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On Common Fixed Point Theorem In Complete Metric Space
Jay G. Mehta and M. L. Joshi
Harish Chandra Research Institute - Allahabad (India) E-mail:jgm [email protected]
M. and N. Virani Science College - Rajkot (india) E-mail:[email protected]
(Received 18.11.2010, Accepted 25.11.2010) Abstract
We prove a unique common fixed-point theorem for two pair of weakly com- patible maps in a complete metric space, which generalizes the result of Brian Fisher by a weaker condition such as weakly compatibility instead of compati- bility and contractive modulus instead of continuity of maps.
Keywords: Common fixed point, Complete metric space, Contractive Mod- ulus, Weakly compatible maps.
2000 MSC No: 47H10, 54H25.
1 Introduction
The concept of the commutativity has generalized in several ways. For this Sessa S [6] has introduced the concept of weakly commuting and Gerald Jungck [2] initiated the concept of compatibility. It can be easily verified that when the two mappings are commuting then they are compatible but not conversely.
In 1998, Jungck and Rhoades [4] introduced the notion of weakly compatible and showed that compatible maps are weakly compatible but not conversely.
Brian Fisher [1] proved an important Common Fixed Point theorem.
The study of common fixed point of mappings satisfying contractive type con- ditions has been a very active field of research activity during the last three decades. In 1922, the Polish mathematician, Banach, proved a theorem which
ensures, under appropriate conditions, the existence and uniqueness of a fixed point. His result is called Banach fixed point theorem or the Banach con- traction principle. This theorem provides a technique for solving a variety of applied problems in mathematical science and engineering. Many authors have extended, generalized and improved Banach fixed point theorem in differ- ent ways. Jungck [2]introduced more generalized commuting mappings, called compatible mappings, which are more general than commuting and weakly commuting mappings.
The main purpose of this paper is to present fixed point results for two pair of maps satisfying a new contractive condition by using the concept of weakly compatible maps in a complete metric space.
2 Preliminaries
we recall the definitions of complete metric space, the notion of convergence and other results that will be needed in the sequel.
Definition 2.1 Let f and g be two self-maps on a set X. Maps f and g are said to be commuting iff gx=gf xfor all x∈X.
Definition 2.2 Let f and g be two self-maps on a set X. If f x=gx , for somex in X then x is called coincidence point of f and g.
Definition 2.3 [4] Let f and g be two self-maps defined on a set, then f and g are said to be weakly compatible if they commute at coincidence points.
that is, if f u=gu for some u∈X , then f gu=gf u.
Lemma 2.4 [2] Let f and g be weakly compatible self mappings of a set X.
If f and g have a unique point of coincidence, that is,w=f x=gx , then w is the unique common fixed point of f and g.
Definition 2.5 A sequence {xn} in a metric space(X, d) is said to be con- vergent to a point x∈X , denoted by lim
n→∞xn=x , if lim
n→∞d(xn, x) = 0.
Definition 2.6 A sequence {xn} in a metric space (X, d) is said to be Cauchy sequence if lim
t→∞d(xn, xm) = 0 for all n, m > t .
Definition 2.7 A metric space(X, d)is said to be complete if every Cauchy sequence in X is convergent.
Definition 2.8 A function φ : [0,∞) → [0,∞) is said to be a contractive modulus if φ: [0,∞)→[0,∞) and φ(t)< t fort >0.
Definition 2.9 A real valued function φ defined on X ⊆ R is said to be upper semi continuous if lim
n→∞supφ(tn) ≤ φ(t), for every sequence {tn} ∈ X with tn→t as n → ∞ .
It is clear that every continuous function is upper semi continuous but converse may not true.
Theorem 2.10 [1] Suppose S, P, T and Q are four self maps of a metric space (X,d) satisfying the following conditions.
1. S(X) ⊆ Q(X) and T(X) ⊆ P(X) . 2. Pairs (S,P) and (T,Q) are commuting.
3. One of S,P,T and Q is continuous.
4. d(Sx, T y)≤cλ(x, y). whereλ(x, y) =max{d(P x, Qy), d(P x, Sx), d(Qy, T y)}
for all x, y ∈X and 0≤c <1.
Further if 5. X is complete.
Then S,P,T and Q have a unique common fixed point z ∈ X. Also z is the unique common fixedpoint of (S,P) and of (T,Q).
3 Main Result
In this section we prove a common fixed point theorem for two pairs of weakly compatible mappings in complete metric spaces using a contractive modulus.
This is the generalization of theorem 2.10 in the sense that instead of taking constantc, we take an upper semi continuous, contractive modulus.
Theorem 3.1 Let (X,d) be a complete metric space. Suppose that the map- pings P, Q, S and T are four self-maps of X satisfying the following conditions:
1. T(X) ⊆ P(X) and S(X) ⊆ Q(X);
2. d(Sx, T y)≤φ(λ(x, y))
where φ is an upper semi continuous, contractive modulus and
λ(x, y) =max{d(P x, Qy), d(P x, Sx), d(Qy, T y),12(d(P x, T y)+d(Qy, Sx))}.
