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Generalised Common Fixed Point Theorem for S - Compatible Mapping in Metric Space

R K Gujetiya, Dheeraj Kumari Mali*, Mala Hakwadiya

Pacific Academy of Higher Education and Research University Udaipur, Rajasthan, India

Abstract: In this paper we prove a common fixed point theorem of eight self mappings satisfying a generalized inequality using the concept of compatibility.

Keywords: common fixed point, compatible mapping, complete metric space

Received June 23, 2014; Revised July 19, 2014; Accepted August 01, 2014

Citation: R K Gujetiya, Dheeraj Kumari Mali, Mala Hakwadiya, Generalised Common Fixed Point Theorem for S - Compatible Mapping in Metric Space. American Journal of Applied Mathematics and Statistics, 2 (4). 220-223. January 2014. doi:10.12691/ajams-2-4-8

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 A-compatible and S-compatible by splitting the definition of compatible mappings of type (A). Fixed point results of compatible mappings are found in [1-8]. Shahidur Rahman [9] proved a common fixed point theorem for A-Compatible and S-Compatible 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) (Non-degenerated) 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 S-compatible 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 S-compatible 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 S-compatible 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 S-compatible 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

[1]   | Bijendra Singh and M.S. Chauhan, “On common fixed points of four mappings”, Bull. Cal. Math. Soc., 88, 451-456, 1996.
[2]   | H.K. Pathak and M.S. Khan, “A comparison of various types of compatible maps and common fixed points”, Indian J. pure appl. Math., 28(4), 477-485, April 1997.
[3]   | H.K. Pathak, M.S. Khan and Reny George, “Compatible mappings of type (A-1) and type (A-2) and common fixed points in fuzzy metric spaces”, International Math. Forum, 2(11), 515-524, 2007.
[4]   | Jungck G., Compatible maps and common fixed points, Inter. J. Math. and Math. Sci. 9, 1986 771-779.
[5]   | Jungck G., Murthy P.P. and Cho Y.J., “Compatible mappings of type (A) and common fixed points”, Math. Japonica 38, 381- 390, 1993.
[6]   | Koireng M., Leenthoi N. and Rohen Y., “Common fixed points of compatible mappings of type (R)”, General Mathematics Notes, 10(1), 58-62, May 2012.
[7]   | P. P. Murthy, Remarks on fixed point theorem of Sharma and Sahu, Bull. of Pure and Appl. Sc., 12 E(1-2), 7-10, 1993.
[8]   | Sharma B. K. and Sahu N. K., “Common fixed point of three continuous mappings”, The Math. Student, 59 (1), 77-80, 1991.
[9]   | Shahidur Rahman, Yumnam Rohen, M. Popeshwar Singh, “Generalised common fixed point theorem of A-Compatible and S-Compatible mappings”. American J. Appl. Math. And Statistics, 1(2), 27-29, April 2013.