Research

# Some coupled coincidence point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions

Saud M Alsulami

Author Affiliations

Department of Mathematics, King Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia

Fixed Point Theory and Applications 2013, 2013:194  doi:10.1186/1687-1812-2013-194

 Received: 1 April 2013 Accepted: 4 July 2013 Published: 22 July 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we present some coupled coincidence point results for mixed g-monotone mappings in partially ordered complete metric spaces involving altering distance functions. Moreover, we present an example to illustrate our main result. Our results extend some results in the field.

MSC: 47H09, 47H10, 49M05.

##### Keywords:
coupled coincidence points; partially metric spaces; contractive mappings; mixed g-monotone property

### 1 Introduction and preliminaries

The existence of a fixed point for contractive mappings in partially ordered metric spaces has attracted the attention of many mathematicians (cf.[1-11] and the references therein). In [3], Bhaskar and Lakshmikantham introduced the notion of a mixed monotone mapping and proved some coupled fixed point theorems for the mixed monotone mapping. Afterwards, Lakshmikantham and Ciric in [11] introduced the concept of a mixed g-monotone mapping and proved coupled coincidence point results for two mappings F and g, where F has the mixed g-monotone property and the functions F and g commute. It is well known that the concept of commuting has been weakened in various directions. One such notion which is weaker than commuting is the concept of compatibility introduced by Jungck [7]. In [5], Choudhury and Kundu defined the concept of compatibility of F and g. The purpose of this paper is to present some coupled coincidence point theorems for a mixed g-monotone mapping in the context of complete metric spaces endowed with a partial order by using altering distance functions which extend some results of [6]. We also present an example which illustrates the results.

Recall that if is a partially ordered set, then f is said to be non-decreasing if for , , we have . Similarly, f is said to be non-increasing if for , , we have . We also recall the used definitions in the present work.

Definition 1.1[11] (Mixed g-monotone property)

Let be a partially ordered set, and . We say that the mapping F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument. That is, for any ,

(1)

and

(2)

Definition 1.2[11] (Coupled coincidence fixed point)

Let , and . We say that is a coupled coincidence point of F and g if and for .

Definition 1.3[11]

Let X be a non-empty set and let and . We say F and g are commutative if, for all ,

Definition 1.4[5]

The mappings F and g, where and , are said to be compatible if

and

whenever and are sequences in X such that and for all .

Definition 1.5 (Altering distance function)

An altering distance function is a function satisfying

1. ψ is continuous and non-decreasing.

2. if and only if .

### 2 Existence of coupled coincidence points

Let be a partially ordered set and suppose that there exists a metric d in X such that is a complete metric space. Also, let φ and ϕ be altering distance functions. Now, we are in a position to state our main theorem.

Theorem 2.1Letbe a mapping having the mixedg-monotone property onXsuch that

(3)

for allwithand. Suppose that, gis continuous, monotone increasing and suppose also thatFandgare compatible mappings. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there existwithand, thenFandghave a coupled coincidence point.

Proof By using , we construct sequences and as follows:

(4)

We are going to divide the proof into several steps in order to make it easy to read.

Step 1. We will show that and for .

We use the mathematical induction to show that. From the assumption of the theorem, it follows that and , so our claim is satisfied for . Now, suppose that our claim holds for some fixed . Since , and F has the mixed g-monotone property, then we get

and

Thus the claim holds for and by the mathematical induction our claim is proved.

Step 2. We will show that .

In fact, using (3), and , we get

(5)

Since ϕ is non-negative, we have

and since φ is non-decreasing, we have

(6)

In the same way, we get the following:

(7)

and hence

(8)

Using (6) and (8), we have

From the last inequality, we notice that the sequence is non-negative decreasing. This implies that there exists such that

(9)

It is easily seen that if is non-decreasing, we have for for . Using this, (5) and (7), we obtain

(10)

Letting in the last inequality and using (6), we have

and this implies . Thus, using the fact that ϕ is an altering distance function, we have . Therefore,

(11)

Hence, and this completes the proof of our claim.

Step 3. We will prove that and are Cauchy sequences.

Suppose that one of the sequences or is not a Cauchy sequence. This implies that or , and hence

This means that there exists , for which we can find subsequences and with , such that

(12)

Further, we can choose corresponding to in such a way that it is the smallest integer with and satisfying (12). Then

(13)

Using (3), and , we get

(14)

and also we get

(15)

Combining (14) and (15), we obtain

(16)

Using the triangular inequality and (13), we get

(17)

and

(18)

Using (12), (17) and (18), we have

Letting in the last inequality and using (11), we have

(19)

Similarly, using the triangular inequality and (13), we have

(20)

and

(21)

Combining (20) and (21), we obtain

(22)

Using the triangular inequality, we have

and

Using the two last inequalities and (12), we have

(23)

Using (22) and (23), we get

Letting in the last inequality and using (11), we obtain

(24)

Finally, letting in (15) and using (18), (23) and the continuity of φ and ϕ, we have

and, consequently, . Since ϕ is an altering distance function, we get , and this is a contradiction. This proves our claim.

