This document presents a common fixed point theorem for six self-maps (A, B, S, T, L, M) on a Menger space using the concept of weak compatibility. It proves that if the maps satisfy certain conditions, including being weakly compatible and their images being complete subspaces, then the maps have a unique common fixed point. The proof constructs sequences to show the maps have a coincidence point, then uses weak compatibility and lemmas to show this point is the unique common fixed point.
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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.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
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.
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A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
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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.
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In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
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Mathematics Subject Classification: 47H10, 54A40
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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.
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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.
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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
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Mathematics Subject Classification: 47H10, 54A40
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Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
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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.
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International Journal of Mathematics and Statistics Invention (IJMSI)
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org ǁ Volume 2 ǁ Issue 1 ǁ January 2014 ǁ PP-11-13
Common fixed point theorem through weak compatibility in
Menger space
Ramesh Bhinde
Govt. P.G. College, Alirajpur, India
ABSTRACT: The object of this paper is removing the condition of continuity in[6]and establish a unique
common fixed point theorem for six mappings using the concept of weak compatibility in Menger space which is
an alternative result of Chandel and Verma[1].
KEYWORDS: Continuous t-norm, PM-space, Menger space, compatible, weak compatible.
I.
INTRODUCTION
In 1942, Menger [4] has introduced the theory of probabilistic metric space in which a distribution
function was used instead of non-negative real number as value of the metric. In 1962,Schweizer and Sklar [8]
studied this concept and gave fundamental result on this space. In 1972,Sehgal and Bharucha–Reid[9] obtained
a generalization of Banach Contraction principle on a complete metric space which is a milestone in developing
fixed point theory in Menger space. In 1982,Sessa [10] introduced weakly commuting mappings in Menger
space.In 1986,Jungck enlarged this concept to Compatible maps .in 1991, Mishra [5] has been introduced the
notion of compatible maps in Memger space.In 1998 Jungck and Rhoades[3] introduced the concept of wealky
compatibility and showed that each pair of compatible maps is wekly compatible but the converse need not to be
true.In 2005 Singh and Jain [11]generalized the result of Mishra[5] using the concept of weak compatibility and
compatibility of pair of self maps.In this paper prove a fixed point theorem for six weakly compatible mappings
in Menger space. First recall some definitions and known result in Menger space.
II.
PRELIMINARY NOTES
Definition 2.1A mapping T: [0,1] × [0,1] →[0,1] is a continuous t-norm if T satisfied the following
conditions:
(1) T(a, 1) = a, T(0,0) = 0
(2) T(a, b) = T(b, a),
(3) T is continuous
(4) T(a, b) ≤ T(c, d), whenever a ≤ c and b ≤ d, and a, b , c , d ∈[0,1]
(5) T(T(a, b), c) = T(a,T(b, c)) for all a,b,c ∈[0,1]
Definition 2.2 A Mapping F:R→ R+ is said to be a distribution function if it is non-decreasing and left
continuous with
Inf {F(t):t∈R} = 0 and Sup {F(t):t∈R} = 1
We will denote the Δ the set of all distribution function defined on [-∞,∞] while H(t) will always denote the
specific distribution function defined by
H(t)=
Definition 2.3(Schweizer and Sklar[8])The ordered pair (X,F) is called a probabilistic metric
space(shortly PM-space) if X is nonempty set and F is a probabilistic distance satisfyin the following condtions:
PM-1
F x,y(t) = 1 if and only if x=y
PM-2
F x,y(0) = 0
PM-3
F x,y(t) = F y,x(t)
PM-4
If F x,z(t) = 1 and F z,y(s) = 1 then F x,y(t+s)=1 for all x,y,z∈X
andt,s> 0
the ordered triple (X,F,T) is called Menger space if (X,F) is PM-space and T is a triangular norm such that for
all x,y,z∈X and t,s> 0
PM-5
F x,y (t+s)≥ F x,z (t) + F z,y (s)
www.ijmsi.org
11 | P a g e
2. Common fixed point theorem through weak compatibility in Menger space
Definition 2.4A Menger space (X, F, T) with the continuous t-norm Tis said to be complete iff every
Cauchy sequence in X converges to a point in X.
Definition 2.5 The self maps A and B of a Menger Space (X,F,T) are said to be compatible ifF
→ 1 for all t>0
Whenever {xn} is a sequence in X such that Axn,Bxn→ x for some x∈X
asn → ∞
ABxn,BAxn(t)
Definition 2.6 Two self-maps A andB of a non-empty set X are saidto be weakly compatible (or
coincidentally commuting) if they commute at theircoincidence points, i.e. if Ax= Bxfor some x ∈ X, then ABx=
BAx.
