This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
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.
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.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
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 ].
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
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 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.
referenciaimovel.com.br Casa em Itaipuaçu Cod 202Referenciaimovel
Casa 1ª Locação com 3 quartos, sendo 1 suíte, sala, cozinha americana, 2 banheiros com box blindex, área de serviço, garagem coberta com portão de alumínio de correr e quintal todo gramado. Nos fundos do terreno uma edícula com área de churrasqueira, lavabo, quarto de ferramenta e chuveirão. O terreno tem 240 m². A casa fica próximo à restaurante, asfalto, condução, supermercado, padaria, escola, pizzaria, dentre outros comércios. Use o seu FGTS. Aceita financiamento.
Aproveite as novas condições de financiamento da Caixa Econômica Federal. Compre seu imóvel podendo financiar até 85% do seu valor, com a melhor taxa de juros do mercado. Podendo ainda ganhar um desconto de 50% no registro do seu imóvel. Agende uma visita conosco.
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.
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.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
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 ].
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
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 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.
referenciaimovel.com.br Casa em Itaipuaçu Cod 202Referenciaimovel
Casa 1ª Locação com 3 quartos, sendo 1 suíte, sala, cozinha americana, 2 banheiros com box blindex, área de serviço, garagem coberta com portão de alumínio de correr e quintal todo gramado. Nos fundos do terreno uma edícula com área de churrasqueira, lavabo, quarto de ferramenta e chuveirão. O terreno tem 240 m². A casa fica próximo à restaurante, asfalto, condução, supermercado, padaria, escola, pizzaria, dentre outros comércios. Use o seu FGTS. Aceita financiamento.
Aproveite as novas condições de financiamento da Caixa Econômica Federal. Compre seu imóvel podendo financiar até 85% do seu valor, com a melhor taxa de juros do mercado. Podendo ainda ganhar um desconto de 50% no registro do seu imóvel. Agende uma visita conosco.
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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.
Some Common Fixed Point Theorems for Multivalued Mappings in Two Metric SpacesIJMER
In this paper we prove some common fixed point theorems for multivalued mappings in two
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Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Journals
Abstract In this paper we proved fixed point theorem of four mapping on fuzzy metric space based on the concept of semi copatibility using implicit relation. These results generalize several corresponding relations in fuzzy metric space. All the results of this paper are new. Keywords: fuzzy metric space, compatibility, semi compatibility, implicit relation. 2000 AMS Mathematics Subject Classification: 47H10, 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
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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
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Mathematics subject classifications: 45H10, 54H25
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Fixed point theorems for four mappings in fuzzy metric space using implicit relation
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
359
Fixed Point Theorems for Four Mappings in Fuzzy Metric Space
using Implicit Relation
Kamal Wadhwa1
, Rashmi Tiwari2
and Ved Prakash Bhardwaj3
1, 2, 3
Govt. Narmada Mahavidyalaya Hoshangabad (M.P.)
Abstract
The present paper deals with proving some common fixed point results in a fuzzy metric space using Implicit
Relation.
Index Terms: Fixed point, fuzzy metric space, occasionally weakly compatible mappings, t-norm, Implicit
Relation.
I. Introduction
It proved a turning point in the development of fuzzy mathematics when the notion of fuzzy set was introduced
by Zadeh [12] in 1965. Following the concept of fuzzy sets, fuzzy metric spaces have been introduced by
Kramosil and Michalek [15] and George and Veeramani [2] modified the notion of fuzzy metric spaces with the
help of continuous t-norm, which shows a new way for further development of analysis in such spaces.
Consequently in due course of time some metric fixed points results were generalized to fuzzy metric spaces by
various authors. Sessa [20] improved commutativity condition in fixed point theorem by introducing the notion
of weakly commuting maps in metric space. Vasuki [20] proved fixed point theorems for R-weakly commuting
mapping. Pant [18, 19] introduced the new concept of reciprocally continuous mappings and established some
common fixed point theorems. The concept of compatible maps by [15] and weakly compatible maps by [9] in
fuzzy metric space is generalized by A.Al Thagafi and Naseer Shahzad [1] by introducing the concept of
occasionally weakly compatible mappings.
