This academic article presents a unique common fixed point theorem for four maps under contractive conditions in cone metric spaces. The authors prove the existence of coincidence points and a common fixed point theorem for four self-maps on a cone metric space that satisfy a contractive condition. They show that if one of the subspaces is complete, then the maps have a coincidence point, and if the maps are commuting, they have a unique common fixed point. This generalizes and improves on previous comparable results in the literature.
Common Fixed Point Theorems in Uniform SpacesIJLT EMAS
In the process of generalization of metric spaces to
Topological spaces, a few aspects of metric spaces are lost.
Therefore, the requirement of generalization of metric spaces
leads to the theory of uniform spaces. Uniform spaces stand
somewhere in between metric spaces and general topological
spaces. Khan[6] extended fixed point theorems due to Hardy and
Rogers[2], Jungck[4] and Acharya[1] in uniform space by
obtaining some results on common fixed points for a pair of
commuting mappings defined on a sequentially complete
Hausdorff uniform space. Rhoades et. al.[7] generalized the
result of Khan[6] by establishing a general fixed point theorem
for four compatible maps in uniform space .
In this paper, a common fixed point theorem in
uniform spaces is proved which generalizes the result of Khan[6]
and Rhoades et al.[7] by employing the less restrictive condition
of weak compatibility for one pair and the condition of
compatibility for second pair, the result is proved for six selfmappings.
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
Common Fixed Point Theorems in Uniform SpacesIJLT EMAS
In the process of generalization of metric spaces to
Topological spaces, a few aspects of metric spaces are lost.
Therefore, the requirement of generalization of metric spaces
leads to the theory of uniform spaces. Uniform spaces stand
somewhere in between metric spaces and general topological
spaces. Khan[6] extended fixed point theorems due to Hardy and
Rogers[2], Jungck[4] and Acharya[1] in uniform space by
obtaining some results on common fixed points for a pair of
commuting mappings defined on a sequentially complete
Hausdorff uniform space. Rhoades et. al.[7] generalized the
result of Khan[6] by establishing a general fixed point theorem
for four compatible maps in uniform space .
In this paper, a common fixed point theorem in
uniform spaces is proved which generalizes the result of Khan[6]
and Rhoades et al.[7] by employing the less restrictive condition
of weak compatibility for one pair and the condition of
compatibility for second pair, the result is proved for six selfmappings.
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
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.
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
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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 ].
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.
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
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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 ].
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.
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
1. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol 3, No 14, 2012
A Unique Common Fixed Point Theorem for Four
Maps under Contractive Conditions in Cone
Metric Spaces
S.Vijaya Lakshmi and J.Sucharita
Department of Mathematics, Osmania University, Hyderabad-500007, Andhra Pradesh, INDIA..
Vijayalakshmi.sandra@gmail.com and sucharitha _gk@yahoo.co.in
Abstract
In this paper, we prove existence of coincidence points and a common fixed point theorem for four maps under
contractive conditions in cone metric spaces for non continuous mappings and relaxation of completeness in the
space. These results extend and improve several well known comparable results in the existing literature.
AMS Subject Classification: 47H10, 54H25.
Keywords: Cone metric space; Common fixed point; Coincidence point.
1. Introduction and preliminaries.
In 2007 Huang and Zhang [3] have generalized the concept of a metric space, replacing the set of real numbers
by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different
contractive conditions. Subsequentl, Abbas and Jungck [1] and Abbas and Rhoades [2] have studied common
fixed point theorems in cone metric spaces (see also [3,4] and the references mentioned therein). Recently,
Abbas and Jungck [1] have obtained coincidence points and common fixed point theorems for two mappings in
cone metric spaces .The purpose of this paper is to extend and improves the fixed point theorem of [6].
Throughout this paper, E is a real Banach space, N = {1,2,3,……} the set of all natural numbers. For the
mappings f,g :X→X, let C(f,g) denotes set of coincidence points of f , g , that is C(f,g):={z ∈ X : fz = gz }.
We recall some definitions of cone metric spaces and some of their properties [3].
Definition1.1. Let E be a real Banach Space and P a subset of E .The set P is Called a cone if and only if :
(a) P is closed, nonempty and P ≠ {0};
(b) a,b ∈ R , a,b ≥ 0 ,x,y ∈ P implies ax + by ∈ P ;
(c) x ∈ P and –x ∈ P implies x = 0.
Definition1.2. Let P be a cone in a Banach Space E , define partial ordering ‘ ≤ ’on E with respect to P by x ≤ y
if and only if y-x ∈ P .We shall write x<y to indicate x ≤ y but x ≠ y while X<<y will stand for y-x ∈ Int P ,
where Int P denotes the interior of the set P. This Cone P is called an order cone.
