Abstract
In this paper, we present some coupled coincidence point results for mixed gmonotone 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 gmonotone property1 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.[111] 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 gmonotone mapping and proved coupled coincidence point results for two mappings F and g, where F has the mixed gmonotone 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 gmonotone 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 nondecreasing if for , , we have . Similarly, f is said to be nonincreasing if for , , we have . We also recall the used definitions in the present work.
Definition 1.1[11] (Mixed gmonotone property)
Let be a partially ordered set, and . We say that the mapping F has the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument. That is, for any ,
and
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 nonempty 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 nondecreasing.
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 mixedgmonotone property onXsuch that
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 nondecreasing sequence, thenfor alln,
(ii) if a nonincreasing sequence, thenfor alln.
If there existwithand, thenFandghave a coupled coincidence point.
Proof By using , we construct sequences and as follows:
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 gmonotone property, then we get
and
Thus the claim holds for and by the mathematical induction our claim is proved.
In fact, using (3), and , we get
Since ϕ is nonnegative, we have
and since φ is nondecreasing, we have
In the same way, we get the following:
and hence
Using (6) and (8), we have
From the last inequality, we notice that the sequence is nonnegative decreasing. This implies that there exists such that
It is easily seen that if is nondecreasing, we have for for . Using this, (5) and (7), we obtain
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,
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
Further, we can choose corresponding to in such a way that it is the smallest integer with and satisfying (12). Then
and also we get
Combining (14) and (15), we obtain
Using the triangular inequality and (13), we get
and
Using (12), (17) and (18), we have
Letting in the last inequality and using (11), we have
Similarly, using the triangular inequality and (13), we have
and
Combining (20) and (21), we obtain
Using the triangular inequality, we have
and
Using the two last inequalities and (12), we have
Using (22) and (23), we get
Letting in the last inequality and using (11), we obtain
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
Since F and g are compatible mappings, we have
and
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 nondecreasing and and is nonincreasing and , by our assumption, and for every .
Applying (3), we have
and as φ is nondecreasing, we obtain
Using the triangular inequality and (28), we get
As and , taking in the last inequality, we have
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 nondecreasing sequence, thenfor alln,
(ii) if a nonincreasing 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 nondecreasing sequence, thenfor alln,
(ii) if a nonincreasing 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 gmonotone property and since g is monotone increasing, we get
and this proves our claim.
Now, since and , using (3), we get
In the same way, we have
Using (29) and (30) and the fact that ϕ is nondecreasing, we get
Using the last inequality and the fact that φ is nondecreasing, we have
Thus the sequence is decreasing and nonnegative, and hence, for certain ,
Using (32) and letting in (31), we have
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 gmonotone property of F, we have
and since g is nondecreasing, this proves our claim.
and since φ is nondecreasing, we get
We notice that the sequence is decreasing. Thus, for certain . Hence,
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
Then, F satisfies the mixed gmonotone property.
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 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. (130037D1433). 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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