This document presents several theorems regarding common fixed points of mappings in fuzzy metric spaces. It begins with definitions of key concepts such as fuzzy sets, continuous t-norms, and fuzzy metric spaces. It then states four fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. The theorems provide conditions in terms of inequalities involving the fuzzy metric that guarantee the existence and uniqueness of a common fixed point. The proofs of the theorems demonstrate that the inequalities imply the mappings have a unique point of coincidence, which must then be their common fixed point.
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Continuous functions play a dominant role in analysis and homotopy theory. They
have applications to image processing, signal processing, information, statistics,
engineering and technology. Recently topologists studied the continuous like functions
between two different topological structures. For example, semi continuity between a
topological structure, α-continuity between a topology and an α-topology.
Nithyanantha Jothi and Thangavelu introduced the concept of binary topology in
2011. Recently the authors extended the notion of binary topology to n-ary topology
where n˃1 an integer. In this paper continuous like functions are defined between a
topological and an n-ary topological structures and their basic properties are
studied.
Ever accelerating pace of globalization has opened a window of opportunity for innovative entrepreneurs to jump from spring board of their locally retained markets into promise lands of globally acclaimed high ranking business heavens. The other name of these business heavens is Emerging Markets. It is now a known fact that the growth advantage in emerging markets, if other things remain the same, is expected to translate into 62% of global growth. Multinationals expect about 70 percent of the world’s growth over the next few years to come from emerging markets, with 40 percent emanating from just two countries: China and India. In addition to growth rate advantage, expanding middle-class consumer base, impressive Doing Business regulatory reforms, more than half of $55 billion of global middle-class spending will come from Asia Pacific.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Continuous functions play a dominant role in analysis and homotopy theory. They
have applications to image processing, signal processing, information, statistics,
engineering and technology. Recently topologists studied the continuous like functions
between two different topological structures. For example, semi continuity between a
topological structure, α-continuity between a topology and an α-topology.
Nithyanantha Jothi and Thangavelu introduced the concept of binary topology in
2011. Recently the authors extended the notion of binary topology to n-ary topology
where n˃1 an integer. In this paper continuous like functions are defined between a
topological and an n-ary topological structures and their basic properties are
studied.
Ever accelerating pace of globalization has opened a window of opportunity for innovative entrepreneurs to jump from spring board of their locally retained markets into promise lands of globally acclaimed high ranking business heavens. The other name of these business heavens is Emerging Markets. It is now a known fact that the growth advantage in emerging markets, if other things remain the same, is expected to translate into 62% of global growth. Multinationals expect about 70 percent of the world’s growth over the next few years to come from emerging markets, with 40 percent emanating from just two countries: China and India. In addition to growth rate advantage, expanding middle-class consumer base, impressive Doing Business regulatory reforms, more than half of $55 billion of global middle-class spending will come from Asia Pacific.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the
conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
https://utilitasmathematica.com/index.php/Index
Utilitas Mathematica journal that publishes original research. This journal publishes mainly in areas of pure and applied mathematics, statistics and others like algebra, analysis, geometry, topology, number theory, diffrential equations, operations research, mathematical physics, computer science,mathematical economics.And it is official publication of Utilitas Mathematica Academy, Canada.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
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On fixed point theorems in fuzzy metric spaces
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
274
On Fixed Point theorems in Fuzzy Metric Spaces
Shailesh T.Patel ,Ramakant Bhardwaj*,Rakesh Shrivastava**,Shyam Patkar*,Sanjay Choudhary***
The Research Scholar of Singhania University, Pacheri Bari (Jhunjhunu)
*Truba Institutions of Engineering & I.T. Bhopal, (M.P.)
**JNCT, Bhopal.
***Prof.&Head Deptt.of Mathematics Govt.NMV Hoshangabad.
Abstract: This paper presents some common fixed point theorems for occasionally weakly compatible mappings
in fuzzy metric spaces.
Keywords: Occasionally weakly compatible mappings,fuzzy metric space.
