Research Article | Open Access

# On -Tupled Coincidence Point Results in Metric Spaces

**Academic Editor:**Qingkai Kong

#### Abstract

We prove some *n*-tupled coincidence point results whenever *n* is even. We give here several new definitions like *n*-tupled fixed point, *n*-tupled coincidence point, and so forth. The main result is supported with the aid of an illustrative example.

#### 1. Introduction and Preliminaries

The classical Banach Contraction Principle proved in complete metric spaces continues to be an indispensable and effective tool in theory as well as applications which guarantees the existence and uniqueness of fixed points of contraction self-mappings besides offering a constructive procedure to compute the fixed point of the underlying map. There already exists an extensive literature on this topic, but keeping in view the relevance of this paper, we merely refer to [1–14].

In 2006, Gnana Bhaskar and Lakshmikantham initiated the idea of coupled fixed point in partially ordered metric spaces and proved some interesting coupled fixed point theorems for mapping satisfying a mixed monotone property. In recent years, many authors obtained important coupled fixed point theorems (e.g., [15–20]). In this continuation, Lakshmikantham and Cirić [21] proved coupled common fixed point theorems for nonlinear -contraction mappings in partially ordered complete metric spaces which indeed generalize the corresponding fixed point theorems contained in Gnana Bhaskar and Lakshmikantham [22].

As usual, this section is devoted to preliminaries which include basic definitions and results on coupled fixed point for nonlinear contraction mappings defined on partially ordered complete metric spaces. In Section 2, we introduce the concepts of -tupled coincidence point and -tupled fixed point for mappings satisfying different contractive conditions and utilize these two definitions to obtain -tupled coincidence point theorems for nonlinear -contraction mappings in partially ordered complete metric spaces.

Now, we present some basic notions and results related to coupled fixed point in metric spaces.

*Definition 1 (see [22]). *Let be a partially ordered set equipped with a metric such that is a metric space. Further, equip the product space with the following partial ordering:

*Definition 2 (see [22]). *Let be a partially ordered set and . One says that enjoys the mixed monotone property if is monotonically nondecreasing in and monotonically nonincreasing in ; that is, for any ,

*Definition 3 (see [22]). *Let be a partially ordered set and . One says that is a coupled fixed point of the mapping if

Theorem 4 (see [22]). *Let be a partially ordered set equipped with a metric such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists a constant with
**
If there exist such that and , then there exist such that and .*

*Definition 5 (see [21]). *Let be a partially ordered set and and two mappings. The mapping is said to have the mixed -monotone property if is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument, that is, if, for all , implies , for any , and, for all , implies , for any .

*Definition 6 (see [21]). *An element is called a coupled coincidence point of mappings and if

Theorem 7 (see [21]). *Let be a partially ordered set equipped with a metric such that is a complete metric space. Assume that there is a function with and for each . Let and be maps such that has the mixed -monotone property and
**
for all for which and . Suppose that is continuous and commutes with besides*(a)*is continuous,*(b)* has the following properties:(i) if nondecreasing sequence , then for all ,(ii)if nonincreasing sequence , then for all .(iii)if a nondecreasing sequence , then for all ,(iv)if a nonincreasing sequence , then for all .*

*If there exist such that*

*then there exist such that*

*That is, and have a coupled coincidence point.*

#### 2. Main Results

Throughout the paper, stands for a general even natural number.

*Definition 8. *Let be a partially ordered set and a mapping. The mapping is said to have the mixed monotone property if is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if, (i)for all , implies (ii)for all , implies (iii)for all , implies

for all , implies .

*Definition 9. *Let be a partially ordered set. Let and be two mappings. Then the mapping is said to have the mixed -monotone property if is -nondecreasing in its odd position arguments and -nonincreasing in its even position arguments, that is, if,(i)for all , implies ,(ii)for all , implies ,(iii)for all , implies ,for all , implies .