3. The pairs (S, P) and (T, Q) are weakly compatible Then P, Q, S and T have a unique common fixed point.
Proof : Supposex0 is an arbitrary point of X and define the sequence {yn}in X such that,
yn =Sxn=Qxn+1 yn+1 =T xn+1 =P xn+2 By (ii), we have,
d(yn, yn+1) = d(Sxn, T xn+1)
≤ φ(λ(xn, xn+1)) where
λ(xn, xn+1)
=max{d(P xn, Qxn+1), d(P xn, Sxn), d(Qxn+1, T xn+1),12(d(P xn, T xn+1)+d(Qxn+1, Sxn))}
=max{d(T xn−1, Sxn), d(T xn−1, Sxn), d(Sxn, T xn+1),12(d(T xn−1, T xn+1)+d(Sxn, Sxn))}
=max{d(T xn−1, Sxn), d(Sxn, T xn+1),12(d(T xn−1, T xn+1) + 0)}
=max{d(yn−1, yn), d(yn, yn+1),12(d(yn−1, yn+1) + 0)}
≤max{d(yn−1, yn), d(yn, yn+1)}
Sinceφ is a contractive modulus, λ(xn, xn+1) =d(yn, yn+1) is not possible.
Thus
d(yn, yn+1)≤φ(d(yn−1, yn)) (1) Sinceφis an upper semi continuous, contractive modulus, equation (1) implies that the sequence {d(yn+1, yn)} is monotonic decreasing and continuous.
Hence there exists a real number, sayr≥0 such that lim
n→∞d(yn+1, yn) =r Therefore as n→ ∞, equation (1) implies that
r≤φ(r)
which is possible only ifr= 0 because φ is a contractive modulus.
Thus lim
n→∞d(yn+1, yn) = 0
Now we show that{yn} is a Chuchy sequence.
Let if possible we assume that {yn}is not a Chuchy sequence.
Then there exists anε >0 and subsequences {ni} and {mi} such that mi < ni < mi+1 and
d(ymi, yni)≥ε and d(ymi, yni−1)< ε (2) So thatε≤d(ymi, yni)≤d(ymi, yni−1) +d(yni−1, yni)< ε+d(yni−1, yni) Therefore lim
i→∞d(ymi, yni) =ε
Now d(ymi−1, yni−1)≤d(ymi−1, ymi) +d(ymi, yni) +d(yni, yni−1)
By taking limit as i→ ∞, we get lim
i→∞d(ymi−1, yni−1) = ε Now by (ii) and (2)
ε≤d(ymi, yni) = d(Sxmi, T xni)≤φ(λ(xmi, xni))
i.e., ε ≤φ(λ(xmi, xni)) (3)
where λ(xmi, xni)
=max{d(P xmi, Qxni), d(P xmi, Sxmi), d(Qxni, T xni),12(d(P xmi, T xni)+d(Qxni, Sxmi))}
=max{d(T xmi−1, Sxni−1), d(T xmi−1, Sxmi), d(Sxni−1, T xni),12(d(T xmi−1, T xni)+
d(Sxni−1, Sxmi))}
=max{d(ymi−1, yni−1), d(ymi−1, ymi), d(yni−1, yni),12(d(ymi−1, yni)+d(yni−1, ymi))}
By taking limit as i→ ∞, we get lim
i→∞λ(xmi, xni) = max{ε,0,0,12(ε+ε)}
Thus we have, lim
i→∞λ(xmi, xni) =ε Therefore from (3) ε≤φ(ε)
This is a contradiction because 0< ε and φ is contractive modulus.
Thus{yn} is a Chuchy sequence in X.
SinceX is complete, there exists a point z inX such that lim
n→∞yn =z T hus lim
n→∞Sxn = lim
n→∞Qxn+1 =z and lim
n→∞T xn+1 = lim
n→∞P xn+2 =z i.e., lim
n→∞Sxn = lim
n→∞Qxn+1 = lim
n→∞T xn+1 = lim
n→∞P xn+2 =z SinceT(X)⊆P(X), there exists a point u∈X such that z =P u.
Then by (ii), we have
d(Su, z) ≤ d(Su, T xn+1) +d(T xn+1, z)
≤ φ(λ(u, xn+1)) +d(T xn+1, z) whereλ(u, xn+1)
= max{d(P u, Qxn+1), d(P u, Su), d(Qxn+1, T xn+1),12(d(P u, T xn+1)+d(Qxn+1, Su)}
= max {d(z, Sxn), d(z, Su), d(Sxn, T xn+1),12(d(z, T xn+1) +d(Sxn, Su))}
Taking the limit as n→ ∞ yields,
λ(u, xn+1) = max {d(z, z), d(z, Su), d(z, z),12(d(z, z) +d(z, Su))} = d(Su, z) Thus as n→ ∞ ,d(Su, z)≤φ(d(Su, z)) +d(z, z) =φ(d(Su, z))
IfSu6=z then d(Su, z)>0 and hence as φ is contractive modulus φ(d(Su, z))< d(Su, z).
Therefored(Su, z)< d(Su, z), which is a contradiction.