1. Introduction
Fuzzy set was defined by Zadeh [7]. Kramosil and Michalek [5] introduced fuzzy metric space, George and
Veermani [2] modified the notion of fuzzy metric spaces with the help of continuous t-norms. Many researchers
have obtained common fixed point theorems for mappings satisfying different types.introduced the new concept
continuous mappings and established some common fixed point theorems.open problem on the existence of
contractive definition which generates a fixed point but does not force the mappings to be continuous at the fixed
point.this paper presents some common fixed point theorems for more general .
2 Preliminary Notes
Definition 2.1 [7] A fuzzy set A in X is a function with domain X and values in [0,1].
Definition 2.2 [6] A binary operation * : [0,1]× [0,1]→[0,1] is a continuous t-norms if *is satisfying conditions:
(1) *is an commutative and associative;
(2) * is continuous;
(3) a *1 = a forall a ϵ [0,1];
(4) a * b ≤ c * d whenever a ≤ c and b ≤ d, and a,b,c,d є [0,1].
Definition 2.3 [2] A 3-tuple (X,M,*) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-
norm and M is a fuzzy set on X2
× (0,∞) satisfying the following conditions, for all x,y,z є X, s,t>0,
(f1)M(x,y,t) > 0;
(f2)M(x,y,t) = 1 if and only if x = y;
(f3) M(x,y,t) = M(y,x,t);
(f4)M(x,y,t)* M(y,z,s) ≤ M(x,z,t+s) ;
(f5)M(x,y,.): (0,∞)→(0,1] is continuous.
Then M is called a fuzzy metric on X.Then M(x,y,t) denotes the degree of nearness between x and y with respect
to t.
Definition 2.4[2]Let (X,d) be a metric space.Denotea * b = ab for all a,bє [0,1] and Md be fuzzy sets onX2
× (0,∞)
defined as follows:
Md(x,y,t)= ),( yxdt
t
+
.
Then (X, Md, *) is a fuzzy metric space.Wecall this fuzzy metric induced by a metric d as the standard
intuitionistic fuzzy metric.
Definition 2.5[2]Let (X, M, *) is a fuzzy metric space.Then
(a) a sequence {xn} in X is said to convers to x in X if for each є>o and each t>o, Nno ∈∃ such
That M(xn,x,t)>1-є for all n≥no.
(b)a sequence {xn} in X is said to cauchy to if for each ϵ>o and each t>o, Nno ∈∃ such
That M(xn,xm,t)>1-є for all n,m≥no.
(c) A fuzzy metric space in which euery Cauchy sequence is convergent is said to be complete.Definition 2.6[3]
Two self mappings f and g of a fuzzy metric space (X,M,*) are called compatible if
1),,(lim =
∞→
tgfxfgxM nn
n
whenever {xn} is a sequencein X such that xgxfx n
n
n
n
==
∞→∞→
limlim
For some x in X.
Definition 2.7[1]Twoself mappings f and g of a fuzzy metric space (X,M,*) are called reciprocally continuous on
X if fxfgxn
n
=
∞→
lim and gxgfxn
n
=
∞→
lim whenever {xn} is a sequence in X such that
xgxfx n
n
n
n
==
∞→∞→
limlim for some x in X.
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
275
Lemma 2.8[4] Let X be a set, f,gowcself maps of X. If f and g have a unique point of coincidence, w = fx = gx,
then w is the unique common fixed point of f and g.
3 Main Results
Theorem 3.1Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) such that
∫
).,(
0
)(
qtRyPxM
dttξ
M(Px,Ry,qt)≥ min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)} ……………(1)
For all x,yєX and for all t>o, then there exists a unique point wєX such that Pw = Sw = w and a unique point zєX
such that Rz = Tz = z. Moreover z = w so that there is a unique common fixed point of P,R,S and T.
