A Common Fixed Point Theorem for Compatible Mappings of Type (C)

The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity during the last two decades. Researchers like R. P. Pant et al. [2,3] have shown that how the three types of contractive conditions (Banach, Meir keeler and contractive gauge function/φ contractive condition) hold simultaneously or independent of each other and as a result of this study they have proved a fixed point theorem using Lipschitz type contractive condition [3] and gauge function [2]. In this paper we generalize the result of K. Jha, R. P. Pant, S. L. Singh [1] and prove a fixed point theorem for six self mappings in a complete metric space.     , , , 0 d Ax By c x y c   1   (1.1)


Introduction
The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity during the last two decades.Researchers like R. P. Pant et al. [2,3] have shown that how the three types of contractive conditions (Banach, Meir keeler and contractive gauge function/φ contractive condition) hold simultaneously or independent of each other and as a result of this study they have proved a fixed point theorem using Lipschitz type contractive condition [3] and gauge function [2].
In this paper we generalize the result of K. Jha, R. P. Pant, S. L. Singh [1] and prove a fixed point theorem for six self mappings in a complete metric space. where,  or a Meir-Keeler type  ,   -contractive condition of the form, given ε > 0, there exists a δ > 0 such that instead of assuming one of the contractive conditions (1.2) or (1.3) with additional conditions on δ and φ.Definition: Two self mappings A and S of a metric space (X, d) are said to be compatible (see Jungck [4] Definition: Two self mappings A and S of a metric space (X, d) are said to be compatible mappings of type (A) (See [5] Definition: Two self mappings A and S of a metric space (X,d) are said to be compatible mappings of type (C) (see [7]) if, Definition: Two self mappings A and S of a metric space (X,d) are said to be compatible mappings of type (P) (see [8]), if whenever . From the propositions given in [4-8] all compatibility conditions are equivalent when A and S are continuous.We observe that they are independent if the functions are discontinuous.

t X 
We give an example which is compatible mapping of type (C) but is neither compatible nor compatible mapping of type (A), compatible mapping of type (B) and compatible mapping of type (P).
Define self maps S and A of X by If one of the mappings A, B, S and T is continuous then A, B, S and T have a unique common fixed point.
We generalise this theorem by extending four self maps to six self maps and replacing the condition of compatibility of self maps by the compatible mapping of type (C).
To prove our theorem we shall use the following lemma.

Lemma
Let A, B, S, T, L and M be self mappings of (X,d) such that (2.2.1) Assume further that given ε > 0 there exists a δ > 0 such that for all ,

 M x y d ABx STy d Lx ABx d My STy d ABx My d Lx STy
for 1, 2, 3 n   Then we have the following for every 0 where p and q are of opposite parity.
(2.2.7) Proof: Since from (2.2.2) for every 0 , 2 Putting p = 2n and in the above inequality, we have Thus the sequence d y y  is non increasing and converges to the greatest lower bound of its range .Now we prove that t = 0  , which contradicts the infrimum nature of t.
Therefore, we have .
In virtue of (2.2.6), it is sufficient to show that   2n y is a cauchy sequence.
Suppose that   2n y is not a cauchy sequence.Then there is an 0   such that for each integer 2k, there exists even integers 2m(k) and 2n(k For each even integer 2k, let 2m(k) be the least even integer exceeding 2n(k) satisfying (2.2.9), that is Then for each even integer 2k, we have From (2.2.6) and (2.2.10), it follows that From the triangle inequality, we have (2.2.13) (Since by (2.2.5) and , , From (2.2.5), (2.2.6) and (2.2.12) as , we get cauchy sequence in X and so is   n y .

Main Theorem
Let A, B, S, T, L and M be self mappings of a complete metric space (X, d) satisfying (2.3.1) The pair (L, AB) and (M, ST) be compatible mappings of type (C) (2.3.5)AB(X) is complete one of the mappings AB,ST,L and M is continuous.(2.3.6) Then AB, ST, L and M have a unique common fixed point.
Further if the pairs (A, B), (A, L), (B, L), (S, T), (S, M) and (T, M) are commuting mappings then A, B, S, T, L and M have a unique common fixed point.Proof: Let 0 x be any point in X. Define sequences n x and in X given by the rule n (2.3.7) 1 and This can be done by virtue of (2.3.2). since the contractive condition (2.3.3) of the theorem implies the contractive condition (2.2.2) and (2.2.3) of the lemma 2.2.1 so by using the lemma 2.2.1 we conclude that {y n } is a Cauchy sequence in X, but by (2.3.6)AB(X) is complete, it converges to a point z = ABu for some u in X.
In view of (2.3.8),(2.3.4) and (2.3.11) (2.3.12) Now,we show that z is a fixed point of  ST.


We prove that .STu Mu  Now, In view of (2.3.8) and (2.3.4) We prove z Lv  , from (2.3.4) we have If the mappings M or ST is continuous instead of L or AB then the proof that z is a Common fixed point of L, M, AB, and ST is similar.
Uniqueness: Let w be another common fixed point of L, M, AB, and ST then Lw = Mw = ABw = STw = w.
From (2.3.4) we have as .

X
The pairs (L, AB) and (M, ST) are compatible mappings of type (c) and also satisfies the conditions (2.3.2),(2.3.3),(2.3.4),(2.3.5) and (2. 1) compatible mappings of type (A) or For this let z is the unique common fixed point of (AB, L) and (ST, M).
2) compatible mappings of type (B) or 3) compatible mappings of type (P) Since (A, B), (A, L), (B, L) are commutative       ; which shows that Az, Bz are common fixed points of (AB, L) yielding there by Az Z Bz Lz ABz     in the view of uniqueness of common fixed point of the pairs (AB, L) .Similarly using the, commutativity of (S,T), (S,M) and (T,M) it can be shown that Sz = z = Tz = Mz = STz.From (2.3.4) we have My d Lz STy k d STy STy d z z d My STy k d STy My d z z

d
Lv z d z z k d z z d Lv z Main theorem remains true if we replace condition compatible mappings of type (C) byFinally we need to show that z is a common fixed point of L, M, A, B, S and T.

[ 3 ]
R. P. Pant, "A Common Fixed Point Theorem for Two Pairs of Maps Satisfying the Condition (E.A)," Journal of Physical Sciences, Vol. 16, No. 12, 2002, pp.77-84.Now, we need to show that Az = Sz (Bz = Tz) also remains a common fixed point of both the pairs (AB, L) and (ST, M).

d
Az Sz d Lz Mz k d ABz STz d Lz ABz k d Mz STz d ABz Mz d Lz STz k d z z d z z d z z k d z z d z z

. Jha, R. P. Pant and S. L. Singh [1] Proved the Following Common Fixed Point. 2.1. Theorem
Now, we show that z is the fixed point of  AB.
Pant and S. L. Singh, "On the Existence of Common Fixed Point for Compatible Mappings," Journal of Mathematics, Vol.37, 2005, pp.39-48.