Definition1.3. Let E be a Banach Space and P ⊂ E be an order cone .The order cone P is called normal if
there exists L>0 such that for all x,y ∈ E,
0 ≤ x ≤ y implies ║x║ ≤ L ║y║.
The least positive number L satisfying the above inequality is called the normal constant of P.
Definition1.4. Let X be a nonempty set of E .Suppose that the map
d: X × X → E satisfies :
(d1) 0 ≤ d(x,y) for all x,y ∈ X and
d(x,y) = 0 if and only if x = y ;
(d2) d(x,y) = d(y,x) for all x,y ∈ X ;
(d3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ X .
Then d is called a cone metric on X and (X,d) is called a cone metric space .
It is obvious that the cone metric spaces generalize metric spaces.
106
2. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol 3, No 14, 2012
Example1.1. ([3]) . Let E = R2, P = {(x, y) ∈ E such that : x, y ≥ 0} ⊂ R2 ,
X = R and d: X × X → E such that d(x, y) = (│x - y│, α│x - y│), where α ≥ 0 is a constant .Then (X,d) is a
cone metric space.
Definition1.5. Let (X,d) be a cone metric space .We say that {xn } is
(a) a Cauchy sequence if for every c in E with 0 << c , there is N such that
for all n , m > N , d(xn, xm) <<c ;
(b) a convergent sequence if for any 0 << c ,there is N such that for all
n > N, d(xn, x) <<c, for some fixed x ∈ X.
A Cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Lemma1.1. ( [3] ) .Let (X,d) be a cone metric space , and let P be a normal cone with normal constant L .Let
{xn } be a sequence in X .Then
(i). {xn } converges to x if and only if d(xn ,x) → 0 (n → ∞ ).
(ii). {xn } is a cauchy sequence if and only if d (xn , xm )→0 (n,m→∞).
Definition 1.6.([5]). Let f, g: X→X. Then the pair (f , g) is said to be
(IT)-Commuting at z ∈ X if f(g(z)) = g(f(z)) with f(z) = g(z ).
(2). Common fixed point theorem
In this section, we obtain existence of coincidence points and a common fixed point Theorem for four maps on a
cone metric space.
The following theorem is extends and improves Theorem 2.1[6].
Theorem2.2. Let (X,d) be a cone metric space and P a normal cone with normal constant L. Suppose that the
mappings S,T ,I and J are four self –maps on X such that T(X) ⊂ I(X) and S(X) ⊂ J(X) and T(X) = S(X) and
satisfy the condition
║d(Sx,Ty)║ ≤ k║d(Ix,Jy)║for all x,y ∈ X. (2.1)
Where k ∈ [0,1) is a constant .
If S(X) = T(X) is a complete subspace of X, then {S,I} and {T,J} have a coincidence point in X. More over,if
{S, I} and {T, J}are (IT)-Commuting then, S,T ,I and J have a unique common fixed point.Proof. For any
arbitrary point x0 in X, construct sequences {xn } and {yn } in X such that
y2n = Sx2n = Jx2n+1
and y2n+1 = Tx2n+1 = Ix2n +2 , for all n = 0,1,2,…..
By (2.1),we have
║d(y2n , y2n+1) ║= ║d(Sx2n ,Tx2n+1)║
≤ k║ d(Ix2n , Jx2n+1)║
≤ k║d(y2n-1, y2n) ║
Similarly, it can be show that
d(y2n+1 , y2n+2) ≤k d(y2n , y2n+1).
Therefore, for all n ,
║d(yn+1 , yn+2)║ ≤k║d(yn, yn+1)║ ≤………≤kn+1 ║d(y0 , y1)║ .
Now, for any m>n,
║d(yn , ym)║ ≤ ║d(yn, yn+1) ║+║ d(yn+1, yn+2) ║+………+║d(ym-1, ym) ║
≤ [kn + kn+1 + …..+ k m-1] ║ d(y1 , y0) ║
107
3. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol 3, No 14, 2012
kn
≤ — ║d(y1 , y0) ║ . From (1.3), we have
1–k
kn
║d(yn , ym)║ ≤ — L║d(y1 , y0) ║.
1–k
Which implies that ║d(yn , ym)║ → 0 as n , m → ∞ .
Hence {yn } is a Cauchy sequence.
Let us suppose that S(X) is complete subspace of X . Completeness on S(X) implies existence of z ∈ S(X) such
that lim y2n = Sx2n = z.
n→∞
limJx2n+1 = limSx2n=limIx2n = lim Tx2n+1 = z. (2.2)
n→∞ n→∞ n→∞ n→∞
That is , for any 0 << c, for sufficiently large n ,we have d (yn y ) <<c.
