𝑛-Tupled Coincidence Point Theorems in Partially Ordered Metric Spaces for Compatible Mappings

Abstract

The intent of this paper is to introduce the notion of compatible mappings for -tupled coincidence points due to (Imdad et al. (2013)). Related examples are also given to support our main results. Our results are the generalizations of the results of (Gnana Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009), Choudhury and Kundu (2010), and Choudhary et al. (2013)).

1. Introduction

Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach fixed point theorem. There exists vast literature on the topic and it is a very active field of research at present. A self-map on a metric space is said to have a fixed point if . Theorems concerning the existence and properties of fixed points are known as fixed point theorems. Such theorems are very important tool for proving the existence and eventually the uniqueness of the solutions to various mathematical models (integral and partial differential equations, variational inequalities).

Existence of a fixed point for contraction type mappings in partially ordered metric spaces and applications has been considered by many authors; for detail, see [1–11]. In particular, Gnana Bhaskar and Lakshmikantham [12], Nieto and Rodriguez-Lopez [8], Ran and Recuring [13], and Agarwal et al. [9] presented some new results for contractions in partially ordered metric spaces.

Coupled fixed point problems belong to a category of problems in fixed point theory in which much interest has been generated recently after the publication of a coupled contraction theorem by Gnana Bhaskar and Lakshmikantham [12]. One of the reasons for this interest is the application of these results for proving the existence and uniqueness of the solution of differential equations, integral equations, the Volterra integral and Fredholm integral equations, and boundary value problems. For comprehensive description of such work, we refer to [1, 3–5, 7, 10–12, 14–18].

Common fixed point results for commuting maps in metric spaces were first deduced by Jungck [19]. The concept of commuting has been weakened in various directions and in several ways over the years. One such notion which is weaker than commuting is the concept of compatibility introduced by Jungck [20]. In common fixed point problems, this concept and its generalizations have been used extensively; for instance, see [3, 8, 9, 13–17, 20].

Most recently, Imdad et al. [21] introduced the notion of -tupled coincidence point and proved -tupled coincidence point theorems for commuting mappings in metric spaces. Motivated by this fact, we introduce the notion of compatibility for -tupled coincidence points and prove -tupled fixed point for compatible mappings satisfying contractive conditions in partially ordered metric spaces.

2. Preliminaries

Definition 1 (see [10]). 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 [10]). Let be a partially ordered set and ; then enjoys the mixed monotone property if is monotonically nondecreasing in and monotonically nonincreasing in ; that is, for any ,

Definition 3 (see [10]). Let be a partially ordered set and ; then is called a coupled fixed point of the mapping if and .

Definition 4 (see [10]). Let be a partially ordered set and and ; then enjoys the mixed -monotone property if is monotonically -nondecreasing in and monotonically -nonincreasing in ; that is, for any ,

Definition 5 (see [10]). Let be a partially ordered set and and ; then is called a coupled coincidence point of the mappings and if and .

Definition 6 (see [10]). Let be a partially ordered set; then is called a coupled fixed point of the mappings and if and .

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

Definition 7 (see [21]). Let be a partially ordered set and ; then is said to have the mixed monotone property if is nondecreasing in its odd position arguments and nonincreasing in its even positions arguments; that is, if,(i)for all , ,(ii)for all , ,(iii)for all , ,for all , .

Definition 8 (see [21]). Let be a partially ordered set and 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 positions arguments; that is, if,(i)for all , ,(ii)for all , ,(iii)for all , ,for all , .

Definition 9 (see [21]). Let be a nonempty set. An element is called an -tupled fixed point of the mapping if

Example 10. Let be a partial ordered metric space under natural setting and let be mapping defined by , for any ; then is an -tupled fixed point of .

Definition 11 (see [21]). Let be a nonempty set. An element is called an -tupled coincidence point of the mappings and if

Example 12. Let be a partial ordered metric space under natural setting and let and be mappings defined by for any ; then is an -tupled coincidence point of and .

Definition 13 (see [21]). Let be a nonempty set. An element is called an -tupled fixed point of the mappings and if

Now, we define the concept of compatible mappings for -tupled mappings.

Definition 14. Let be a partially ordered set; then the mappings and are called compatible if whenever are sequences in such that for some .

3. Main Results

Recently, Imdad et al. [21] proved the following theorem.

Theorem 15. Let be a partially ordered set equipped with a metric d such that is a complete metric space. Assume that there is a function with and for each . Further let and be two mappings such that has the mixed -monotone property satisfying the following conditions:(i),(ii) is continuous and monotonically increasing,(iii) is a commuting 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; that is, there exist such that

Now, we prove our main results.

Theorem 16. 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 let and be two mappings such that has the mixed -monotone property satisfying the following conditions:(1),(2) is continuous and monotonically increasing,(3)the pair is compatible,(4),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; that is, there exist such that

Proof. Starting with , we define the sequences in as follows: Now, we prove that, for all , So (15) holds for . Suppose (15) holds for some . Consider Thus by induction (15) holds for all . Using (14) and (15) Similarly, we can inductively write Therefore, by putting we have Since for all , for all so that is a nonincreasing sequence. Since it is bounded below, there are some such that We will show that . Suppose, if possible, . Taking limit as of both sides of (21) and keeping in mind our supposition that for all , we have and this contradiction gives and hence 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 exist and sequences of positive integers and such that, for all positive integers , , Now, That is, Taking in the above inequality and using (24), we have Again, Taking limit as in the above inequality and using (24) and (28), we have Now, Letting in the above inequality and using (28), (30), and the property of , we get which is a contradiction. Therefore, are Cauchy sequences. Since the metric space is complete, there exist such that As is continuous, from (33), we have By the compatibility of and , we have Now, we show that and have an -tupled coincidence point. To accomplish this, suppose (a) holds. That is, is continuous. Then using (35) and (15), we see that which implies . Similarly, we can easily prove that , . Hence is an -tupled coincidence point of the mappings and .
If (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, Now, using triangle inequality together with (15), we get Therefore,. Similarly, we can prove , . Thus the theorem follows.

Now, we furnish an illustrative example to support our theorem.

Example 17. Let be complete metric space under usual metric and natural ordering of real numbers. Define the mappings and as follows: Set ; then we see that Also, the pair is compatible. Thus all the conditions of our Theorem 16 are satisfied (without order) and is an -tuple coincidence point of and .

Corollary 18. Let be a partially ordered set equipped with a metric d such that is a complete metric space. Further let and be two mappings satisfying all the conditions of Theorem 15 with a suitable replacement of condition (4) of Theorem 16 by Then and have an -tupled coincidence point.

Proof. If we put where in Theorem 15, then the result follows immediately.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors would like to express their sincere thanks to the guest editor Professor Jen-Chih Yao and the anonymous reviewers for their valuable suggestions.