In this paper we give a fixed point theorem on fuzzy metric space with an implicit relation defined in [14]. Our
results extend and generalize the result of Wadhwa and Dubey [11] using Chouhan, Khanduja and Singh [14].
We start with some preliminaries:
II. Preliminaries
Definition 2.1 [4]: A binary operation *: [0, 1] × [0, 1]→[0, 1] is continuous t-norm if satisfies the following
conditions:
(1) * is commutative and associative,
(2) * is continuous,
(3) a*1 = a for all a є [0, 1],
(4) a*b ≤ c*d whenever a ≤ c and b ≤ d for all a, b, c, d є [0, 1].
Examples of t- norm are a*b = min {a, b} and a*b = ab.
Definition 2.2 [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: (The functions M(x, y, t) denote
the degree of nearness between x and y with respect to t, respectively.)
1) M(x, y, 0) = 0 for all x, y ∈ X
2) M(x, y, t) = 1 for all x, y ∈ X and t>0 if and only if x = y
3) M(x, y, t) = M(y, x, t) for all x, y ∈ X and t>0
4) M(x, y, t)*M(y, z, s) ≤ M(x, z, t+s) for all x, y, z ∈ X and t, s > 0
5) For all x, y ∈ X, M(x, y, .) : [0, ∞) → [0, 1] is left continuous.
Remark 2.1: In a FM M(X, M, *), M(x, y, .) is non- decreasing for all x.
Example 2.3 (Induced fuzzy metric [2]): – Let (X, d) be a metric space. Denote a*b =ab for all a, b∈[0, 1] and
let
Md be fuzzy sets on X2
× [0, ∞) defined as follows:
Md (x, y, t) =
( , )
Then (X, Md,*) is a fuzzy metric space. We call this fuzzy metric induced by a metric d as the standard
intuitionistic fuzzy metric.
Definition 2.4: Let M(X, M, *) be a FM - space. Then
(i) A sequence{xn} in X is said to be Cauchy Sequence if for all t>0 and p>0
lim →∞ ( xn+p, xn, t) = 1
(ii) A sequence {xn} in X is said to be convergent to a point x ∈ X if for all t>0,
lim →∞ ( xn, x, t) = 1
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
360
Since * is continuous, the limit is uniquely determined from (5) and (11) respectively.
Definition 2.5: A FM-Space M(X, M, *) is said to be complete if and only if every Cauchy sequence in X is
convergent.
Definition 2.6: Let A and S be maps from a fuzzy metric M(X, M, *) into itself. The maps A and S are said to be
weakly commuting if M( ASz, SAz, t) ≥ M( Az, Sz, t) for all z ∈ X and t>0
Definition 2.7 [10]: Let f and g be maps from an FM-space M(X, M, *) into itself .The maps f and g are said to
be compatible if for all t>0, lim →∞ ( fgxn, gfxn, t) = 1 whenever {xn} is a sequence in X such that
lim →∞ fxn = lim →∞ gxn = z for some z ∈ X.
Definition 2.8: Two mappings A and S of a fuzzy metric space M(X, M, *) will be called reciprocally
continuous if ASun→Az and SAun→Sz, whenever {un} is a sequence such that for some Aun, Sun→z for some z
∈ X.
Definition 2.9: Let M(X, M, *) be a fuzzy metric space. f and g be self maps on X. A point x in X is called a
coincidence point of f and g iff fx =gx. In this case w = fx =g x is called a point of coincidence of f and g.
Definition 2.10[8]: A pair of maps S and T is called weakly compatible pair if they commute at the coincident
points i.e., if Su = Tu for some u in X then STu = TSu.
Definition 2.11[1]: Two self maps f and g of a set X are called occasionally weakly compatible (owc) iff there is
a point x in X which is coincidence point of f and g at which f and g commute.