Since z ∈ T(X) ⊆ I(X) , then there exists a point u ∈ X such that z = Iu.
Let us prove that z = Su. By the triangle inequality ,we have .
║d(Su,z)║ ≤ ║d(Su,Tx2n+1)║+║d(Tx2n+1 , z)║
≤ k ║d(Iu, Jx2n+1)║+║d(Tx2n+1 , z)║.
Letting n→∞, we get
║d(Su,z)║ ≤ k║ d(z,z) ║ + ║d(z , z)║.
≤ k(0) + 0 = 0.That is Su =z.
Therefore z = Su = Iu , that is u is a coincidence point of S and I. (2.3)
Since ,z = S(X) ⊆ J(X), there exists a point v ∈ X such that z = Jv .We shall show that Tv = z . We have
║d(Tv ,z )║ ≤║d(Su,Tv)║
≤ k║d(Iu, Jv)║
≤ k║d(z, z)║ = 0.
Implies Tv = z.
Therefore z = Tv = Jv , that is v is a coincidence point of T and J. (2.4)
From (2.3) and (2.4) it follows
Su = Iu = Tv = Jv (=z). (2.5)
Since (S,I ) and (T,J) are (IT) – commuting .
║d(SSu , Su)║ = ║d(SSu , Iu)║
= ║ d(SSu , Tv)║
≤ k ║d(ISu, Jv)║
= k║d(SIu , Su)║
= k║ d(SSu,Su)║ <║ d(SSu,Su)║, (since k<1) ,a contradiction
SSu = Su(= z).
Su = SSu= SIu = ISu.
That is SSu = ISu= Su (= z).
Therefore Su = z is a common fixed point of S and I. (2.6)
Similarly,Tv = TTv = TJv = JTv
108
4. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol 3, No 14, 2012
Implies TTv = JTv = Tv( = z).
Therefore Tv (=z) is a common fixed point of T and J. (2.7)
In view of (2.6) and (2.7) it follows S,T ,I and J have a common fixed point namely z.
Uniqueness , let z1 be another common fixed point of S , T , I and J .
Then ║d(z, z1)║ = ║d(Sz, Tz)║
≤ k ║d(Iz , Jz1)║
≤ k║d(z, z1)║
< ║d(z, z1)║. (since k<1 ).
Which is a contradiction,
Implies z = z1 .
Therefore S, T, I and J have a unique common fixed point .
Remark2.1. If S = T and I = J then the theorem reduces to the theorem 2.1 of Stojan Radenovic [6] with I(X)
complete ,which is an improvement of Theorem 2.1.of [6]. Since in this paper J(X) is complete which is a super
space of I(X)
References
[1] M.Abbas and G.Jungck, common fixed point results for non commuting mappings without continuity in
cone metric spaces. J.math.Anal.Appl. 341(2008) 416-420.
[2] M.Abbas and B.E.Rhoades, Fixed and periodic point results in cone metric Spaces.
Appl.Math .Lett.,22(2009),511-515.
[3] L.G.Huang, X.Zhang, cone metric spaces and fixed point theorems of contractive mappings,
J.Math.Anal.Appl.332(2)(2007)1468-1476
[4] S.Rezapour and Halbarani, some notes on the paper “cone metric spaces and fixed point theorem of
contractive mappings “, J.Math. Anal. Appl. 345(2008), 719-724.
[5] S.L.Singh, Apichai Hematulin and R.P.Pant, new coincidence and common fixed point theorem, Applied
General Topology 10(2009), no.1, 121-130.
[6] Stojan Radenovi, common fixed points under contractive conditions in cone metric spaces, Computer and
Mathematics with Applications 58 (2009) 1273- 1278.
109
5. This academic article was published by The International Institute for Science,
Technology and Education (IISTE). The IISTE is a pioneer in the Open Access
Publishing service based in the U.S. and Europe. The aim of the institute is
Accelerating Global Knowledge Sharing.
More information about the publisher can be found in the IISTE’s homepage:
http://www.iiste.org
CALL FOR PAPERS
The IISTE is currently hosting more than 30 peer-reviewed academic journals and
collaborating with academic institutions around the world. There’s no deadline for
submission. Prospective authors of IISTE journals can find the submission
instruction on the following page: http://www.iiste.org/Journals/
The IISTE editorial team promises to the review and publish all the qualified
submissions in a fast manner. All the journals articles are available online to the
readers all over the world without financial, legal, or technical barriers other than
those inseparable from gaining access to the internet itself. Printed version of the
journals is also available upon request of readers and authors.
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar