Abstract
In this paper we present some coupled coincidence point results for -weakly contractive mappings in the setup of partially ordered -metric spaces. Our results extend the results of Cho et al. (Fixed Point Theory Appl. 2012:8, 2012) and the results of Choudhury and Maity (Math. Comput. Model. 54:73-79, 2011). Moreover, examples of the main results are given.
MSC: 47H10, 54H25.
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1 Introduction
Existence of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [1]. Also, Bhaskar and Lakshmikantham [2] established some coupled fixed point theorems for a mixed monotone mapping in partially ordered metric spaces.
Recently, Lakshmikantham and Ćirić [3] introduced the notions of mixed g-monotone mapping and coupled coincidence point and proved some coupled coincidence point and common coupled fixed point theorems in partially ordered complete metric spaces.
Definition 1.1 [3]
Let be a partially ordered set, and let and be two mappings. 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 all , implies for any and for all , implies for any .
Definition 1.2 [2]
An element is called a coupled fixed point of the mapping if and .
Definition 1.3 [3]
An element is called
-
(1)
a coupled coincidence point of mappings and if and ;
-
(2)
a common coupled fixed point of mappings and if and .
Recently, Abbas et al. [4] introduced the concept of w-compatible mappings to obtain some coupled coincidence point results in a cone metric space.
Definition 1.4 [4]
Two mappings and are called w-compatible if , whenever and .
The concept of generalized metric space, or a G-metric space, was introduced by Mustafa and Sims [5]. Mustafa and others studied fixed point theorems for mappings satisfying different contractive conditions (see [5–22]).
Definition 1.5 (G-metric space [5])
Let X be a nonempty set, and let be a function satisfying the following properties:
-
(G1)
iff ;
-
(G2)
for all with ;
-
(G3)
for all with ;
-
(G4)
(symmetry in all three variables);
-
(G5)
for all (rectangle inequality).
Then the function G is called a G-metric on X and the pair is called a G-metric space.
Definition 1.6 [5]
Let be a G-metric space, and let be a sequence of points of X. A point is said to be the limit of the sequence if and one says that the sequence is G-convergent to x. Thus, if in a G-metric space , then for any , there exists a positive integer N such that for all .
Definition 1.7 [5]
Let be a G-metric space. A sequence is called G-Cauchy if for every , there is a positive integer N such that for all , that is, if as .
Lemma 1.8 [5]
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x.
-
(2)
as .
-
(3)
as .
Lemma 1.9 [23]
If is a G-metric space, then is a G-Cauchy sequence if and only if for every , there exists a positive integer N such that for all .
Definition 1.10 [5]
A G-metric space is said to be G-complete if every G-Cauchy sequence in is convergent in X.
Definition 1.11 [5]
Let and be two G-metric spaces. Then a function is G-continuous at a point if and only if it is G-sequentially continuous at x, that is, whenever is G-convergent to x, is -convergent to .
Definition 1.12 [23]
Let be a G-metric space. A mapping is said to be continuous at if for any two G-convergent sequences and converging to x and y, respectively, is G-convergent to .
Definition 1.13 [3]
Let X be a nonempty set. We say that the mappings and are commutative if for all .
Choudhury and Maity [23] established some coupled fixed point results for mappings with mixed monotone property in partially ordered G-metric spaces. They obtained the following results.
Theorem 1.14 ([23], Theorem 3.1)
Let be a partially ordered set, and let G be a G-metric on X such that is a complete G-metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists such that
for all and , where either or .
If there exist such that and , then F has a coupled fixed point in X, that is, there exist such that and .
Theorem 1.15 ([23], Theorem 3.2)
If in the above theorem, in place of the continuity of F, we assume the following conditions, namely,
-
(i)
if a non-decreasing sequence , then for all n, and
-
(ii)
if a non-increasing sequence , then for all n,
then F has a coupled fixed point.
The concept of an altering distance function was introduced by Khan et al. [24] as follows.
Definition 1.16 The function is called an altering distance function if the following properties are satisfied:
-
1.
ψ is continuous and non-decreasing.
-
2.
if and only if .
In [25], Cho et al. studied coupled coincidence and coupled common fixed point theorems in ordered generalized metric spaces for a nonlinear contractive condition related to a pair of altering distance functions.