Since X is a complete metric space, there exist such that

(25)

Since F and g are compatible mappings, we have

(26)

and

(27)

We now show that and . Suppose that assumption (a) holds. For all , we have

Taking the limit as , using (3), (25), (26) and the fact that F and g are continuous, we have . Similarly, using (3), (25), (27) and the fact that F and g are continuous, we have . Hence, we get

Finally, suppose that (b) holds. In fact, since is non-decreasing and and is non-increasing and , by our assumption, and for every .

Applying (3), we have

and as φ is non-decreasing, we obtain

(28)

Using the triangular inequality and (28), we get

As and , taking in the last inequality, we have

and, consequently, .

Using a similar argument, it can be proved that and this completes the proof. □

Corollary 2.1[6]

Letbe a partially ordered set and suppose that there exists a metricdinXsuch thatis a complete metric space. Letbe a mapping having the mixed monotone property onXsuch that

for allwithand, whereφandϕare altering distance functions. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there existwithand, thenFhas a coupled fixed point.

Corollary 2.2[3]

Letbe a partially ordered set and suppose that there exists a metricdinXsuch thatis a complete metric space. Letbe a mapping having the mixed monotone property onXsuch that

for allwithand. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there existwithand, thenFhas a coupled fixed point.

Proof Let and and g is the identity function. Then applying Theorem 2.1, we get Corollary 2.2. □

### 3 Uniqueness of the coupled coincidence point

In this section, we prove the uniqueness of the coupled coincidence point. Note that if is a partially ordered set, then we endow the product with the following partial order relation, for all ,

Theorem 3.1In addition to the hypotheses of Theorem 2.1, suppose that for every, in, there exists ainthat is comparable toand, thenFandghave a unique coupled coincidence point.

Proof Suppose that and are coupled coincidence points of F, that is, , , and .

Let be an element of comparable to and . Suppose that (the proof is similar in the other case).

We construct the sequences and as follows:

We claim that for each . In fact, we will use mathematical induction.

For , as , this means and and, consequently, . Suppose that , then since F has the mixed g-monotone property and since g is monotone increasing, we get

and this proves our claim.

Now, since and , using (3), we get

(29)

In the same way, we have

(30)

Using (29) and (30) and the fact that ϕ is non-decreasing, we get

(31)

Using the last inequality and the fact that φ is non-decreasing, we have

Thus the sequence is decreasing and non-negative, and hence, for certain ,

(32)

Using (32) and letting in (31), we have

This gives and hence .

Finally, since , we have and . Using a similar argument for , we can get and , and the uniqueness of the limit gives and . This completes the proof. □

Theorem 3.2Under the assumptions of Theorem 2.1, suppose thatandare comparable, then the coupled coincidence pointsatisfies.

Proof Assume (a similar argument applies to ).

We claim that for all n, where and .

Obviously, the inequality is satisfied for . Suppose . Using the mixed g-monotone property of F, we have

and since g is non-decreasing, this proves our claim.

Now, using (3) and , we get

(33)

and since φ is non-decreasing, we get

We notice that the sequence is decreasing. Thus, for certain . Hence,

and this gives us .

Since , and , we have

and thus . This completes the proof. □

### 4 Example

The following example illustrates our main result.

Example 4.1 Let . Then is a partially ordered set with the natural ordering of real numbers. Let

Then is a complete metric space. Let be defined as

and let be defined as

Then, F satisfies the mixed g-monotone property.

Let be defined as

and let be defined as

Let and be two sequences in X such that , , and . Then, obviously, and . Now, for all ,

and

Then it follows that

and

Hence, the mappings F and g are compatible in X. Also, and () are two points in X such that

and

We next verify the contraction of Theorem 2.1. We take such that and , that is, and .

We consider the following cases.

Case 1. , . Then

Case 2. , Then

Case 3. and . Then

Case 4. and with and . Then and , that is,

Obviously, the contraction of Theorem 2.1 is satisfied.

### Competing interests

The author declares that he has no competing interests.

### Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-037-D1433). The author, therefore, acknowledges with thanks DSR technical and financial support. Also, the author would like to thank Prof. Abdullah Alotaibi for useful discussion on this paper. Moreover, many thanks to the referees and the editor for their helpful comments.

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