Fixed Point Theory and Applications

, 2010:983802 | Cite as

Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces

Open Access
Research Article

Abstract

We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces.

Keywords

Banach Space Point Theorem Closed Subset Fixed Point Theorem Fixed Function 

1. Introduction and Preliminaries

Recently, Huang and Zhang [1] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [2, 3, 4, 5, 6, 7, 8]. The aim of this paper is to prove a common fixed point theorem for four non-self-mappings on cone metric spaces in which the cone need not be normal. This result generalizes the result of Radenović and Rhoades [5].

Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.

Let Open image in new window be a real Banach space. A subset Open image in new window of Open image in new window is called a cone if and only if

(a) Open image in new window is closed, nonempty and Open image in new window ;

(b) Open image in new window , Open image in new window , Open image in new window implies Open image in new window ;

(c) Open image in new window .

Given a cone Open image in new window , we define a partial ordering Open image in new window with respect to Open image in new window by Open image in new window if and only if Open image in new window . A cone Open image in new window is called normal if there is a number Open image in new window such that for all Open image in new window ,

The least positive number satisfying the above inequality is called the normal constant of Open image in new window , while Open image in new window stands for Open image in new window (interior of Open image in new window ).

Definition 1.1 (see [1]).

Let Open image in new window be a nonempty set. Suppose that the mapping Open image in new window satisfies

(d1) Open image in new window for all Open image in new window and Open image in new window if and only if Open image in new window ;

(d2) Open image in new window for all Open image in new window ;

(d3) Open image in new window for all Open image in new window .

Then Open image in new window is called a cone metric on Open image in new window , and Open image in new window is called a cone metric space.

The concept of a cone metric space is more general than that of a metric space.

Definition 1.2 (see [1]).

Let Open image in new window be a cone metric space. One says that Open image in new window is
  1. (e)
     
  2. (f)

    a Convergent sequence if for every Open image in new window with Open image in new window , there is an Open image in new window such that for all Open image in new window , Open image in new window for some fixed Open image in new window .

     

A cone metric space Open image in new window is said to be complete if every Cauchy sequence in Open image in new window is convergent in Open image in new window . It is known that Open image in new window converges to Open image in new window if and only if Open image in new window as Open image in new window . It is a Cauchy sequence if and only if Open image in new window .

Remark 1.3 (see [9]).

Let Open image in new window be an ordered Banach (normed) space. Then Open image in new window is an interior point of Open image in new window if and only if Open image in new window is a neighborhood of Open image in new window .

Remark 1.5 (see [5, 11]).

If Open image in new window , Open image in new window , and Open image in new window , then there exists an Open image in new window such that for all Open image in new window we have Open image in new window .

Remark 1.6 (see [6, 10]).

If Open image in new window is a real Banach space with cone Open image in new window and if Open image in new window where Open image in new window and Open image in new window , then Open image in new window .

We find it convenient to introduce the following definition.

Definition 1.7 (see [5]).

Let Open image in new window be a complete cone metric space and Open image in new window a nonempty closed subset of Open image in new window , and Open image in new window satisfying

for all Open image in new window , Open image in new window , Open image in new window , then Open image in new window is called a generalized Open image in new window -contractive mapping of Open image in new window into Open image in new window .

Definition 1.8 (see [2]).

Let Open image in new window and Open image in new window be self-maps on a set Open image in new window (i.e., Open image in new window ). If Open image in new window for some Open image in new window in Open image in new window , then Open image in new window is called a coincidence point of Open image in new window and Open image in new window , and Open image in new window is called a point of coincidence of Open image in new window and Open image in new window . Self-maps Open image in new window and Open image in new window are said to be weakly compatible if they commute at their coincidence point; that is, if Open image in new window for some Open image in new window , then Open image in new window .

2. Main Result

The following theorem is Radenović and Rhoades [5] generalization of Imdad and Kumar's [12] result in cone metric spaces.

Theorem 2.1.

Let Open image in new window be a complete cone metric space and Open image in new window a nonempty closed subset of Open image in new window such that for each Open image in new window and Open image in new window there exists a point Open image in new window (the boundary of Open image in new window ) such that

Suppose that Open image in new window are such that Open image in new window is a generalized Open image in new window -contractive mapping of Open image in new window into Open image in new window , and

(i) Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window is closed in Open image in new window .