Theorem 1.17 ([25], Theorem 3.1)
Let be a partially ordered set, and let be a complete G-metric space. Let and be continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there are altering distance functions ψ and φ such that
for all with and . Also, suppose that . If there exist such that and , then F and g have a coupled coincidence point.
Definition 1.18 [25]
Let be a partially ordered set, and let G be a G-metric on X. We say that is regular if the following conditions hold:
-
(i)
If is a non-decreasing sequence with , then for all .
-
(ii)
If is a non-increasing sequence with , then for all .
Theorem 1.19 ([25], Theorem 3.2)
Let be a partially ordered set, and let G be a G-metric on X such that is regular. Assume that there exist altering distance functions ψ, φ and mappings and satisfying (1.2) for all with and . Suppose also that is G-complete, F has the mixed g-monotone property and . If there exist such that and , then F and g have a coupled coincidence point.
So far, many authors have discussed fixed point results, periodic point results, coupled and tripled fixed point results and many other related topics in fixed point theory in different extensions of the concept of metric spaces such as b-metric spaces, partial metric spaces, cone metric spaces, G-metric spaces, etc. (see, e.g., [6, 14, 20, 26–33]). Motivated by the work in [34], Aghajani et al., in a submitted paper [35], extended the notion of G-metric space to the concept of -metric space (see Section 2). In this paper, we obtain some coupled coincidence point theorems for nonlinear -weakly contractive mappings in partially ordered -metric spaces. These results generalize and modify several comparable results in the literature.
2 Mathematical preliminaries
Aghajani et al. in [35] introduced the concept of generalized b-metric spaces (-metric spaces) and then they presented some basic properties of -metric spaces.
The following is their definition of -metric spaces.
Definition 2.1 [35]
Let X be a nonempty set, and let be a given real number. Suppose that a mapping satisfies:
-
() if ,
-
() for all with ,
-
() for all with ,
-
() , where p is a permutation of (symmetry),
-
() for all (rectangle inequality).
Then G is called a generalized b-metric and the pair is called a generalized b-metric space or a -metric space.
It should be noted that the class of -metric spaces is effectively larger than that of G-metric spaces given in [5]. Indeed, each G-metric space is a -metric space with (see also [36]).
The following example shows that a -metric on X need not be a G-metric on X.
Example 2.2 [35]
Let be a G-metric space, and let , where is a real number.
Note that is a -metric with . Obviously, satisfies conditions ()-() of Definition 2.1, so it suffices to show that () holds. If , then the convexity of the function () implies that . Thus, for each , we obtain
So, is a -metric with .
Also, in the above example, is not necessarily a G-metric space. For example, let and G-metric G be defined by
for all (see [5]). Then is a -metric on ℝ with , but it is not a G-metric on ℝ. To see this, let , , and . Hence, we get , , . Therefore, .
Example 2.3 Let and . We know that is a b-metric space with . Let , we show that is not a -metric space. Indeed, () is not true for , and . To see this, we have
and
So, .
However, is a -metric on ℝ with .
Now we present some definitions and propositions in a -metric space.
Definition 2.4 [35]
Let be a -metric space. Then, for any and any , the -ball with center and radius r is
For example, let and consider the -metric G defined by
for all . Then
By some straightforward calculations, we can establish the following.
Proposition 2.5 [35]
Let X be a -metric space. Then, for each , it follows that:
-
(1)
if , then ,
-
(2)
,
-
(3)
,
-
(4)
.
Definition 2.6 [35]
Let X be a -metric space. We define for all . It is easy to see that defines a b-metric d on X, which we call the b-metric associated with G.
Proposition 2.7 [35]
Let X be a -metric space. Then, for any and any , if , then there exists such that .
Proof For , see Proposition 4 in [5]. Suppose that and let . If , then we choose . If , then . Let . Since , hence, for , there exists such that or . Hence, and A is a nonempty set, then by the well-ordering principle, A has a least element m. Since , we have . Now, if , then we choose and if , we choose . □
From the above proposition, the family of all -balls
is a base of a topology on X, which we call -metric topology.
Now, we generalize Proposition 5 in [5] for a -metric space as follows.
Proposition 2.8 [35]
Let X be a -metric space. Then, for any and , we have
Thus, every -metric space is topologically equivalent to a b-metric space. This allows us to readily transport many concepts and results from b-metric spaces into the -metric space setting.