Example 2.11.1[1]: Let R be the usual metric space. Define S, T: R→R by Sx=2x and Tx=x2
for all x∈R. Then
Sx=Tx for x=0, 2, but ST0=TS0, and ST2≠TS2. S and T are occasionally weakly compatible self maps but not
weakly compatible.
Definition 2.12 (Implicit Relation)[14]: Let φ5 be the set of all real and continuous function φ : (R+
)5
→R and
such that
2.12 (i) φ non increasing in 2nd
and 4th
argument and
2.12 (ii) for u, v ≥0, φ(u,v,v,v,v)≥0 ⇒ u ≥ v
Example: φ(t1, t2, t3, t4, t5) = t1 – max {t1, t2, t3, t4}.
Lemma 2.1: Let {un} be a sequence in a fuzzy metric space M(X, M, *). If there exist a constant k ∈ (0, 1) such
that
M( un, un+1, kt) ≥ M(un-1, un, t) for all t>0 and n = 1 , 2 , 3….. Then {un} is a Cauchy sequence in X.
Lemma 2.2: Let M(X, M, *) be a FM space and for all x, y ∈ X, t>0 and if for a number q ∈ (0, 1),
M( x, y, qt) ≥ M(x, y, t) then x = y.
Lemma 2.3[9]: Let X be a set, f and g be owc self 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 Result
Theorem 3.1: Let M(X, M, *) be a complete fuzzy metric space and let A, B, S and T be self mappings of X.
Let the pairs {A, S} and {B, T} be owc. If for φ∈φ5 there exist q ∈ (0, 1) such that
φ
M(Ax, By, qt), M(Sx, Ax, t),
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+
. ≥ 0 ……………….. (1)
for all x, y ∈ X and t>0, and a,b,c,d≥0 with a&b and c&d cannot be simultaneously 0 , there exist a unique point
w ∈ X such that Aw = Sw = w and a unique point z ∈ X such that Bz = Tz = z. Morever z = w, so that there is a
unique common fixed point of A, B, S and T.
Proof: Let the pairs {A, S} and {B, T} be owc, so there are points x, y ∈ X such that Ax = Sx and By = Ty. We
claim that Ax = By. If not by inequality (1)
φ
M(Ax, By, qt), M(Sx, Ax, t),
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+
. ≥ 0
φ
M(Ax, By, qt), M(Ax, Ax, t),
,$!,") # ($!, ,")&
#
,
' , ,") ( ($!,$!,")&
' (
,
)
* ( , ,")
* ($!,$!,")
+ . )
( ,$!,") ($!, ,")
-
+
. ≥ 0
φ M(Ax, By, qt), 1, M(Ax, By, t), 1, M(Ax, By, t)& ≥ 0
φ M(Ax, By, qt), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t)& ≥ 0
Since, φ is non-increasing in 2nd
and 4th
argument therefore by 2.12 (i) and 2.12 (ii)
M(Ax, By, qt) ≥ M(Ax, By, t)
Therefore Ax = By i.e. Ax = Sx = By =Ty.
Suppose that there is a unique point z such that Az = Sz then by (1) we have
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
361
φ
M(Az, By, qt), M(Sz, Az, t),
2, !,") # ($!,%2,")&
#
,
' %2, 2,") ( ($!, !,")&
' (
,
)
* ( 2,%2,")
* ($!, !,")
+ . )
( 2, !,") ($!,%2,")
-
+
. ≥ 0
φ M(Az, By, qt), 1, M(Az, By, t), 1, M(Az, By, t)& ≥ 0
φ M(Az, By, qt), M(Az, By, t), M(Az, By, t), M(Az, By, t), M(Az, By, t)& ≥ 0
Since, φ is non-increasing in 2nd
and 4th
argument therefore 2.12 (i) and 2.12 (ii)
M(Az, By, qt) ≥ M(Az, By, t)
Az = By = Sz =Ty, So Ax = Az and w = Ax = Sx the unique point of coincidence of A and S. By lemma (2.3) w
is the only common fixed point of A and S. Similarly there is a unique point z ∈ X such that z = Bz =Tz.
