**INTRODUCTION**

The concept of C-contraction was defined by Chatterjea
(1972) as follows.

**Definition 1:** A mapping T:X→X where (X, d) is a metric space is said to be a C-contraction if there exists αε(0,1/2) such that for all x, yεX the following inequality holds:

Chatterjea (1972) has proved that, if (X, d) is a complete
metric space, then every C-contraction on X has a unique fixed point. Choudhury
(2009) introduced a generalization of C-contraction by the following definition.

**Definition 2:** A mapping T:X→X, where (X, d) is a metric space is said to be a weakly C-contractive mapping if for all x, yεX:

where, φ:[0,∞)^{2}→[0,∞) is a continuous function such that φ(x, y) = 0 if and only if x = y = 0.

Choudhury (2009) has proved that, if (X, d) is a complete
metric space, then every weak C-contraction on X has a unique fixed point.

For a survey of fixed point theory and related results we refer to Mujahid
and Dragan (2010), Zhang and Song (2009), Moradi
*et al*. (2011), Doric (2009), Nashine
and Samet (2011), Mohamadi *et al*. (2009),
Okoroafor and Osu (2006), Olaleru
(2006) and Tiwari *et al*. (2012).

Let us note that the beautiful theory of fixed point is used frequently in
other branches of mathematics and engineering sciences (Shakeri
*et al*., 2009).

The purpose of this study is to obtain a common fixed point theorem for four
maps satisfying a certain contractive condition. Our result generalized the
results of Chatterjea (1972) and Choudhury
(2009).

Throughout this paper, let:

Ω = {φ|φ:[0,∞)^{2}→{0,∞) is a continuous function such that φ(x,y) = 0 iff x = y = 0}.

**Definition 3:** (a) Let (X, d) be a metric space and T,S:X→X. If
w = Tx = Sx, for some x∈X, then x is called a coincidence point of T and
S and w is called a point of coincidence of T and S, (b) Let T and S be two
self-mappings of a metric space (X, d). T and S are said to be weakly compatible
if for all x∈X the equality Tx = Sx implies TSx = STx (Beg
and Abbas, 2006).

**MAIN RESULTS**

**Definition 1:** Two mappings T,S:X→X, where (X,d) is a metric space are called weakly C_{f,g}-contractive (or weak C_{f,g}-contraction) if for all x, y∈X,

where, φεΩ.

Following is the main result of this study.

**Theorem 1:** Let (X, d) be a complete metric space and let E be a nonempty
closed subset of X. Let T,S:X→X be two weakly C_{f,g}-contractive
mappings (condition 1):

II |
The pairs (S, f) and (T, g) be weakly compatible. |

Assume that f and g also are continuous functions on X. In addition, for all
x∈X:

and for all x, y∈X

then, T, f, S and g have a unique common fixed point.

**Proof:** Let x_{0}∈E be arbitrary. Using (I), there exist
tow sequences
such that y_{0} = Tx_{0} = gx_{1}, y_{1} = Sx_{1}
= fx_{2}, y_{2} = Tx_{2} = gx_{3},..., y_{2n}
= Tx_{2n} = gx_{1n+1}, y_{2n+1} = Sx_{2n+1}
= fx_{2n+2},….

We complete the proof in two steps:

**Step 1**: {y_{n}} is Cauchy.

Consider two cases as follows:

• |
If for some n, y_{n} = y_{n+1}, then y_{n+1}
= y_{n+2}. If not, then y_{n+1}≠y_{n+2}. Let
n = 2k Therefore, using condition (1), we have: |

which is a contradiction. Hence, we must have y_{n+1} = y_{n+2},
when, n is even. In a same way we can show that this equality holds, when n
is odd. Therefore, in any case, if for an n, y_{n} = y_{n+1},
we always obtain y_{n} = y_{n+2}. Repeating the above process
inductively, we obtain that y_{n} = y_{n+k} for all k≥1 Therefore,
in this case {y_{n}} is a constant sequence and hence is a Cauchy one.

• |
If y_{n}≠y_{n+1}, for every positive integer
n, then for n = 2k, using condition (1), we obtain that: |

Hence,

If, n = 2k+1, similarly we can prove that:

That is:

Therefor, in general, d(y_{n+1}, y_{n}) is a decreasing sequence of nonnegative real numbers and bounded from below and hence it is convergent.

Assume that:

From the above argument,

and if k→∞, we have:

Therefore:

We have proved that:

Now, if k→∞ and using the continuity of φ we obtain

and consequently, φ(2r, 0) = 0. This gives us that,

by our assumption about φ.

Now, it is sufficient to show that the subsequence {y_{2n}} is a Cauchy sequence. Suppose opposite, that is {y_{2n}} is not a Cauchy sequence. Then there exists ε >0 for which we can find subsequences y_{2m(k)} and y_{2n(k)} of y_{2n} such that n(k) is smallest index for which n(k)>m(k) and:

This means that:

From (5) and triangle inequality:

Letting k→∞ and using Eq. 4 we can conclude that:

Moreover, we have:

and

and

Using Eq. 6, 7, 8, 9
and 10, we get:

Using Eq. 1 we have:

Making k→∞ in the above inequality and taking into account (10) and by the continuity of φ, we have:

and hence, φ(ε, ε) = 0. By our assumption about φ, we have ε = 0 which is a contradiction.

**Step 2:** Existence of coincidence point and common fixed point.

Since, (X, d) is complete and {y_{n}} is Cauchy, there exists z∈X such that lim_{n→∞} y_{n} = z. Since, E is closed and {y_{n}}⊆E, we have z∈E.