Definition 2.9 [35]
Let X be a -metric space. A sequence in X is said to be:
-
(1)
-Cauchy if for each there exists a positive integer such that for all , ;
-
(2)
-convergent to a point if for each there exists a positive integer such that for all , .
Using the above definitions, we can easily prove the following two propositions.
Proposition 2.10 [35]
Let X be a -metric space. Then the following are equivalent:
-
(1)
The sequence is -Cauchy.
-
(2)
For any , there exists such that for all .
Proposition 2.11 [35]
Let X be a -metric space. The following are equivalent:
-
(1)
is -convergent to x.
-
(2)
as .
-
(3)
as .
Definition 2.12 [35]
A -metric space X is called -complete if every -Cauchy sequence is -convergent in X.
Definition 2.13 Let and be two -metric spaces. Then a function is -continuous at a point if and only if it is -sequentially continuous at x, that is, whenever is -convergent to x, is -convergent to .
Definition 2.14 Let be a -metric space. A mapping is said to be continuous if for any two -convergent sequences and converging to x and y, respectively, is -convergent to .
Mustafa and Sims proved that each G-metric function is jointly continuous in all three of its variables (see Proposition 8 in [5]). But, in general, a -metric function for is not jointly continuous in all its variables. Now, we present an example of a discontinuous -metric.
Example 2.15 Let and let be defined by
Then it is easy to see that for all , we have
Thus, is a b-metric space with (see corrected Example 3 from [37]).
Let . It is easy to see that G is a -metric with . Now, we show that is not a continuous function. Take and . Then we have , and . Also,
and
On the other hand,
and
Hence, .
So, from the above discussion, we need the following simple lemma about the -convergent sequences in the proof of our main result.
Lemma 2.16 Let be a -metric space with , and suppose that , and are -convergent to x, y and z, respectively. Then we have
In particular, if , then we have .
Proof Using the triangle inequality in a -metric space, it is easy to see that
and
Taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the desired result. □
3 Main results
Our first result is the following.
Theorem 3.1 Let be a partially ordered set, and let G be a -metric on X such that is a complete -metric space. Let and be two mappings such that
for every pair such that and , or and , where are altering distance functions.
Also, suppose that:
-
1.
.
-
2.
F has the mixed g-monotone property.
-
3.
F is continuous.
-
4.
g is continuous and commutes with F.
If there exist such that and , then F and g have a coupled coincidence point in X.
Proof Let be such that and . Since , we can choose such that and . Then, and . Since F has the mixed g-monotone property, we have and , that is, and . In this way, we construct the sequences and as and for all , inductively.
One can easily show that for all , and .
We complete the proof in three steps.
Step I. Let
we shall prove that .
Since and , using (3.1) we obtain that
If for an , , then the conclusion of the theorem follows. So, we assume that
Let, for some n, . So, from (3.2) as ψ is non-decreasing, we have
that is, . By our assumptions, we have , which contradicts (3.3). Therefore, for all , we deduce that , that is, is a non-increasing sequence of nonnegative real numbers. Thus, there exists such that .
Letting in (3.2), we get that
So, . Thus,
Step II. We shall show that and are -Cauchy sequences in X. So, we shall show that for every , there exists such that for all ,
Suppose that the above statement is false. Then there exists , for which we can find subsequences and of and and of such that
Further, corresponding to we can choose in such a way that it is the smallest integer with satisfying (3.6). So,
From the rectangle inequality we have
and
So,
If , as , from (3.6) and (3.7) we conclude that
On the other hand, we have
Similarly,
So, we have
Taking the upper limit as in the above inequality and using (3.6), we obtain
Consequently, from (3.4) we get
As and , putting , , , , and in (3.1), for all , we have
Taking the upper limit as in the above inequality and using (3.7) and (3.9), we have
which implies that
or, equivalently, , which is a contradiction to (3.8). Consequently, and are -Cauchy.
Step III. We shall show that F and g have a coupled coincidence point.
Since X is -complete and is -Cauchy, there exists such that
Similarly, there exists such that
Now, we prove that is a coupled coincidence point of F and g.