Proof :Let the pairs {P,S} and {R,T} be owc, so there are points x,yϵX such that Px=Sx andRy=Ty. We claim
thatPx=Ry. If not, by inequality (1)
M(Px,Ry,qt) ≥ min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)}
M(Px,Ry,qt) ≥ min{ M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t), M(Ry,Px,t),
M(Px,Ry,t), M(Px,Ry,t)* M(Px,Px,t)}
≥ min{ M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t),
M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t)*1}
=M(Px,Ry,t).
Therefore Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz = Sz then by
(1) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P and S.By
Lemma 2.8 w is the only common fixed point of P and S.Similarly there is a unique point zєX such that z = Rz =
Tz.
Assume that w ≠ z. we have
M(w,z,qt) = M(Pw,Rz,qt)
≥min{ M(Sw,Tz,t), M(Sw,Pw,t), M(Rz,Tz,t), M(Pw,Tz,t), M(Rz,Sw,t),
M(Pw,Rz,t), M(Sw,Tz,t)* M(Pw,Pw,t)}
≥ min{ M(w,z,t), M(w,w,t), M(z,z,t), M(w,z,t), M(z,w,t),
M(w,z,t), M(w,z,t)* M(w,w,t)}
=M(w,z,t).
Therefore we have z = w and z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds.
Theorem 3.2 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) such that
M(Px,Ry,qt) ≥ Ø( min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)}) ……………(2)
For all x,yєXand Ø: [0,1]→[0,1] such that Ø(t) > t for all 0<t<1, then there existsa unique common fixed point of
P,R,S and T.
Proof :Let the pairs {P,S} and {R,T} be owc, so there are points x,yєX such that Px = Sx and Ry = Ty. We
claim that Px = Ry. If not, by inequality (2)
M(Px,Ry,qt) ≥ Ø( min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)})
>Ø(M(Px,Ry,t)). From Theorem 3.1
=M(Px,Ry,t).
Assume that w ≠ z. we have
M(w,z,qt) = M(Pw,Rz,qt)
≥min{ M(Sw,Tz,t), M(Sw,Pw,t), M(Rz,Tz,t), M(Pw,Tz,t), M(Rz,Sw,t),
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
276
M(Pw,Rz,t), M(Sw,Tz,t)* M(Pw,Pw,t)}
=M(w,z,t). From Theorem 3.1
Therefore we have z = w and z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds.
Theorem 3.3 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) such that
M(Px,Ry,qt) ≥ Ø(M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)) ……………(3)
For all x,yєXand Ø: [0,1]7
→[0,1] such that Ø(t,1,1,t,t,1,t) > t for all 0<t <1, then there exists a unique common
fixed point of P,R,S and T.
Proof: Let the pairs {P,S} and {R,T} be owc, so there are points x,yϵX such that Px = Sx and Ry = Ty. We
claim that Px = Ry. If not, by inequality (3)
M(Px,Ry,qt) ≥ Ø(M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t))
M(Px,Ry,qt) ≥ Ø(M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t), M(Ry,Px,t),
M(Px,Ry,t), M(Px,Ry,t)* M(Px,Px,t))
= Ø(M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t),
M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t)*1)
= Ø(M(Px,Ry,t), 1, 1, M(Px,Ry,t), M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t))
>M(Px,Ry,t).
A contradiction, therefore Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz =
Sz then by (3) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P
and S.By Lemma 2.8 w is the only common fixed point of P and S.Similarly there is a unique point zϵX such
that z = Rz = Tz.Thus z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds from
(3).
Theorem 3.4 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yєX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,t)*
M(Px,Ry,t)* M(Sx,Ty,t) ………………… (4)
Then there existsa unique common fixed point of P,R,S and T.
Proof: Let the pairs {P,S} and {R,T} be owc, so there are points x,yєX such that Px = Sx and Ry = Ty. We
claim that Px = Ry. If not, by inequality (4)
We have
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,t)*
M(Px,Ry,t)* M(Sx,Ty,t)
= M(Px,Ry,t)* M(Px,Px,t)* M(Ty,Ty,t)* M(Px,Ry,t)* M(Ry,Px,t)*
M(Px,Ry,t)* M(Px,Ry,t)
= M(Px,Ry,t)* 1* 1* M(Px,Ry,t)* M(Ry,Px,t)*
M(Px,Ry,t)* M(Px,Ry,t)
>M(Px,Ry,t).