ThusSu=z. So P u=Su=z.
Sou is a coincidence point of P and S.
Since the pair of maps S and P are weakly compatible, SP u=P Su, i.e.,Sz =P z.
Again Since S(X)⊆Q(X), there exists a pointv ∈X such that z =Qv.
Then by (ii), we have
d(z, T v) = d(Su, T v)
≤ φ(λ(u, v)) where
λ(u, v) = max{d(P u, Qv), d(P u, Su), d(Qv, T v),1
2(d(P u, T v) +d(Qv, Su))}
= max{d(z, z), d(z, z), d(z, T v),1
2(d(z, T v) +d(z, z))}
= d(z, T v)
Thusd(z, T v)≤φ(d(z, T v))
IfT v 6=zthend(z, T v)>0 and hence asφis contractive modulusφ(d(z, T v))<
d(z, T v).
Therefored(z, T v)< d(z, T v), which is a contradiction.
ThereforeT v =Qv =z.
Sov is a coincidence point of Qand T.
Since the pair of maps Q and T are weakly compatible, QT v = T Qv, i.e., Qz=T z.
Now we show thatz is a fixed point ofS.
By (ii), we have
d(Sz, z) = d(Sz, T v)
≤ φ(λ(z, v)) where
λ(z, v) = max{d(P z, Qv), d(P z, Sz), d(Qv, T v),1
2(d(P z, T v) +d(Qv, Sz))}
= max{d(Sz, z), d(Sz, Sz), d(z, z),1
2(d(Sz, z) +d(z, Sz))}
= d(Sz, z)
Thusd(Sz, z)≤φ(d(Sz, z))
IfSz 6=zthend(Sz, z)>0 and hence asφis contractive modulusφ(d(Sz, z))<
d(Sz, z).
Therefored(Sz, z)< d(Sz, z), which is a contradiction.
ThereforeSz =z.
Hence Sz=P z =z.
Now, we show that z is a fixed point ofT. By (ii), we have
d(z, T z) = d(Sz, T z)
≤ φ(λ(z, z)) where
λ(z, z) = max{d(P z, Qz), d(P z, Sz), d(Qz, T z),1
2(d(P z, T z) +d(Qz, Sz))}
= max{d(z, T z), d(z, z), d(T z, T z),1
2(d(z, T z) +d(T z, z))}
= d(z, T z)
Thusd(z, T z)≤φ(d(z, T z))
Ifz 6=T zthend(z, T z)>0 and hence asφis contractive modulusφ(d(z, T z))<
d(z, T z).
Therefored(z, T z)< d(z, T z), which is a contradiction.
Hence z =T z.
ThereforeT z =Qz =z.
ThereforeSz =P z =T z =Qz =z. i.e. z is a common fixed point of P, Q, S and T.
Uniqueness : For uniqueness ofz let if possible, we assume that z and w,(z 6=
w) are common fixed points of P, Q, S and T. By (ii), we have
d(z, w) = d(Sz, T w)
≤ φ(λ(z, w)) where
(z, w) = max{d(P z, Qw), d(P z, Sz), d(Qw, T w),1
2(d(P z, T w) +d(Qw, Sz))}
= max{d(z, w), d(z, z), d(w, w),1
2(d(z, w) +d(w, z))}
= d(z, w)
Thusd(z, w)≤φ(d(z, w))
Sincez 6=wthend(z, w)>0 and hence asφis contractive modulusφ(d(z, w))<
d(z, w).
Therefored(z, w)< d(z, w), which is a contradiction.
Thereforez =w.
Thusz is the unique common fixed point ofP, Q, S and T. Hence the theorem.
Corollary 3.2 Let (X,d) be a complete metric space. Suppose that the map- pings P, S and T are self-maps of X satisfying the following conditions:
1. T(X) ⊆ P(X) and S(X) ⊆ P(X);
2. d(Sx, T y)≤φ(λ(x, y))
where φ is an upper semi continuous, contractive modulus and
λ(x, y) =max{d(P x, P y), d(P x, Sx), d(P y, T y),12(d(P x, T y)+d(P y, Sx))}.
3. The pairs (S, P) and (T, P) are weakly compatible Then P, S and T have a unique common fixed point.
proof: By taking P =Q in theorem 3.1 we get the proof.
Corollary 3.3 Let (X,d) be a complete metric space. Suppose that the map- pings P and S are self-maps of X satisfying the following conditions:
1. S(X) ⊆ P(X);
2. d(Sx, Sy)≤φ(λ(x, y))
where φ is an upper semi continuous, contractive modulus and
λ(x, y) =max{d(P x, P y), d(P x, Sx), d(P y, Sy),12(d(P x, Sy)+d(P y, Sx))}.
3. The pair (S, P) is weakly compatible
Then P and S have a unique common fixed point.
proof: By taking P =Q and S =T in theorem 3.1 we get the proof.
ACKNOWLEDGEMENTS.The authors are grateful to the referees for careful readings and corrections. The authors are also grateful to Professor L.
N. Joshi and J. N. Chauhan for his cooperation and valuable suggestions in the preparation of this paper.
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