

mappings in Menger and uniform spaces, Riv. Mat. Univ. Parma

14 (4) (1988), 81-85.

References

132

143. Singh, S.L. and Pant, B.D., Coincidence and fixed point

theorems for a family of mappings on Menger spaces and

extension to uniform spaces, Math. Japon. 33 (6) (1988), 957-

973.

144. Singh, S.L., Pant, B.D. and Talwar, R., Fixed points of weakly

commuting mappings on Menger spaces, Jñānãbha 23 (1993),

115-122.

145. Singh, S.L. and Tomar, A., Weaker forms of commuting maps

and existence of fixed points, J. Korean Soc. Math. Edu. Ser. B:

Pure Appl. Math. 10 (3) (2003), 145-161.

146. Stojaković, M., A common fixed point theorem for the

commuting mapping, Indian J. Pure Appl. Math. 17 (4) (1986),

466-475.

147. Stojaković, M., Common fixed point theorems in complete

metric space and probabilistic metric spaces, Bull. Austral.

Math. Soc. 36 (1987), 73-88.

148. Stojaković, M., On some classes of contraction mappings,

Math. Japon. 33 (2) (1988), 311-318.

149. Stojaković, M., A common fixed point theorem in probabilistic

metric spaces and its applications, Glasnik Mat. 23 (43) (1988),

203-211.

References

133

150. Sumitra and Chugh, R., A generalization of Banach contraction

principle in Menger spaces, Indian J. Math. 45 (3) (2003), 369-

372.

151. Tan, D.H., On probabilistic condensing mappings, Rev.

Roumaine Math. Pures Appl. 26 (1981), 1305-1317.

152. Taniguchi, T., Common fixed point theorems on expansion type

mappings on complete metric spaces, Math. Japon. 34 (1)

(1989), 139-142.

153. Tardiff, R.M., Contraction maps on probabilistic metric spaces,

J. Math. Anal. Appl. 165 (1992), 517-523.

154. Tivari, B.M.L. and Pant, B.D., Fixed points of a pair of mappings

on probabilistic metric spaces, Jñānãbha 13 (1983), 13-25.

155. Vauski, R., A fixed point theorem for a sequence of maps

satisfying a new contractive type condition in Menger spaces,

Math. Japon. 35 (6) (1990), 1099-1102.

156. Vasuki, R., Fixed point and common fixed point theorems for

expansive maps in Menger spaces, Bull. Cal. Math. Soc. 83

(1991), 565-570.

157. Vasuki, R., Common fixed points for R-weakly commuting maps

in fuzzy metric spaces, Indian J. Pure Appl. Math. 30 (4) (1999),

419-423.

References

134

158. Xieping, D., A common fixed point theorem of commuting

mappings in probabilistic metric spaces, Kexue Tongbao 29 (2)

(1984), 147-150.

159. Wang, S.Z., Li, B.Y., Gao, Z.M. and Iseki, K., Some fixed point

theorems on expansion mappings, Math. Japon. 29 (4) (1984),

631- 636.

160. Wenzhi, Z., Probabilistic 2-metric spaces, J. Math. Research

Expo. 2 (1987), 241-245.

161. Zadeh, L.A., Fuzzy sets, Inform. Control 8 (1965), 338-353.

LIST OF PUBLICATIONS


theorems for mappings satisfying a new contraction condition

in Menger spaces, Varāhmihir J. Math. Sci. 5 (1) (2005), 227-

234. MR2204665.


expansion mappings in probabilistic metric spaces. Ganita 57

(1) (2006), 89-95. MR2290267


theorems for commuting mappings in Menger spaces, J.

Natur. Phy. Sci. 19 (1) (2005), 29-37.

4. Pant, B.D. and Kumar, Suneel, A related fixed point theorem

for two pairs of mappings in two Menger spaces, Varāhmihir J.

Math. Sci. 6 (2) (2006), 471-476.

5. Pant, B.D., Kumar, Suneel and Aalam, I., Coincidence and

fixed points of probabilistic densifying mappings, Ganita 58 (1)

(2007), 9-15.

List of publications

136

6. Dimri, R.C., Pant, B.D. and Kumar, Suneel, Fixed point of a

pair of non-surjective expansion mappings in Menger spaces,

Stud. Cerc. St. Ser. Matematica Universitatea Bacau. 18

(2008) (Accepted for publication ).

7. Pant, B.D. and Kumar, Suneel, Related fixed point theorems in

three Menger spaces, Varāhmihir J. Math. Sci.

(Communicated).

8. Pant, B.D. and Kumar, Suneel, A common fixed point theorem

in PM-spaces using implicit relation and application, Acta

Math. Hungar. (Communicated).