Also, we know that

Since, f and g are continuous,

On the other hand, from 2 and 3 we conclude that:

Therefore, from 4 and 11:

Also, using 2 we have,

Therefore, from 4 and 11:

From Eq. 1:

If in the above inequality, n→∞, from 11 and 13 we have:

So:

and hence, Sz = fz. We can analogously prove that Tz = gz.

Also:

consequently: fz = gz, therefore Tz = gz = fz = sz = t

Now we show that z is a common fixed point.

Using weak compatibility of the pair (T, f) and (S,g) we have Tt = ft and gt = St. So,

That is, φ(d(T, St), d(t, Tt)) =0 and this implies that Tt = t. Therefore ft = Tt = t.

Analogously,

That is, φ(d(t, St), d(St, t)) =0 and this implies that St = t. Therefore, gt = St = t.

Hence, gt = St = t = ft = Tt.

It is easy to show that t is unique.

**Example 1:** Let X = R(The set of all real numbers) be endowed with the Euclidean metric. Suppose that T:X→X is defined by:

and Sx = 0 for all x∈R.

We define functions f, g: X→X by:

and:

and function φ:[0,∞)^{2}→[0,∞)by φ(t,s) = t+s/8.

One can easily obtains that for all x∈X,

and for all x,y∈X,

Now, we have the following four cases:

• |
x,y∈X(-∞, 0). Then we have |

• |
x∈(-∞, 0) and y∈[0, ∞). Then we have |

• |
x,y∈[0, ∞) . Then we have |

• |
x∈[0, ∞) and y∈(-∞, 0). Then we have: |

So mappings T and S satisfy relation (1) and all conditions of Theorem 1 are hold and T, S, f and g have a unique common fixed point (x = 0).

Taking f = g in Theorem 1, we obtain the following.

**Corollary 1:** Let (X, d) be a complete metric space and let E be a nonempty closed subset of X. Let T, S are such that for all x,y∈X:

where, T, S and f be such that:

• |
TE⊆fE and SE⊆fE. |

• |
The pairs (T, f) and (T, g) be weakly compatible. |

Assume that f is a continuous function on X. In addition, for all x∈X:

and for all x,y∈X:

then, T, f and S have a unique common fixed point.

Taking T = S in Theorem 1, we have the following result.

**Corollary 2:** Let (X, d) be a complete metric space and let E be a nonempty closed subset of X. Let T:X→X be such that for all x,y∈X:

where, T, f and g be such that:

• |
TE⊆fE and SE⊆gE. |

• |
The pairs (T,f) and (T,g) be weakly compatible. |

Assume that f and g also are continuous functions on X. In addition, for all x∈X:

and for all x,y∈X:

then, T, f and g have a unique common fixed point.

Taking T = S and f = g in Theorem 2.4, the following result is obtained.

**Corollary 3:** Let (X,d) be a complete metric space and let E be a nonempty closed subset of X. Let T:X→X be such that for all x,y∈X:

where, T and f be such that:

• |
TE⊆ fE. |

• |
The pair (T, f) be weakly compatible |

Assume that T also is continuous on X. In addition, for all x∈X:

and for all x,y∈X,

Then, T and f have a unique common fixed point.

**Remark 1:** Taking T = S and f = g = l_{x} (the identity mapping
on X) and X = E in Theorem 1, we obtain the result of Choudhury
(2009) which has been mentioned above.

**APPLICATIONS**

In this part, from previous obtained results, we will deduce some common fixed point results for mappings satisfying a contraction condition of integral type in a complete metric space.

Branciari (2002) obtained a fixed point result for
a single mapping satisfying an integral type inequality. Afterwards, Altun
*et al*. (2007) established a fixed point theorem for weakly compatible
mappings satisfying a general contractive inequality of integral type.

Similar to Nashine and Samet (2011), we denote by
the set of all functions φ:[0,+∞)→[0,+∞) satisfying the
following conditions:

• |
φ is a Lebesgue integrable mapping on each compact subset
of [0,+∞) |

• |
For all ε>0, we have: |

**Corollary 4:** Let T and S satisfy the conditions of Theorem 1, except that condition (1) be replaced by the following:

There exists a ø∈
such that:

Then, T, S, f and g have a unique common fixed point.

**Proof:** Consider the function .
Then Eq. 23 changes to the following:

and putting Ψ = Φφ and applying Theorem 1, we obtain the proof (it is easy to verify that Ψ∈Ω).

**Corollary 5:** If in the above corollary, Eq. 23 be replaced by the following:

then the result of corollary 4 is also hold.

**Proof:** Assume that:

Then the above condition will be the following:

Taking,

and applying Theorem 1, we obtain the proof (it is obvious that Ψ∈Ω).

As in Nashine and Samet (2011), let N∈N be fixed.
Let {φ_{i}}_{1≤i≤N} be a family of N functions which
belong to .
For all t≥0, we define:

We have the following result.

**Corollary 6:** Let T and S satisfy the conditions of Theorem 1 and condition (1) be substituted by the following:

There exists a φ∈
such that:

Then, T, S, f and g have a unique common fixed point.

**Proof:** Consider the function Ψ(x,y) = I_{N}φ(x,y). Then the inequality 24 will be:

applying Theorem 1, we obtain the desired result (it is easy to verify that Ψ∈Ω).

**Corollary 7:** Let T and S satisfy the conditions of Theorem 1, except that condition (1) be replaced by the following:

There exists a φ∈
such that:

Then, T, S, f and g have a unique common fixed point.

**Proof:** Let Ψ(x,y) = φ(I_{N}(x),I_{N}(y)). Then the above inequality will be changed to:

Using Theorem 1, we obtain the proof (it is easy to show that Ψ∈Ω).