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

Abstract

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

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 [2–8]. 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 be a real Banach space. A subset of is called a cone if and only if

(a) is closed, nonempty and ;

(b), , implies ;

(c).

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

(1.1)

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

Definition 1.1 (see [1]).

Let be a nonempty set. Suppose that the mapping satisfies

(d1) for all and if and only if ;

(d2) for all ;

(d3) for all .

Then is called a cone metric on , and 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 be a cone metric space. One says that is

• (e) a Cauchy sequence if for every with , there is an such that for all , ;

• (f) a Convergent sequence if for every with , there is an such that for all , for some fixed .

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

Remark 1.3 (see [9]).

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

Corollary 1.4 (see [10]).

• (1) If and , then .Indeed, implies .

(2)If and , then .Indeed, implies .

• (3) If for each , then .

Remark 1.5 (see [5, 11]).

If , , and , then there exists an such that for all we have .

Remark 1.6 (see [6, 10]).

If is a real Banach space with cone and if where and , then .

We find it convenient to introduce the following definition.

Definition 1.7 (see [5]).

Let be a complete cone metric space and a nonempty closed subset of , and satisfying

(1.2)

where

(1.3)

for all , , , then is called a generalized -contractive mapping of into .

Definition 1.8 (see [2]).

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

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 be a complete cone metric space and a nonempty closed subset of such that for each and there exists a point (the boundary of ) such that

(2.1)

Suppose that are such that is a generalized -contractive mapping of into , and

(i),

(ii),

(iii) is closed in .

Then the pair has a coincidence point. Moreover, if pair is weakly compatible, then and 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 be a complete cone metric space and a nonempty closed subset of , and satisfying

(2.2)

where

(2.3)

for all , , , then is called a generalized -contractive mappings pair of into .

Notice that by setting and 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 be a complete cone metric space and a nonempty closed subset of such that for each and there exists a point (the boundary of ) such that

(2.4)

Suppose that are such that is a generalized -contractive mappings pair of into , and

(I),

(II),

(III) and (or and ) are closed in .Then

(IV) has a point of coincidence,

(V) has a point of coincidence.

Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

Proof.

Firstly, we proceed to construct two sequences and in the following way.

Let be arbitrary. Then (due to ) there exists a point such that . Since , one concludes that . Thus, there exists such that . Since there exists a point such that

(2.5)

Suppose that . Then which implies that there exists a point such that . Otherwise, if , then there exists a point such that

(2.6)

Since there exists a point with , so that

(2.7)

Let be such that . Thus, repeating the foregoing arguments, one obtains two sequences and such that

(a), ,

(b) or ,

(2.8)

(c) or ,

(2.9)

We denote that

(2.10)

Note that , as if , then , and one infers that which implies that . Hence . Similarly, one can argue that .

Now, we distinguish the following three cases.

Case 1.

If , then from (2.2)

(2.11)

where

(2.12)

Clearly, there are infinite many such that at least one of the following four cases holds:

• (1)
  

  (2.13)

• (2)
  

  (2.14)

• (3)
  

  (2.15)

• (4)
  

  (2.16)

which implies , that is,

(2.17)

From (1), (2), (3), and () it follows that

(2.18)

Similarly, if , we have

(2.19)

If , we have

(2.20)

Case 2.

If , then and

(2.21)

which in turn yields

(2.22)

and hence

(2.23)

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

If , then . We show that

(2.24)

Using (2.21), we get

(2.25)

By noting that , one can conclude that

(2.26)

in view of Case 1.

Thus,

(2.27)

and we proved (2.24).

Case 3.

If , then . We show that

(2.28)

Since , then

(2.29)

From this, we get

(2.30)

By noting that , one can conclude that

(2.31)

in view of Case 1.

Thus,

(2.32)

and we proved (2.28).

Similarly, if , then , and

(2.33)

From this, we have

(2.34)

By noting that , one can conclude that

(2.35)

in view of Case 1.

Thus, in all Cases 1–3, there exists such that

(2.36)

and there exists such that

(2.37)

Following the procedure of Assad and Kirk [13], it can easily be shown by induction that, for , there exists such that

(2.38)

From (2.38) and by the triangle inequality, for , we have

(2.39)

From Remark 1.5 and Corollary 1.4(1), .

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

(2.40)

where

(2.41)

Let . Clearly at least one of the following four cases holds for infinitely many :

• (1)
  

  (2.42)

• (2)
  

  (2.43)

• (3)
  

  (2.44)

• (4)
  

  (2.45)

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

Further, since Cauchy sequence converges to and , , there exists such that . Again using (2.2), we get

(2.46)

where

(2.47)

Hence, we get the following cases:

(2.48)

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

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

Suppose now that and are weakly compatible pairs, then

(2.49)

Then, from (2.2),

(2.50)

where

(2.51)

Hence, we get the following cases:

(2.52)

Since , using Remark 1.6 and Corollary 1.4(3), it follows that . Thus, .

Similarly, we can prove that . Therefore , that is, is a common fixed point of , , , and .

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

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

Example 2.4.

Let , , , , and defined by , where is a fixed function, for example, . Then is a complete cone metric space with a nonnormal cone having the nonempty interior. Define , , , and as

(2.53)

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

Also,

(2.54)

Moreover, for each ,

(2.55)

that is, (2.2) is satisfied with .

Evidently, and . Notice that two separate coincidence points are not common fixed points as and , 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.

Let , , , and defined by , where is a fixed function, for example, . Then is a complete cone metric space with a nonnormal cone having the nonempty interior. Define , , , and as

(2.56)

Since . Clearly, for each and there exists a point such that . Further, , , and .

Also,

(2.57)

Moreover, if and , then

(2.58)

Next, if , then

(2.59)

Finally, if , then

(2.60)

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

Remark 2.6.

• (1) Setting and in Theorem 2.3, one deduces Theorem 2.1 due to [5].

• (2) Setting and in Theorem 2.3, we obtain the following result.

Corollary 2.7.

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

(2.61)

Suppose that satisfies the condition

(2.62)

where

(2.63)

for all , , , and has the additional property that for each , , has a unique fixed point.