SOME FIXED POINT THEOREMS IN

MENGER SPACES AND APPLICATIONS

A SUMMARY

SUBMITTED TO THE

KUMAUN UNIVERSITY, NAINITAL

FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

SUPERVISOR: SUBMITTED BY:

Dr. B.D. PANT SUNEEL KUMAR

DEPARTMENT OF MATHEMATICS MOH - GUJRATIYAN

R.H. GOVT. POSTGRADUATE COLLEGE JASPUR (U.S. NAGAR)

KASHIPUR (UTTARAKHAND) 244713 UTTARAKHAND 247712

ENROLMENT NO. KU 979220 MARCH 2008

SUMMARY

‘Fixed Point Theory’ is a beautiful mixture of analysis (pure

and applied), topology and geometry. Fixed point theorems give the

conditions under which mappings (single or multivalued) have

solutions. Fixed point theory in probabilistic metric spaces can be

considered as a part of Probabilistic Analysis, which is a very

dynamic area of mathematical research.

The idea of introducing probabilistic notions into geometry

was one of the great thoughts of Karl Menger. His motivation came

from the idea that positions, distances, areas, volumes, etc., all are

subject to variation in measurement in practice. And, as, e.g.,

quantum mechanics implies, even in theory some measurements

are necessarily inexact. In 1942, Menger [27] published a note

entitled Statistical Metrics. The idea of Menger was to use

distribution functions instead of nonnegative real numbers as values

of the metric. The notion of a probabilistic metric space corresponds

to the situations when we do not know exactly the distance between

two points, we know only probabilities of possible values of this

distance. In this note he explained how to replace the numerical

Summary

2

distance between two points u and v by a function vuF , whose

value )(, xF vu at the real number x is interpreted as the probability

that the distance between u and v is less than x . Schweizer and

Sklar [38] took up the work, initiated by Menger [27] and developed

what is now called the theory of probabilistic metric spaces [see,

39].

Chapter I is introductory in nature. In this chapter, some

probabilistic topological preliminaries are collected and a brief

survey of the development of fixed point theory in PM-spaces is also

presented.

Chapter II is devoted to fixed points of contraction mappings.

The first effort of study of contraction mappings in PM-spaces was

made by Sehgal [40] in his doctoral dissertation in 1966. Studies by

several fixed point theorist have culminated in an elegant theory of

fixed point theorems in probabilistic metric spaces which have far

reaching consequences and are useful in the study of existence of

solutions of operator equations in probabilistic metric spaces and

probabilistic functional analysis. In this chapter, we obtain some

Summary

3

common fixed point theorems for triplet and quadruplet of mappings

satisfying new contraction conditions in Menger spaces. These

results improve and extend some well-known result of Singh and

Pant [42] and Singh, Mishra and Pant [41]. In the end of this

chapter, we prove a common fixed point theorem for pointwise R -

weakly commuting mappings having reciprocal continuity and

satisfying an implicit relation. Theorems 2.1-2.2 have been

published in [32] and Theorems 2.3-2.4 have been published in [25].

Chapter III is intended to the study of fixed points of

expansion mappings. Banach contraction principle also yields a

fixed point theorem for a diametrically opposite class of mappings

viz. expansion mappings. The study of fixed point of single

expansion mapping in a metric space is initiated by Wang, Li, Gao

and Iseki [43]. In 1987, Pant, Dimri and Singh [31] initiated the study

of fixed points of expansion mappings in Menger spaces. The first

result in this chapter is for four expansion mappings, two of them

being surjective, via compatibility of mappings in Menger spaces.

Theorem 3.3 is an improvement of Kumar [23, Theorem 3.2]. Also,

Theorems 3.3-3.6 have been proved for non-surjective expansion

Summary

4

mappings. Theorem 3.1 has been published in [24]. Theorems 3.3-

3.4 have been accepted for publication in [5].

The purpose of chapter IV is to establish coincidence and

common fixed point theorems for certain classes of nearly

densifying mappings in complete Menger space. The concept of

probabilistic densifying mappings was introduced by Bocşan [1]. In

[15], Ganguly, Rajput and Tuteja introduced the notion of

probabilistic nearly densifying mappings. Our results extend the

results of Khan and Liu [21] to PM-spaces and of Ganguly, Rajput

and Tuteja [15] as well. Theorems 4.1-4.3 have been published in

[34].

Chapter V is devoted to study of related fixed point theorems.

In 2002, Pant [30] initiated the study of the relation between the

fixed points of two contraction mappings in two different Menger

spaces by generalizing the results of Fisher [12, 13] to PM-spaces.