*Definition 10. *Let be a nonempty set. An element is called an -tupled fixed point of the mapping if

*Definition 11. *Let be a nonempty set. An element is called an -tupled coincidence point of the mappings and if

*Definition 12. *Let be a nonempty set. The mappings and are said to be commutating if
for all .

Now, we are equipped to prove our main result as follows.

Theorem 13. *Let be a partially ordered set equipped with a metric such that is a complete metric space. Assume that there is a function with and for each . Further, suppose that and are two maps such that has the mixed -monotone property satisfying the following conditions:*(i)*,*(ii)* is continuous and monotonically increasing,*(iii)* is a commutating pair,*(iv)*for all with . Also, suppose that either*(a)* is continuous or*(b)* has the following properties:
**If there exist such that
**
then and have a -tupled coincidence point; that is, there exist such that
*

*Proof. *Starting with in , we define the sequences in as follows:
Now, we prove that for all ,
So (17) holds for . Suppose (17) holds for some . Consider
Then, by induction, (17) holds for all .

Using (16) and (17), we have
Similarly, we can inductively write
Therefore, by putting
we have
Since for all , therefore, for all so that is a nonincreasing sequence. Since it is bounded below, there is some such that
We shall show that . Suppose, on the contrary that . Taking the limits as of both the sides of (23) and keeping in mind our supposition that for all , we have
which is a contradiction so that yielding thereby
Next we show that all the sequences , and are Cauchy sequences. If possible, suppose that at least one of and is not a Cauchy sequence. Then there exists and sequences of positive integers and such that for all positive integers ,
Now,
that is,
Letting in the above inequality and using (26), we have
Again,
Letting in the above inequalities, using (26) and (30), we have
Now,
Letting in the above inequality, using (30), (33), and the property of , we have
which is a contradiction. Therefore, , and are Cauchy sequences in . Since the metric space is complete, so there exist such that
By the continuity of and (36), we can have
Using (16) and the commutativity of with , we get
Now, we show that and have an -tupled coincidence point. To accomplish this, suppose (a) holds, then using (16), (37), and the continuities of and , we obtain
Similarly, we can also show that
Hence the element is a -tupled coincidence point of the mappings and . Next, assume that (b) holds. Since is nondecreasing or nonincreasing as is odd or even and as , we have , when is odd while , when is even.

Since is monotonically increasing, therefore

On using triangle inequality together with (16), we get
Letting in the above inequality and using (37), we have . Similarly, we can also show that
which shows that and have an -tupled coincidence point. This completes the proof.

Corollary 14. *Let be a partially ordered set equipped with a metric such that is a complete metric space. Assume that there is a function with and for each . Further, suppose that is a mapping such that has the mixed monotone property satisfying the following conditions:
**
for all with . Also, suppose that 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*

*then has an -tupled fixed point in ; that is, there exist such that*

*Proof. *Setting , the identity mapping, in Theorem 13, we obtain Corollary 14.

Also, Theorem 13 immediately yields the following corollary.

Corollary 15. *Let be a partially ordered set equipped with a metric such that is a complete metric space. Suppose that and are two maps such that has the mixed -monotone property satisfying the following conditions:*(i)*,*(ii)* is continuous and monotonically increasing,*(iii)* is a commutating pair,*(iv)*for all with . Also, suppose that 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*

*then and have an -tupled coincidence point in*

*X*;*that is, there exist such that*

*Proof. *Setting with in Theorem 13, we obtain Corollary 15.

The following example illustrates Theorem 13.

*Example 16. *Let . Then is a complete metric space under natural ordering of real numbers and natural metric for all . Define as wherein is fixed and for all . Also, define by
for all . Define as , where is fixed as earlier. Then has all the properties mentioned in Theorem 13. Also and are commutating mapping in .

Next, we verify inequality (12) (of Theorem 13)
Thus all the conditions of Theorem 13 (without order) are satisfied and is a -tupled coincidence point of and .

#### Acknowledgments

All the authors are grateful to both the learned referees for their fruitful suggestions and remarks towards the improvement of this paper.

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#### Copyright

Copyright © 2013 Mohammad Imdad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.