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) (Nondegenerated) 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
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