Theorem 5.2 extends the results of Pant [30] to two pairs of

mappings. Theorem 5.3 is an interesting generalization of the

results of Fisher and Murthy [14], Jain [18] and Jain, Sahu and

Fisher [19] to PM-spaces. Theorem 5.4 is an extension of Pant [30,

Summary

5

Theorem 2] for three mappings in three different Menger spaces and

is a generalization of the result of Nung [29] to PM-spaces. Theorem

5.2 has been published in [33].

In the last chapter, applications of fixed point theory

(especially, Menger probabilistic metric spaces) are mentioned. The

concept of PM-spaces may have very important applications in

quantum particle physics particularly in connections with both string

and ∞e theory, which were introduced and studied by a well-known

scientist, Mohamed Saladin El Naschie [6–9]. It is also of

fundamental importance in probabilistic functional analysis,

nonlinear analysis and applications [2, 16]. In the theory of PM-

spaces, contraction is one of the main tools to prove the existence

and uniqueness of a fixed point. In 1996, a group of mathematicians

Chang, Lee, Cho, Chen, Kang and Jung [3] presented a research

paper in which they obtained a generalized contraction mapping

principle in PM-spaces and applied it to prove the existence

theorems of solutions to differential equations in these spaces. In

1968, the concept of fuzzy sets was introduced by Zadeh [44].

Various authors, for example, Deng [4], Ereeg [10], Fang [11],

Kaleva and Seikkala [20], Kramosil and Michalek [22] have

Summary

6

introduced the concept of fuzzy metric spaces in different ways.

Fixed-point theory in fuzzy metric spaces for different contractive-

type mappings is closely related to that in probabilistic metric spaces

(refer [2, Chapters VIII, IX], [16, Chapters 3–5], [28]). Various

authors, for example, Hadžić and Pap [17], Razani and Shirdaryazdi

[37], Razani and Kouladgar [36] and Liu and Li [26] have studied the

applications of fixed point theorems in PM-spaces to fuzzy metric

spaces. As an application of some of our results namely, Theorems

2.1, 2.3, 2.5 and 3.3 to fuzzy metric spaces, we give here Theorems

6.1, 6.2, 6.3 and 6.4 respectively. It is worth mentioning that

Theorem 6.3 is an extension of the result of Pant and Jha [35] to

implicit relation and Theorem 6.4 is an improvement of Kumar [23,

Theorem 4.1] in the sense that we have taken completeness of one

of the subspaces, not the whole space.

REFERENCES


metric spaces, Universitatea Din Timisoara, Facultatea De Ale

Naturii 24 (1974), 1-7.

2. Chang, S.S., Cho, Y.J. and Kang, S.M., Probabilistic Metric

Spaces and Nonlinear Operator Theory, Sichuan Univ. Press

(Chengdu), 1994.

3. Chang, S.S., Lee, B.S., Cho, Y.J., Chen, Y.Q., Kang, S.M., and

Jung, J.S., Generalized contraction mapping principle and

differential equations in probabilistic metric spaces, Proc. Amer.

Math. Soc.124 (1996), 2367-2376.

4. Deng Z.K., Fuzzy pseudo-metric spaces. J. Math. Anal. Appl. 86

(1982), 74–95.

5. Dimri, R.C., Pant, B.D. and Kumar, Suneel, Fixed point of a pair

of non-surjective expansion mappings in Menger spaces, Stud.

Cerc. St. Ser. Matematica Universitatea Bacau. 18 (2008) (To

Appear).

6. El Naschie M.S., On the uncertainty of Cantorian geometry and

two-slit experiment, Chaos, Solitons & Fractals 9 (1998), 517–

529.

References

8

7. El Naschie M.S., A review of E-infinity theory and the mass

spectrum of high energy particle physics, Chaos, Solitons &

Fractals 19 (2004), 209–236.

8. El Naschie M.S., On a fuzzy Kahler-like Manifold which is

consistent with two-slit experiment, Int. J. Nonlinear Sci. and

Numerical Simulation 6 (2005), 95–98.

9. El Naschie M.S., The idealized quantum two-slit gedanken

experiment revisited-Criticism and reinterpretation, Chaos,

Solitons & Fractals 27 (2006), 9–13.

10. Ereeg M.A., Metric spaces in fuzzy set theory. J. Math. Anal.

Appl. 69 (1979), 338–353.

11. Fang, J.X., On fixed point theorems in fuzzy metric spaces,

Fuzzy Sets and Systems 46 (1992) 107–113.

12. Fisher, B., Fixed point on two metric spaces, Glasnik Mat. 16

(1981), 333-337.

13. Fisher, B., Related fixed point on two metric spaces, Math. Sem.

Notes Kobe Univ. 10 (1982), 17-26.

