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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 486384, 6 pages

http://dx.doi.org/10.1155/2014/486384

## On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces

^{1}Department of Mathematics, Faculty of Science and Arts, Celal Bayar University Muradiye Campus, 45047 Manisa, Turkey^{2}Hasan Türek Anatolian High School, 45200 Manisa, Turkey

Received 13 September 2013; Accepted 31 October 2013; Published 29 January 2014

Academic Editor: Abdon Atangana

Copyright © 2014 Ali Mutlu 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.

#### Abstract

Common coupled fixed point theorems are examined in this paper for comparable mappings ensuring nonlinear contraction in ordered partial metric spaces. Given theorems enlarge and universalize some conclusions of Gnana Bhaskar and Lakshmikantham (2006).

#### 1. Introduction

The contraction method presented the fixed point theory on partially metric spaces. It is enlarged to nonlinear contraction mapping, which is attributed by many authors. (cf. [1–25]). Particularly a partial metric space is a universalized metric space. Some further generalizations of the conclusions in [16] are demonstrated by Valero [25], Oltra and Valero [18], Shatanawi et al. [23], and Altun and Erduran [5]. Additionally, Caristi type fixed point theorem on a partial metric space was introduced by Romaguera [21].

Existence of fixed points was introduced in ordered metric spaces by Ran and Reurings [19]. Some applications of fixed points are also shown for linear and nonlinear equations. Fixed and common fixed point theorems are searched recently by many authors on this topic. Moreover coupled coincidence and coupled fixed point theorems for two mappings and such that has to be mixed g-monotone property are stated by Lakshmikantham and Ćirić [15].

The authors propose to give more information about couple fixed point theory exists in the theory [1–25] for the reader.

Let us give some necessarily definitions related to mixed monotone maps and common coupled fixed point of a mapping.

*Definition 1. *Suppose that () is a partially ordered set and also . Assume that is monotone nondecreasing pursuant and also is monotone nonincreasing according to , for any , at the time the map is named to have mixed monotone property:

*Definition 2. *If and , then is defined as an a coupled fixed point of a mapping [15].

*Definition 3. *Suppose that is a nonempty set. A partial metric on is a real function of of ordered pairs of elements of which satisfies the following four conditions: ,,, [16]. A metric space consists of two objects: a set and partial metric on , and also the elements of are called the point of the metric space () (see [16]).

Notice that the span of any point to itself need not be null; so universalizing metrics, namely, a metric on a set , are named to be a partial metric on providing for any . We refer the reader to check some results and related examples on partial metric spaces in the theory [1–25].

Each partial metric on generates a topology on , which has a base of the family of open p-balls , where

If is a partial metric on , then the function given by is a metric on .

*Definition 4. *Assume that is a partial metric space and also is a sequence in .

At the time,(i) converges to a point ,(ii)if there exists , then is a Cauchy sequence [5].

*Definition 5. *A partial metric space is named to be complete if every Cauchy sequence in converges,

in accordance with , to a point , with [5].

Lemma 6. *Suppose that is a partial metric space. At the time *(i)*the sequence is Cauchy sequence in it is a Cauchy sequence in the metric space ,*(ii)* is complete the metric space is complete. Besides, [16].*

Theorem 7. *Assume that is a complete partial metric space and also suppose that is a mapping to itself. Then there exists a constant providing
**
for all . So has an individual fixed point [16].*

Recently, Gnana Bhaskar and Lakshmikantham [8] obtained the following nice result for possessing the mixed monotone property mapping, which universalizes Theorem 7 of Matthews [16].

Theorem 8. *Suppose that is a continuous mapping possessing the mixed monotone property on . There exists a such that
**
If there exist with
**
then, there exist with
*

The goal of the paper is to build coupled and common fixed point theorems in partially ordered partial metric spaces with a function providing conditions , nonincreasing, and for each . Offered theorems universalize and enlarge to a pair of mappings which are conclusions of Gnana Bhaskar and Lakshmikantham [8] and some other theorems related to them.

#### 2. Main Result

*Definition 9. *Assume that () is a partially ordered set and . and mappings have the following properties: if is even, then and ; if is odd, then and .

Theorem 10. *Suppose that () is a partially ordered set and is a partial metric on with being a complete partial metric space. Assume that are satisfied by Definition 2 and also are continuous mappings possessing the mixed monotone property on . Let there be a non-increasing function such that , and for all and also having and , with
**
for . If there exists with and , at the time with and .*

*Proof. *Suppose with and . Define sequences and in in the following way:
We are to prove that sequence is nondecreasing and sequence is nonincreasing. That is, for all
For this, mathematical induction method is used.

Firstly suppose . Having and , because and and as and , so (10) is verified for .

Assume that (10) is satisfied for a constant ; then, because and , from Definition 9 we have
Thus we get and .

Hereby, by the induction method we conclude that (10) hold for all . Thereof,

Denote
showing sequence is nonincreasing. From (10) and (8) we have
Similarly, we can obtain
Thus, using properties of function we get
Similarly one can show that
Then, we obtain

Thus a sequence is nonincreasing. Thence, there is a is obtained with

Now, we claim that
we substitute in (14). Then we can get
Letting in (22), we get
Hence . That is

Now we show that
Suppose the contrary. At the time there exists when obtaining two subsequences and of with is the smallest index where
This means that
By in Definition 3 and (27), we have
Similarly, we can obtain that
Adding (28) and (29) and also from (27) and (26) we get
Taking the limit as in (30) and by (26) we get
Employing the triangle inequality,
Similarly, we get
As in (33) and (32) and from (31) and (26) we can obtain

Since from (12) we have and and also by (8) and (10),
Similarly, we get
Thus
As in (37) we get , which is a contrast. Whence (25) is verified, possessing
By (3), we have
and are Cauchy sequences in the metric space . Because is complete, it is also the case for , then there exist with
On the other hand, we have
Getting the limit as in the upward equation and utilizing (40) and (38), we attain
in other words, possessing for all . On letting , we achieve
Using (42) and (43), we get that
Analogously, one can show that
exposing , , , and . To do that we prove the following steps.

*Step 1. *Demonstrate that and .

Since and , we have
Letting in (46) we get . The same one can demonstrate that .

*Step 2. *We show that and .

We have . Since and as in and is continuous as in , then we get
That is,
Similarly one can show that .

*Step 3. *Indicating and , we have
While in (49) and employing (46) and Steps 1 and 2, we obtain . By in Definition 3, we have . Similarly one can show that , , and .

Theorem 11. *Intercalarily to the supposition of Theorem 10 assume that there exist , such that is compared with . Then for is couple common fixed point. To wit, and possess a couple common fixed point and .*

*Proof. *If is comparable to , at the time is comparable to . So if we substitute , , , and in (8), then we obtain
Therefore .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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