Journal of Applied Sciences

Year: 2012 | Volume: 12 | Issue: 7 | Page No.: 661-667
DOI: 10.3923/jas.2012.661.667

Some Common Fixed Point Theorems for Weakly C_f,g-contractive Mappings in Complete Metric Spaces

V. Parvaneh and H. Hosseinzadeh

Abstract: The aim of this paper is to present some common fixed point theorems for C_f,g-contractions in a complete metric space. Finally, some results for contractions of integral type are given.

Keywords: Common fixed point, complete metric space and weak C-contraction

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):

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 x₀∈E be arbitrary. Using (I), there exist tow sequences such that y₀ = Tx₀ = gx₁, y₁ = Sx₁ = fx₂, y₂ = Tx₂ = gx₃,..., 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:

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.

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:

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:

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:

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:

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:

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_Nφ(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).

REFERENCES