Abstract

Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example. The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

1. Introduction and Preliminaries

In recent times coupled fixed point theory has experienced a rapid growth in partially ordered metric spaces. The speciality of this line of research is that the problems herein utilize both order theoretic and analytic methods. References [119] are some instances of these works.

Definition 1.1 (see [14]). A function is said to be monotone nondecreasing (or increasing) if implies .

Definition 1.2 (see [14]). Let be a nonempty set. Let be a mapping. An element is called a coupled fixed point of if

Recently, Berinde and Borcut [20] extended the idea of coupled fixed points to tripled fixed points. The definition is as follows.

Definition 1.3 (see [20]). Let be a nonempty set. Let be a mapping. An element is called a tripled fixed point of if
They also extended the mixed monotone property to functions with three arguments.

Definition 1.4 (see [20]). Let be a partially ordered set and . The mapping is said to have the mixed monotone property if for any

Our purpose here is to establish tripled coincidence point results in metric spaces with partial ordering. For that purpose we define mixed -monotone property in the following. Mixed g-monotone property was already defined in the context of coupled fixed points [14]. Here in the spirit of Definition 1.4 we have made an extension of that.

Definition 1.5. Let be a partially ordered set. Let and . The mapping is said to have the mixed -monotone property if for any .

Coupled coincidence point was defined by Lakshmikantham and Ćirić [14]. We also extend the concept of coupled coincidence point to tripled coincidence point in the following.

Definition 1.6. Let be any nonempty set. Let and . An element is called a tripled coincidence point of and if

We extend the concept of commuting mappings given by Lakshmikantham and Ćirić [14], in the following definition.

Definition 1.7. Let be a nonempty set. Then one says that the mappings and are commuting if for all

The following is the definition of compatible mappings which is an extension of the compatibility defined by Choudhury and Kundu in [8].

Definition 1.8 (see [8]). Let be a metric space. The mappings and , where and are said to be compatible if whenever are sequences in such that

2. Main Results

Theorem 2.1. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose and are such that, is monotone increasing, has the mixed -monotone property and for all for which , and , where is such that is monotone, and for all . Suppose , is continuous, and is a compatible pair. Suppose either (a)is continuous or (b) has the following properties: (i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that , , and , then there exist such that that is, and have a tripled coincidence point.

Proof. By a condition of the theorem, there exist such that , , and . Since , we can choose such that Continuing this process, we can construct sequences , and in such that Next we will show that, for , Since, , and , by (2.3), we get that is, (2.5) holds for .
We presume that (2.5) holds for some . As has the mixed -monotone property and and , we obtain Thus, (2.5) holds for . Then, by induction, we conclude that (2.5) holds for .
If for some , then, by (2.4), is a tripled coincidence point of and . Therefore we assume, for any , Set .
Then Then, by (2.1), (2.4) and (2.5), we have Thus, from (2.13) we obtain that It then follows from (2.12) and a property , that for all , Thus, is a monotone decreasing sequence of nonnegative real numbers. So, there exist a such that Suppose . Letting in (2.14), using (2.15), (2.16), and a property of , we get which is a contradiction. Thus , or or Now, we will prove that , , and are Cauchy sequences. Suppose, to the contrary, that at least one of , , and is not a Cauchy sequence. So, there exists an for which we can find subsequences of , of , and of with such that Additionally, corresponding to , we may choose such that it is the smallest integer satisfying (2.20). Then, for all , By using (2.20) and (2.21) we have for , Letting in (2.22), and using (2.19), we get Let, for ,
Again, for all , Analogously we have for , Letting in (2.25) and (2.26), we get that Since , for , we have Then from (2.1), (2.4), and (2.28), we have for , From (2.29) for , we get Letting in (2.30), using (2.20), (2.27), and a property of , we get which is a contradiction. This shows that , , and are Cauchy sequences.
Since is complete, there exist such that From (2.4) and (2.32), using the continuity of , we have Now we will show that , and .
Since and are compatible, in addition with (2.33), (2.34), and (2.35), respectively imply Suppose now the assumption holds, that is, is continuous.
For all , we have Taking the limit as , using (2.32), (2.33), (2.36), and the facts that and are continuous, we have .
Similarly, by using (2.32), (2.34), and (2.37) and (2.32), (2.35), and (2.38), respectively, and also the facts that and are continuous, we have and .
Thus we have proved that and have a tripled coincidence point.
Suppose that the assumption holds. Since are nondecreasing and with and also is nonincreasing with , by assumption we have for all By virtue of monotone increasing property of we have Now using (2.4) we have Taking the limit as in the above inequality, using (2.33), (2.36), and (2.41) we have By (2.33), (2.34), (2.35), and the property of , we have that is In a similar manner using (2.33), (2.34), (2.35), and (2.36), (2.37), (2.38), respectively, we obtain Thus, we proved that and have a tripled coincidence point.
This completes the proof of the theorem.