Lemma 2.7(Singh and Jain [11]) Let {xn} be a sequence in a Mengerspace (X, F, T) with continuous tnorm T and T(a,a) ≥ a. If there exists a constant k ∈ (0, 1) such that
Fxn,xn+1(kt) ≥ Fxn−1,xn(t) for all t > 0 and n = 1, 2,..........
then {xn} is a Cauchy sequence in X.
Lemma 2.8 (Singh and Jain [11]) Let (X, F, ∗) be a Menger space. If there exists k ∈(0, 1) such that
Fx,y(kt) ≥Fx,y(t) for all x, y ∈ X and t > 0, then x = y.
III.
MAIN RESULT
Theorem 3.1: Let A, B, S, T, L And M be self maps on a Menger space (X, F, t) with continuous t-norm
and defined by t(a, b)= min{a,b} for alla∈[0,1] and satisfying the following:
(3.1.1) AB (X)⊆M(X) and ST(X) ⊆L(X)
(3.1.2) M(X) and L(X) are complete subspace of X
(3.1.3) (AB,L) and (ST,M) are weakly compatible
(3.1.4) For all x, y∈X ,k∈(0,1) , t>0
𝐹3ABx,STy (kt) ≥min{ 𝐹3Lx,My(t) 𝐹3ABx,Lx(t) 𝐹3STy,Lx (t) 𝐹ABx,My (2t)
𝐹STy,Lx(t) 𝐹3STy,My(t)}
Then AB, ST, L and M have a unique common fixed point in X.
Proof: Let 𝑥0be any arbitrary pointof X.
Since AB (X) ⊆ M(X) and ST(X) ⊆ L(X).there exist 𝑥1, 𝑥2∈𝑋 such that
AB 𝑥0 = M 𝑥1, and ST 𝑥1 = L 𝑥2,
Inductively we construct the sequence { xn }and {yn }in X such that
M 𝑥2n−1 = AB 𝑥2n−2= y2n-1and y2n = L 𝑥2n= ST 𝑥2n−1for n=1,2,3,…………
Step 1: By taking x= 𝑥2n andy= 𝑥2n+1 in (iv), we have from(3.1.4)
𝐹3ABx2n ,STx2n+1 (kt) ≥min{ 𝐹3Lx2n,M x2n+1 (t) 𝐹3ABx2n,Lx2n (t) 𝐹3ST x2n+1,Lx2n (t)
𝐹ABx2n,M x2n+1 (2t) 𝐹ST x2n+1,Lx2n(t) 𝐹3ST x2n+1,M x2n+1 (t)}
⇒ 𝐹3y2n+1,y2n+2 (kt) ≥min{ 𝐹3y2n,y2n+1 (t) 𝐹3y2n+1, y2n (t) 𝐹3 y2n+1, y2n (t)
𝐹y2n+1, y2n+1(2t) 𝐹 y2n+1, y2n(t) 𝐹3 y2n+1, y2n+1(t)}
⇒ 𝐹y2n+1, y2n+2 (kt)≥ 𝐹y2n, y2n+1 (t)
Similarly we can written as,
𝐹 y2n, y2n+1 (kt)≥ 𝐹 y2n-1, y2n (t)
In general, for all n even or odd ,we have
𝐹yn, yn+1 (kt)≥ 𝐹 yn-1, yn (t) for k∈(0,1) and all t>0
Thus by lemma 2.7 {yn}is a Cauchy sequence in X and subsequence are also Cauchy sequence in X.
Step 2.Since M(X) is a complete subspace of X.Therefore { y2n+1 } converges to z∈Xthen Mu=z
Now takingx= 𝑥2n-2 , y = u, we havefrom(3.1.4)
𝐹3ABx2n-2,STu (kt) ≥ min{ 𝐹3L x2n-2,Mu (t) 𝐹3AB x2n-2,L x2n-2 (t) 𝐹3STu,L x2n-2 (t)
𝐹AB x2n-2,Mu (2t) 𝐹STu,L x2n-2 (t) 𝐹3STu,Mu (t)}
taking limit n → ∞,we have
𝐹3z,STu (kt) ≥ min{ 𝐹3z, z (t) 𝐹3 z,z (t) 𝐹3STu,z(t) 𝐹z,z (2t) 𝐹STu, z (t) 𝐹3STu,z (t)}
This gives, 𝐹3z,STu (kt) ≥ 𝐹3STu,z (t)
Hence by lemma 2.8 ,STu = z .Therefore STu=Mu=z
We can say,u is a coincidence point of ST and M.
