A unique common fixed point theorem for four
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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.
![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](https://image.slidesharecdn.com/auniquecommonfixedpointtheoremforfour-121204000559-phpapp01/85/A-unique-common-fixed-point-theorem-for-four-1-320.jpg)
![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](https://image.slidesharecdn.com/auniquecommonfixedpointtheoremforfour-121204000559-phpapp01/85/A-unique-common-fixed-point-theorem-for-four-2-320.jpg)
![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](https://image.slidesharecdn.com/auniquecommonfixedpointtheoremforfour-121204000559-phpapp01/85/A-unique-common-fixed-point-theorem-for-four-4-320.jpg)
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A unique common fixed point theorem for four
- 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.. [email protected] and sucharitha [email protected] 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
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