Corollary 2.2. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose and are such that has the mixed -monotone property and for any for which , and , where be such that is monotone, and for all . Suppose , is continuous, and and are commuting. Suppose either (a) is continuous, or   (b) has the  following property: (i)if a nondecreasing sequence ,  then for all ,(ii)if a nonincreasing sequence  , then for all .
If there exist such that , , and , then there exist such that that is, and have a tripled coincidence point.

Proof. Since a commuting pair is also a compatible pair, the result of the Corollary 2.2 follows from Theorem 2.1.

Later, by an example, we will show that the Corollary 2.2 is properly contained in Theorem 2.1.

Corollary 2.3. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose be such that has the mixed monotone property and for any for which and , where be such that is monotone, and for all . Suppose(a) is continuous, or(b) has the following property: (i)if a nondecreasing sequence , then for all ,   (ii)if a nonincreasing sequence , then for all .
If there exist such that , , and , then there exist such that that is, has a tripled fixed point.

Proof. Taking in Theorem 2.1 we obtain Corollary 2.3.

Corollary 2.4. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose and are such that has the mixed monotone property and for any for which and , where . Suppose either(a) is continuous,  or(b) has the following  property:(i)if a nondecreasing sequence , then   for all ,(ii)if a nonincreasing sequence , then   for all .
If there exist such that , , and , then there exist such that that is, has a tripled coincidence point.

Proof. Taking , where , in Corollary 2.3 we obtain Corollary 2.4.

The following corollary is the result of Berinde and Borcut in [20].

Corollary 2.5. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose be such that has the mixed monotone property and for any for which and , where . Suppose either(a) is continuous, or(b) has the following property:(i)if a nondecreasing sequence , then for all (ii)if a nonincreasing sequence , then for all .
If there exist such that , , and , then there exist such that that is, has a tripled fixed point.

Proof. The proof follows from Corollary 2.4, since the inequality in Corollary 2.5 implies that Corollary 2.4.

Remark 2.6. The method used in the proof of Corollary 2.5 is different from that used by Berinde and Borcut [20].

Next we discuss an example.

Example 2.7. Let . Then is a partially ordered set with the partial ordering defined by if and only if and .

Let for . Then is a complete metric space.

Let be defined as for all .

Let be defined as Then F obeys the mixed -monotone property.

Let be defined as for all .

Let, , and be three sequences in such that Then explicitly, Again, And Then from the above relations we have, , and .

Therefore, Hence, the pair is compatible in .

Also, , and are three points in such that , and .

We next verify inequality (2.1) of Theorem 2.1. We take , such that and , that is, , and .

Let .

Then , .

Thus it is verified that the functions , and satisfy all the conditions of Theorem 2.1. Here is the tripled coincidence point of and in .

Remark 2.8. It is observed that in Example 2.7 the function and do not commute, but they are compatible. Hence Corollary 2.2 cannot be applied to this example. This shows that Theorem 2.1 properly contains Corollary 2.2. Also , so the results of Berinde and Borcut [20] cannot be applied to this example. This shows that result in [20] is effectively generalised.