Step 3. Since L(X) is a complete subspace of X.Therefore {y2n+1 } converges to
z∈X then Lv=z.Nowusing x=v and y= 𝑥2n-1we have from(3.1.4)
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3. Common fixed point theorem through weak compatibility in Menger space
𝐹3ABv,STx2n-1 (kt) ≥ min{ 𝐹3Lv,M x2n-1 (t) 𝐹3ABv,Lv (t) 𝐹3ST x2n-1,Lv (t)
𝐹ABv,M x2n-1y (2t) 𝐹ST x2n-1,Lv (t) 𝐹3ST x2n-1,My (t)}
taking limit n → ∞,we have
𝐹3ABv,z (kt) ≥ min{ 𝐹3z,z (t) 𝐹3ABv,z (t) 𝐹3z,z (t) 𝐹ABv,z (2t) 𝐹z,z(t) 𝐹3z,z(t)}
⇒ 𝐹3ABv,z (kt) ≥ 𝐹3ABv,z (t)
Hence by lemma 2.8 ,ABv = z . Therefore ABv=Lv=z
We can say, v is a coincidence point of AB and L.
Step 4.Since the pair{ST,M} is weakly compatible for some u∈𝑋
(ST)Mu=M(ST)u whenever STz=Mz
Now using x= 𝑥2n-2 andy=z,we havefrom(3.1.4)
𝐹3ABx2n-1,STz (kt) ≥ min{ 𝐹3L x2n-1,Mz (t) 𝐹3AB x2n-1,L x2n-1 (t) 𝐹3STz,L x2n-1 (t)
𝐹AB x2n-1,Mz (2t) 𝐹STz,L x2n-1 (t) 𝐹3STz,Mz (t)}
taking limit n → ∞,we have
𝐹3z,STz (kt) ≥ min{ 𝐹3z,z (t) 𝐹3z,z (t) 𝐹3STz,z (t) 𝐹z,z (2t) 𝐹STz,z (t) 𝐹3STz,z (t)}
Thus, 𝐹3z,STz (kt) ≥ 𝐹3STz,z (t)
Hence by lemma 2.8 ,STz = z . Since Mz=STz .Therefore STz=Mz=z
Step5:Since the pair{ST,M} is weakly compatible for some v∈𝑋
(AB)Lv = L(AB)v whenever ABz=Lz
now x= z and y= 𝑥2n-1 ,we havefrom (3.1.4)
𝐹3ABz,STx2n-1 (kt) ≥ min{ 𝐹3Lz,M x2n-1 (t) 𝐹3ABz,Lz (t) 𝐹3ST x2n-1,Lz (t) 𝐹ABz,M x2n-1 (2t)
𝐹ST x2n-1,Lz (t) 𝐹3ST x2n-1,M x2n-1 (t)}
Taking limit n → ∞,we have
𝐹3ABz,z (kt) ≥ min{ 𝐹3z,z (t) 𝐹3ABz,z (t) 𝐹3z,z (t) 𝐹ABz,z (2t) 𝐹z,z (t) 𝐹3z,z(t)}
⇒ 𝐹3ABz,z (kt) ≥ 𝐹3ABz,z (t)
Hence by lemma 2.8 ,ABz = z .
Since Lz=ABz ,Therefore ABz=Lz=z
Step6.Uniqueness Let w (w≠z) be another common fixed point of AB, ST, L and M , then w=ABw =STw
=Lw=Mw taking x=z and y=w thenfrom (3.1.4)
𝐹3ABz,STw (kt) ≥ min{ 𝐹3Lz,Mw (t) 𝐹3ABz,Lz (t) 𝐹3STw,Lz (t) 𝐹ABz,Mw (2t)
𝐹STw,Lz (t) 𝐹3STw,Mw (t)}
⇒ 𝐹3z,w (kt) ≥ min{ 𝐹3z,w (t) 𝐹3z,z (t) 𝐹3w,z (t)
𝐹z,w (2t) 𝐹w,z (t) 𝐹3w,w (t)}
𝐹3z,w (kt) ≥ 𝐹3z,w (t)
Hence by lemma 2.8, z=w which is a contradiction of our hypothesis .
Therefore,z is a common fixed point of AB,ST,L and M.
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