1. Introduction

The first important result in the theory of fixed point of compatible mappings was obtained by Gerald Jugck in 1986 ^[2] as a generalization of commuting mappings. Pathak, Chang and Cho ^[3] in 1994 introduced the concept of compatible mappings of type(P). In 2004 Rohen, Singh and shambhu ^[5] introduced the concept of compatible mappings of type(R) by combining the definitions of compatible mappings and compatible mappings of type(P). The aim of this paper is to prove a common fixed point theorem of compatible mappings of type(R) in metric space by considering eight self mappings.

2. Preliminaries

Definition 2.1: ^[2] A metric space is given by a set X and a distance function such that

(i) (Positivity) For all

(ii) (Non-degenerated) For all

(iii) (Symmetry) For all

(iv) (Triangle inequality) For all

Definition 2.2: ^[4] Let S and T be mappings from a complete metric space X into itself. The mappings S and T are said to be compatible if whenever is a sequence in X such that for some

Definition 2.3: ^[4] Let S and T be mappings from a complete metric space X into itself. The mappings S and T are said to be compatible of type (P) if whenever is a sequence in X such that for for some

Definition 2.4: ^[4] Let S and T be mappings from a complete metric space X into itself. The mappings S and T are said to be compatible of type (R) if and whenever is a sequence in X such that for for some

Proposition 2.5. ^[4] Let S and T be mappings from a complete metric space (X, d) into itself. If a pair {S, T} is compatible of type (R) on X and Sz = Tz for z ∈ X,Then STz = TSz = SSz = TTz.

Proposition 2.6. ^[4] Let S and T be mappings from a complete metric space (X, d) into itself. If a pair {S, T} is compatible of type (R) on X and for some then we have

(i) as n → ∞ if S is continuous,

(ii) as n → ∞ if T is continuous and

(iii) STz = TSz and Sz = Tz if S and T are continuous at z.

Lemma 2.7. ^[4] Let A, B, S and T be mapping from a metric space(X, d) into itself satisfying the following conditions:

(1)

(2)

(3) Let then by (1) there exists such that and for there exists such that and so on. Continuing this process we can define a sequence in X such that

Then the sequence is Cauchy sequence in X.

Theorem: ^[4] Let A, B, S and T be mapping from a metric space (X, d) into itself satisfying the following conditions:

(1)

(2)

(3) Let then by (1) there exists such that and for there exists such that and so on. Continuing this process we can define a sequence in X such that

Then the sequence is Cauchy sequence in X.

(4) One of A, B, S or T is continuous.

(5) [A, S] and [B, T] are compatible of type (R) on X.

Then A, B, S and T have a unique common fixed point in X.

3. Main Result

Lemma 3.1: Let C, D, E, F, K, M, P and V be self maps of a complete metric space (X, d) satisfying the following conditions:

(1)

(2)

Where

(3) Let then by (1) there exists such that and for there exists such that and so on.continuing this process we candefine a sequence in X such that

Then the sequence is a Cauchy sequence in X.

Proof: By condition (2) and (3), we have

Hence is Cauchy sequence.

Theorem 3.2: Let C, D, E, F, K, M, P and V be self maps of a complete metric space (X, d) satisfying the following conditions:

(1)

(2)

Where

(3) Let then by (1) there exists such that and for there exists such that and so on.continuing this process we candefine a sequence in X such that

Then the sequence is a Cauchy sequence in X.

(4) One of C, E, FKM, DPV is continuous.

(5) [C, FKM] and [E, DPV] are compatible of type (R) on X.

Then C, D, E, F, K, M, P and V have a unique common fixed point in X.

Proof: By lemma 3.1, is Cauchy sequence. and since X is complete so there exists a point such that as

Consequently subsequences and converges to z. Let FKM be continuous. Since C and FKM are compatible of type (R) on X. Then by proposition 2.6, We have and as

Now by condition (2), we have

Letting , we have

Which is a contradiction. Hence

(3.1)

Now by putting x = z and in condition (2), then we have

(3.2)

Letting , we have

Which is a contradiction. Hence

(3.3)

Now since , by condition (1) Also DPV is self map of X, so there exists a point such that

(3.4)

Moreover by putting Cz = z and in condition (3.2), we obtain

Which is a contradiction.

Hence Eu = z, i.e., z = DPVu = Eu.

By condition (5), we have

Hence d(DPVz, Ez) = 0 i.e., DPVz = Ez.

Now

Which is a contradiction.

(3.5)

Now to prove Vz = z, put x = z and y = Vz in (1) and using (3.1), (3.3) and (3.5), we have

Which is a contradiction.

Hence z = Vz. Since DPVz = z, implies that DPz = z.

Now to prove Pz = z, put x = z and y = Pz in (1) and using (3.1), (3.3) and (3.5), we have

Which is a contradiction.

Hence Pz = z. Since DPz = z, implies that Dz = z.

Now to prove Mz = z, put x = Mz and y = z in (1) and using (3.1), (3.3) and (3.5), we have

Which is a contradiction.

Hence Mz = z. Since FKMz = z, implies that FKz = z.

Now to prove Kz = z, put x = Kz and y = z in (1) and using (3.1), (3.3) and (3.5), we have

Which is a contradiction.

Hence Kz = z. Since FKz =z, implies that Fz = z.Thus Cz = Dz = Ez = Fz = Kz = Mz = Pz = Vz = z. Therefore z is common fixed point of C, D, E, F, K, M, P and V. Similarly we can prove this any one of C, D, E, F, P and V is continuous.

4. Uniqueness

Suppose w be another common fixed point of C, D, E, F, K. M, P and V. Then we have

Which is a contradiction.

Hence z = w. Therefore z is a unique common fixed point of C, D, E, F, K, M, P and V.

Corollary: Let C, D, E, K, M and V be self maps of a complete metric space (X, d) satisfying the following conditions:

(1)

(2)

Where

(3) Let then by (1) there exists such that and for there exists such that and so on.continuing this process we candefine a sequence in X such that

Then the sequence is a Cauchy sequence in X.

(4) One of C, E, KM, DV is continuous.

(5) [C, KM] and [E, DV] are compatible of type (R) on X.

Then C, D, E, K, M and V have a unique common fixed point in X.

5. Conclusion

In this paper, we have presented common fixed point theorem for eight 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]	Singh, B. and Chauhan, M.S., “On common fixed points of four mappings”, Bull. Cal. Math. Soc., 88, 451-456, 1996.
	In article
[2]	Jungck, G. “Compatible maps and common fixed points”, Inter. J. Math. And Math. Sci., 9, 771-779, 1986.
	In article		CrossRef
[3]	Pathak, H.K., Chang, S.S. and Cho, Y.J., “Fixed point theorem for compatible mappings of type (P)”, Indian J. Math. 36(2), 151-166, 1994.
	In article
[4]	Koireng Meitei, M., Ningombam, L. and Rohen, Y. “common fixed point theorem of compatible mappings of type(R)”, Gen.Math. Notes, 10(1), 58-62, May 2012.
	In article
[5]	Rohen, Y., Singh, M.R. and Shambhu, L., “Common fixed points of compatible mapping of type (C) in Banach Spaces”, Proc. of Math.Soc., 11(2), 42-50, 2011.
	In article