Then the pair Open image in new window has a coincidence point. Moreover, if pair Open image in new window is weakly compatible, then Open image in new window and Open image in new window have a unique common fixed point.

The purpose of this paper is to extend the above theorem for four non-self-mappings in cone metric spaces. We begin with the following definition.

Definition 2.2.

Let Open image in new window be a complete cone metric space and Open image in new window a nonempty closed subset of Open image in new window , and Open image in new window satisfying

for all Open image in new window , Open image in new window , Open image in new window , then Open image in new window is called a generalized Open image in new window -contractive mappings pair of Open image in new window into Open image in new window .

Notice that by setting Open image in new window and Open image in new window in (2.2), one deduces the slightly generalized form of (1.3).

We state and prove our main result as follows.

Theorem 2.3.

Let Open image in new window be a complete cone metric space and Open image in new window a nonempty closed subset of Open image in new window such that for each Open image in new window and Open image in new window there exists a point Open image in new window (the boundary of Open image in new window ) such that

Suppose that Open image in new window are such that Open image in new window is a generalized Open image in new window -contractive mappings pair of Open image in new window into Open image in new window , and

(I) Open image in new window ,

(II) Open image in new window ,

(III) Open image in new window and Open image in new window (or Open image in new window and Open image in new window ) are closed in Open image in new window .Then

(IV) Open image in new window has a point of coincidence,

(V) Open image in new window has a point of coincidence.

Moreover, if Open image in new window and Open image in new window are weakly compatible pairs, then Open image in new window , Open image in new window , Open image in new window , and Open image in new window have a unique common fixed point.

Proof.

Firstly, we proceed to construct two sequences Open image in new window and Open image in new window in the following way.

Let Open image in new window be arbitrary. Then (due to Open image in new window ) there exists a point Open image in new window such that Open image in new window . Since Open image in new window , one concludes that Open image in new window . Thus, there exists Open image in new window such that Open image in new window . Since Open image in new window there exists a point Open image in new window such that
Suppose that Open image in new window . Then Open image in new window which implies that there exists a point Open image in new window such that Open image in new window . Otherwise, if Open image in new window , then there exists a point Open image in new window such that

Let Open image in new window be such that Open image in new window . Thus, repeating the foregoing arguments, one obtains two sequences Open image in new window and Open image in new window such that

(a) Open image in new window , Open image in new window ,

We denote that

Note that Open image in new window , as if Open image in new window , then Open image in new window , and one infers that Open image in new window which implies that Open image in new window . Hence Open image in new window . Similarly, one can argue that Open image in new window .

Now, we distinguish the following three cases.

Case 1.

Clearly, there are infinite many Open image in new window such that at least one of the following four cases holds:
which implies Open image in new window , that is,
From (1), (2), (3), and ( Open image in new window ) it follows that
Similarly, if Open image in new window , we have

Case 2.

which in turn yields
and hence

Now, proceeding as in Case 1, we have that (2.18) holds.

Using (2.21), we get
By noting that Open image in new window , one can conclude that

in view of Case 1.

and we proved (2.24).

Case 3.

From this, we get
By noting that Open image in new window , one can conclude that

in view of Case 1.

and we proved (2.28).

From this, we have
By noting that Open image in new window , one can conclude that

in view of Case 1.

Thus, in all Cases 1–3, there exists Open image in new window such that
and there exists Open image in new window such that
Following the procedure of Assad and Kirk [13], it can easily be shown by induction that, for Open image in new window , there exists Open image in new window such that
From (2.38) and by the triangle inequality, for Open image in new window , we have

From Remark 1.5 and Corollary 1.4(1), Open image in new window .

Thus, the sequence Open image in new window is a Cauchy sequence. Then, as noted in [14], there exists at least one subsequence Open image in new window or Open image in new window which is contained in Open image in new window or Open image in new window , respectively, and finds its limit Open image in new window Furthermore, subsequences Open image in new window and Open image in new window both converge to Open image in new window as Open image in new window is a closed subset of complete cone metric space Open image in new window . We assume that there exists a subsequence Open image in new window for each Open image in new window , then Open image in new window . Since Open image in new window as well as Open image in new window are closed in Open image in new window , and Open image in new window is Cauchy in Open image in new window , it converges to a point Open image in new window . Let Open image in new window , then Open image in new window . Similarly, Open image in new window a subsequence of Cauchy sequence Open image in new window also converges to Open image in new window as Open image in new window is closed. Using (2.2), one can write
Let Open image in new window . Clearly at least one of the following four cases holds for infinitely many Open image in new window :

In all cases we obtain Open image in new window for each Open image in new window . Using Corollary 1.4(3) it follows that Open image in new window or Open image in new window . Thus, Open image in new window , that is, Open image in new window is a coincidence point of Open image in new window , Open image in new window .