Thus we have Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz = Sz then by
(4) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P and
S.Similarly there is a unique point zϵX such that z = Rz = Tz.Thus w is a common fixed point of P,R,S and T.
Corollary 3.5 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yϵX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,2t)*
M(Px,Ry,t)* M(Sx,Ty,t) …………………(5)
Then there existsa unique common fixed point of P,R,S and T.
Proof: We have
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,2t)*
M(Px,Ry,t)* M(Sx,Ty,t)
≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Sx,Ty,t)* M(Ty,Ry,t)*
M(Px,Ry,t)* M(Sx,Ty,t)
≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t) * M(Px,Ry,t)*
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
277
M(Sx,Ty,t)
= M(Px,Ry,t)* M(Px,Px,t)* M(Ty,Ty,t)* M(Px,Ry,t)* M(Ry,Px,t)*
M(Px,Ry,t)* M(Px,Ry,t)
= M(Px,Ry,t)* 1* 1* M(Px,Ry,t)* M(Ry,Px,t)*
M(Px,Ry,t)* M(Px,Ry,t)
>M(Px,Ry,t).
And therefore from theorem 3.4, P,R,S and T have a common fixed point.
Corollary 3.6 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yєX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t) …………………(6)
Then there existsa unique common fixed point of P,R,S and T.
Proof: The Proof follows from Corollary 3.5
Theorem 3.7 Let (X, M, *) be a complete fuzzy metric space.Then continuous self-mappings S and T of X have
a common fixed point in X if and only if there exites a self mapping P of X such that the following conditions
are satisfied
(i) PX ⊂ TX I SX
(ii) The pairs {P,S} and {P,T} are weakly compatible,
(iii) There exists a point qє(0,1) such that for all x,yєX and t > 0
M(Px,Py,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Py,Ty,t)* M(Px,Ty,t)* M(Py,Sx,t)
…………………(7)
Then P,S and T havea unique common fixed point.
Proof: Since compatible implies ows, the result follows from Theorem 3.4
Theorem 3.8 Let (X, M, *) be a complete fuzzy metric space and let P and R be self-mappings of X. Let the P
and R areowc.If there exists qє(0,1) for all x,yєX and t > 0
M(Sx,Sy,qt) ≥αM(Px,Py,t)+β min{M(Px,Py,t), M(Sx,Px,t), M(Sy,Py,t), M(Sx,Py,t)}
…………………(8)
For all x,yϵ X where α,β> 0, α+β> 1. Then P and S have a unique common fixed point.
Proof: Let the pairs {P,S} be owc, so there are points x єX such that Px = Sx. Suppose that exist another point y
єX for whichPy = Sy. We claim that Sx = Sy. By inequality (8)
We have
M(Sx,Sy,qt) ≥αM(Px,Py,t)+ β min{M(Px,Py,t) , M(Sx,Px,t), M(Sy,Py,t), M(Sx,Py,t)}
=αM(Sx,Sy,t)+ β min{M(Sx,Sy,t) , M(Sx,Sx,t), M(Sy,Sy,t), M(Sx,Sy,t)}
=(α+β)M(Sx,Sy,t)
A contradiction, since (α+β)> 1.Therefore Sx = Sy. Therefore Px = Py and Px is unique.
From lemma2.8 , P and S have a unique fixed point.
Acknowledgement: One of the author (Dr. R.K. B.) is thankful to MPCOST Bhopal for the project No 2556
References
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[2]A.George, P.Veeramani,”On some results in fuzzy metric spaces”,Fuzzy Sets and Systems, 64 (1994), 395-
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[3]G.Jungck,”Compatible mappings and common fixed points (2)”,Internat.J.Math.Sci. (1988), 285-288.
[4]G.Jungck and B.E.Rhoades,”Fixed Point Theorems for Occasionally Weakly compatible Mappings”,Fixed
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