14. Fisher, B. and Murthy, P.P., Related fixed point theorems for two

pairs of mappings on two metric spaces, Kyungpook Math. J. 37

(1997), 343-347.

References

9

15. Ganguly, A., Rajput, A.S. and Tuteja, B.S., Fixed points of

probabilistic densifying mappings, J. Indian Acad. Math. 13 (2)

(1991), 110-114.

16. Hadžić, O. and Pap, E., Fixed point theory in probabilistic metric

spaces. Dordrecht: Kluwer Academic publishers 2001.

17. Hadžić, O. and Pap, E., A fixed point theorem for multivalued

mappings in probabilistic metric spaces and an application in

fuzzy metric spaces, Fuzzy Sets and Systems 127 (2002), 333–

344.

18. Jain, R.K., Fixed points on three metric spaces, Bull. Cal. Math.

Soc. 87 (1995), 463-466.

19. Jain, R.K., Sahu, H.K. and Fisher, B., Related fixed point

theorems for three metric spaces, Novi Sad J. Math. Soc. 26

(1996), 11-17.

20. Kaleva, O. and Seikkala, S., On fuzzy metric spaces. Fuzzy Set

and Systems 12 (1984), 215–229.

21. Khan, M.S. and Liu, Z., On coincidence points of densifying

mappings, Turkish J. Math. 21 (1997), 269-276.

22. Kramosil, I. and Michalek, J., Fuzzy metric and statistical metric

spaces, Kybernetika 11 (1975) 336–344.

References

10

23. Kumar, S., Common fixed point theorems for expansion

mappings in various spaces, Acta Math. Hungar. (2007), doi:

10.1007/s10474-007-6142-2.


expansion mappings in probabilistic metric spaces. Ganita 57 (1)

(2006), 89-95. MR2290267


theorems for mappings satisfying a new contraction condition in

Menger spaces, Varāhmihir J. Math. Sci. 5 (1) (2005), 227-234.

MR2204665.

26. Liu, Y. and Li, Z., Coincidence point theorems in probabilistic and

fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007) 58–

70.

27. Menger, K., Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28

(1942), 535-537.

28. Mihet, D., A Banach contraction theorem in fuzzy metric spaces,

Fuzzy Sets and Systems 144 (2004), 431-439.

29. Nung, N.P., A fixed point theorem in three metric spaces, Math.


30. Pant, B.D., Relation between fixed points in Menger spaces, J.

Indian Acad. Math. 24 (2002), 135-142.

References

11

31. Pant, B.D., Dimri, R.C. and Singh, S.L., Fixed point theorems for

expansion mapping on probabilistic metric spaces, Honam Math.

J. 9 (1987), 77-81.


theorems for commuting mappings in Menger spaces, J. Natur.

Phy. Sci. 19 (1) (2005), 29-37.

33. Pant, B.D. and Kumar, Suneel, A related fixed point theorem for

two pairs of mappings in two Menger spaces, Varāhmihir J.

Math. Sci. 6 (2) (2006), 471-476.

34. Pant, B.D., Kumar, Suneel and Aalam, I., Coincidence and fixed

points of probabilistic densifying mappings, Ganita 58 (2007), 9-

15.

35. Pant, R.P. and Jha, K., A remark on common fixed points of four

mappings in a fuzzy metric space, J. Fuzzy Math. 12 (2) (2004),

433-437.

36. Razani, A. and Fouladgar, K., Extension of contractive maps in

the Menger probabilistic metric space, Chaos, Solitons & Fractals

34 (2007), 1724–1731.

37. Razani, A. and Shirdaryazdi, M., A common fixed point theorem

of compatible maps in Menger space, Chaos, Solitons & Fractals

32 (1), (2007), 26-34.

References

12

38. Schweizer, B. and Sklar, A., Statistical metric spaces, Pacific J.

Math. 10 (1960), 313-334.

39. Schweizer. B. and Sklar, A., Probabilistic Metric Spaces, North

Holland Series 1983.

40. Sehgal, V.M., Some fixed point theorems in functional analysis

and probability, Ph. D. Dissertation, Wayne State Univ. 1966.

41. Singh, S.L., Mishra, S.N. and Pant, B.D., General fixed point

theorems in probabilistic metric and uniform space, Indian

J. Math. 29 (1987), 9-21.

42. Singh, S.L. and Pant, B.D., Common fixed point theorems for

commuting mappings in probabilistic metric spaces, Honam

Math. J. 5 (1983), 139-150.

43. Wang, S.Z., Li, B.Y., Gao, Z.M. and Iseki, K., Some fixed point

theorems on expansion mappings, Math. Japon. 29 (1984), 631-

636.

44. Zadeh, L.A., Fuzzy sets, Inform. Control 8 (1965), 338-353.

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