Further, since Cauchy sequence Open image in new window converges to Open image in new window and Open image in new window , Open image in new window , there exists Open image in new window such that Open image in new window . Again using (2.2), we get
Hence, we get the following cases:

Since Open image in new window , using Remark 1.6 and Corollary 1.4(3), it follows that Open image in new window ; therefore, Open image in new window , that is, Open image in new window is a coincidence point of Open image in new window .

In case Open image in new window and Open image in new window are closed in Open image in new window , Open image in new window or Open image in new window . The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence Open image in new window with Open image in new window as well Open image in new window being closed in Open image in new window , then noting that Open image in new window is a Cauchy sequence in Open image in new window , foregoing arguments establish (IV) and (V).

Suppose now that Open image in new window and Open image in new window are weakly compatible pairs, then
Then, from (2.2),
Hence, we get the following cases:

Since Open image in new window , using Remark 1.6 and Corollary 1.4(3), it follows that Open image in new window . Thus, Open image in new window .

Similarly, we can prove that Open image in new window . Therefore Open image in new window , that is, Open image in new window is a common fixed point of Open image in new window , Open image in new window , Open image in new window , and Open image in new window .

Uniqueness of the common fixed point follows easily from (2.2).

The following example shows that in general Open image in new window , Open image in new window , Open image in new window , and Open image in new window satisfying the hypotheses of Theorem 2.3 need not have a common coincidence justifying two separate conclusions (IV) and (V).

Example 2.4.

Since Open image in new window . Clearly, for each Open image in new window and Open image in new window there exists a point Open image in new window such that Open image in new window . Further, Open image in new window , Open image in new window , Open image in new window , and Open image in new window , Open image in new window , Open image in new window , and Open image in new window are closed in Open image in new window .

that is, (2.2) is satisfied with Open image in new window .

Evidently, Open image in new window and Open image in new window . Notice that two separate coincidence points are not common fixed points as Open image in new window and Open image in new window , which shows necessity of weakly compatible property in Theorem 2.3.

Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [12].

Example 2.5.

Since Open image in new window . Clearly, for each Open image in new window and Open image in new window there exists a point Open image in new window such that Open image in new window . Further, Open image in new window , Open image in new window , and Open image in new window .

Therefore, condition (2.2) is satisfied if we choose Open image in new window . Moreover Open image in new window is a point of coincidence as Open image in new window as well as Open image in new window whereas both the pairs Open image in new window and Open image in new window are weakly compatible as Open image in new window and Open image in new window . Also, Open image in new window , Open image in new window , Open image in new window , and Open image in new window are closed in Open image in new window . Thus, all the conditions of Theorem 2.3 are satisfied and Open image in new window is the unique common fixed point of Open image in new window , Open image in new window , Open image in new window , and Open image in new window . One may note that Open image in new window is also a point of coincidence for both the pairs Open image in new window and Open image in new window .

Remark 2.6.
  1. (1)

    Setting Open image in new window and Open image in new window in Theorem 2.3, one deduces Theorem 2.1 due to [5].

     
  1. (2)

    Setting Open image in new window and Open image in new window in Theorem 2.3, we obtain the following result.

     

Corollary 2.7.

Let Open image in new window be a complete cone metric space and Open image in new window a nonempty closed subset of Open image in new window such that for each Open image in new window and Open image in new window there exists a point Open image in new window (the boundary of Open image in new window ) such that
Suppose that Open image in new window satisfies the condition

for all Open image in new window , Open image in new window , Open image in new window , and Open image in new window has the additional property that for each Open image in new window , Open image in new window , Open image in new window has a unique fixed point.

Notes

Acknowledgments

The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments. This project was supported by the National Natural Science Foundation of China (10461007 and 10761007) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2008GZS0076 and 2009GZS0019).

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© The Author(s). 2010

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Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina
  2. 2.Department of Computer ScienceNanchang UniversityNanchangChina

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