Keywords: common fixed point, compatible mapping, complete metric space
American Journal of Applied Mathematics and Statistics, 2014 2 (4),
pp 220223.
DOI: 10.12691/ajams248
Received June 23, 2014; Revised July 19, 2014; Accepted August 01, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
A metric space is just a set X equipped with a function d of two variables which measures the distance between points: d(x, y) is the distance between two points x and y in X. It turns out that if we put mild and natural conditions on the function d, we can develop a general notion of distance that covers distances between number, vectors, sequences, functions, sets and much more. The first important result in the theory of fixed point of compatible mappings was obtained by Gerald Jungck in 1986 ^{[4]} as a generalization of commuting mappings. In 1993 Jungck, Murthy and Cho ^{[5]} introduced the concept of compatible mappings of type (A) by generalizing the definition of weakly uniformly contraction maps. Pathak and Khan ^{[2]} introduced the concept of Acompatible and Scompatible by splitting the definition of compatible mappings of type (A). Fixed point results of compatible mappings are found in [18]^{[1]}. Shahidur Rahman ^{[9]} proved a common fixed point theorem for ACompatible and SCompatible mapping. We will generalized the result of Shahidur Rahman ^{[9]}.
2. Preliminaries
Definition 2.1: A metric space is given by a set X and a distance function such that
(i) (Positivity) For all
(ii) (Nondegenerated) For all implais that x = y.
(iii) (Symmetry) For all
(iv) (Triangle inequality) For all
Definition 2.2: ^{[4]} Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be compatible if whenever is a sequence in X such that for some
Definition 2.3: ^{[5]} Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be compatible of type (A) if and whenever is a sequence in X such that for for some
Definition 2.4: ^{[7]} Let A and S be mappings from a complete metric space X into itself. The mappings A and S are said to be Scompatible if whenever is a sequence in X such that for for some
Proposition 2.5: ^{[2]} Let A and S be mappings from a complete metric space (X, d) into itself. If a pair (A, S) is Scompatible on X and for , then if S is continuous at t.
3. Main Result
Lemma: Let A, B, S, T, D and V be self maps of a complete metric space
satisfying the following conditions:
(1) and
(2)
Where, and
(3) Let then by (1) there exists such that and for there exists such that and so on. Continuing this process we define a sequence in X such that
and Then the sequence is Cauchy sequence in
By condition (2) and (3),we have
Where
Since and
In order to satisfy the inequation, one value of λ will be positive and the other will be negative. We also note that the sum and product of the two values of λ is less than 1 and 1 respectively. Neglecting the negative value, we have < p where 0 < p < 1.
Hence is Cauchy sequence.
Now we prove the following theorem.
Theorem 3.1: Let A, B, S, T, D and V be self maps of a complete metric space satisfying the following conditions:
(1) and
(2)
Where, and
(3) one of AD, BV, S or T is continuous.
(4) [AD,S] and [BV,T] are Scompatible mapping on X.
(5) Let then by (1) there exists such that and for there exists such that and so on. Continuing this process we define a sequence in X such that
Then the sequence is Cauchy sequence in X.
Then A, D, B, V, S and T have a unique common fixed point.
Proof: By lemma, we have
is Cauchy sequence. Since X is complete, there exists a point such that as Consequently subsequence and converges to z.
Let S be a continuous mapping, since AD and S are Scompatible mapping on Then by proposition 2.5 we have and as
Now by condition (2) of lemma, we have
Letting , we have
Which is a contradiction.
Hence Now by (2), we have
Letting , we have
Hence
Now since by condition (1), Also T is self map of X so there exists a point such that Moreover by condition (2) we obtain,
Hence i.e.,
By condition (4), we have
Hence i.e.,
Now by condition (2), we have
Which is a contradiction.
Hence i.e.,
Now putting x = Dz and in (2),then we have
Letting n→∞,we have
Which is a contradiction.
Hence . Since which implies that
Now putting and in (2), then we have
Letting n→∞,we have
Which is a contradiction.
Hence . Since BVz = z which implies that Bz = z.
Therefore z is a common fixed point of A, D, B, V, S and T.
Uniqueness: Finally, to prove the uniqueness of z, suppose w be another common fixed point of A, D, B, V, S and T. Then we have,
Which is a contradiction. Hence
Thus z is a unique common fixed point of A, D, B, V, S and T.
4. Conclusion
In this paper, we have presented common fixed point theorem for six self mappings in metric spaces through concept of compatibility.
Acknowledgement
The Authors are thankful to the anonymous referees for his valuable suggestions for the improvement of this paper.
References
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