Assume that w ≠ z we have
φ
M(Aw, Bz, qt), M(Sw, Aw, t),
4, 2,") # ($2,%4,")&
#
,
' %4, 4,") ( ($2, 2,")&
' (
,
)
* ( 4,%4,")
* ($2, 2,")
+ . )
( 4, 2,") ($2,%4,")
-
+
. ≥ 0
φ
M(Aw, Bz, qt), M(w, w, t),
4,2,") # (2,4,")&
#
,
' 4,4,") ( (2,2,")&
' (
,
)
* (4,4,")
* (2,2,")
+ . )
(4,2,") (2,4,")
-
+
. ≥ 0
φ M(Aw, Bz, qt), M(w, w, t) , M(w, z, t), 1, M(w, z, t)& ≥ 0
φ M(Aw, Bz, qt), 1, M(w, z, t), 1, M(w, z, t)& ≥ 0
φ M(Aw, Bz, qt), M(w, z, t) , M(w, z, t), M(w, z, t), M(w, z, t)& ≥ 0
Since, φ is non-increasing in 2nd
and 4th
argument
M(Aw, Bz, qt) ≥ M(w, z, t)
Since, w is the only common fixed point of A and we have z = Bz.
M(Aw, Bz, qt) = M(w, z, qt) ≥ M(w, z, t)
We have M(w, z, qt) ≥ M(w, z, t)
Hence z = w by Lemma (2.2) and z is a common fixed point of A, B, S and T. The uniqueness of the fixed point
holds from (1).
Definition 3.11 (Implicit Relation) [14]: Let φ6 be the set of all real and continuous function φ : (R+
)6
→R and
such that
3.11 (i) φ non increasing in 2nd
and 4th
argument and
3.11 (ii) for u, v ≥0, φ(u,v,v,v,v,v)≥0 ⇒ u ≥ v
Theorem 3.2: Let M(X, M, *) be a complete fuzzy metric space and let A, B, S and T be self mappings of X.
Let the pairs {A, S} and {B, T} be owc. If for φ∈φ6 there exist q ∈ (0, 1) such that
φ
M(Ax, By, qt), M(Sx, Ax, t),
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+ , M(By, Sx, t)
. ≥ 0 ……………….. (2)
for all x, y ∈ X and t>0, and a,b,c,d≥0 with a&b and c&d cannot be simultaneously 0 , there exist a unique point
w ∈ X such that Aw = Sw = w and a unique point z ∈ X such that Bz = Tz = z. Morever z = w, so that there is a
unique common fixed point of A, B, S and T.
Proof: Let the pairs {A, S} and {B, T} be owc, so there are points x, y ∈ X such that Ax = Sx and By = Ty. We
claim that Ax = By. If not by inequality (2)
φ
M(Ax, By, qt), M(Ax, Ax, t),
,$!,") # ($!, ,")&
#
,
' , ,") ( ($!,$!,")&
' (
,
)
* ( , ,")
* ($!,$!,")
+ . )
( ,$!,") ($!, ,")
-
+ , M(By, Ax, t)
. ≥ 0
φ(M(Ax, By, qt), 1, M(Ax, By, t), 1, M(Ax, By, t), M(By, Ax, t)) ≥ 0
φ(M(Ax, By, qt), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(By, Ax, t)) ≥ 0
Since, φ is non-increasing in 2nd
and 4th
argument therefore 3.11 (i) and 3.11 (ii)
M(Ax, By, qt) ≥ M(Ax, By, t)
Therefore Ax = By i.e. Ax = Sx = By = Ty. Suppose that there is another point z such that A z =S z then by (2)
we have A z = S z = Ty, So Ax = A z and w =Ax = Tx is the unique point of coincidence of A and T. By lemma
(2.2) w is a unique point z ∈ X such that w = Bz = Tz. Thus z is a common fixed point of A, B, S and T. The
uniqueness of fixed point holds by (2).
4. Mathematical Theory and Modeling www.iiste.org
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Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
362
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