The continuity of g and Lemma 2.16 yield that
Hence,
and, similarly, we get
Since and , the commutativity of F and g yields that
and
From the continuity of F, is -convergent to and is -convergent to . By uniqueness of the limit, we have and . That is, g and F have a coupled coincidence point. □
In the following theorem, we omit the continuity and commutativity assumptions of g and F.
Theorem 3.2 Let be a partially ordered set, and let G be a -metric on X such that is a regular -metric space. Suppose that and are two mappings satisfying (3.1) for every pair such that and , or and , where ψ and φ are the same as in Theorem 3.1.
Let , is a -complete subset of X and F has the mixed g-monotone property.
If there exist such that and , then F and g have a coupled coincidence point in X.
Moreover, if and are comparable, then , and if F and g are w-compatible, then F and g have a coupled coincidence point of the form .
Proof Following the proof of the previous theorem, as is a -complete subset of X and , there exist such that
Now, we prove that and .
Since is non-decreasing and is non-increasing, from regularity of X we have and for all .
Using (3.1), we have
In the above inequality, by using Lemma 2.16, if , we have
and hence, and .
Now, let . Then for all . We shall show that .
From (3.1), we have
Therefore, . Hence, we get that , and this means that .
Now, let . Since F and g are w-compatible, then . Thus, F and g have a coupled coincidence point of the form . □
Remark 3.3 In Theorems 3.1 and 3.2, we have extended the results of Cho et al. [25] (Theorems 1.17 and 1.19).
Note that if is a partially ordered set, then we can endow with the following partial order relation:
for all [3].
In the following theorem, we give a sufficient condition for the uniqueness of the common coupled fixed point. Similar conditions were introduced by many authors (see, e.g., [2, 3, 9, 20, 38–45]).
Theorem 3.4 Let all the conditions of Theorem 3.1 be fulfilled, and let the following condition hold:
For arbitrary two points , , there exists such that is comparable with and .
Then F and g have a unique common coupled fixed point.
Proof Let and be two coupled coincidence points of F and g, i.e.,
and
We shall show that and .
Suppose that and are not comparable. Choose an element such that is comparable with and .
Let , and choose so that and . Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences and such that and . Since and are comparable, we may assume that . Then and . Using the mathematical induction, it is easy to prove that and for all .
Let . We shall show that . First, assume that for an .
Applying (3.1), as and , one obtains that
So, from the properties of ψ and φ, we deduce that . Repeating this process, we can show that for all . So, .
Now, let for all n, and let for some n.
As ψ is an altering distance function, from (3.10)
This implies that , which is a contradiction.
Hence, for all . Now, if we proceed as in Theorem 3.1, we can show that
So, and .
Similarly, we can show that
that is, and . Finally, since the limit is unique, and .
Since and , by the commutativity of F and g, we have and . Let and . Then and . Thus, is another coupled coincidence point of F and g. Then and . Therefore, is a coupled common fixed point of F and g.
To prove the uniqueness of a coupled common fixed point, assume that is another coupled common fixed point of F and g. Then and . Since is a coupled coincidence point of F and g, we have and . Thus, and . Hence, the coupled common fixed point is unique. □
The following corollary can be deduced from our previous obtained results.
Theorem 3.5 Let be a partially ordered set, and let be a complete -metric space. Let be a mapping with the mixed monotone property such that
for every pair such that and , or and , where are altering distance functions.
Also, suppose that either
-
(a)
F is continuous, or
-
(b)
X is regular.
If there exist such that and , then F has a coupled fixed point in X.
Proof If F satisfies (3.11), then F satisfies (3.1). So, the result follows from Theorems 3.1 and 3.2. □
In Theorems 3.1 and 3.2, if we take and for all , where , we obtain the following result.
Theorem 3.6 Let be a partially ordered set, and let be a complete -metric space. Let be a mapping having the mixed monotone property such that
for every pair such that and , or and , where .
Also, suppose that either
-
(a)
F is continuous, or
-
(b)
X is regular.
If there exist such that and , then F has a coupled fixed point in X.
The following corollary is an extension of the results by Choudhury and Maity (Theorems 1.14 and 1.15).
Theorem 3.7 Let be a partially ordered set, and let be a complete -metric space. Let be a mapping with the mixed monotone property such that
for every pair such that and , or and .
Also, suppose that either
-
(a)
F is continuous, or
-
(b)
X is regular.
If there exist such that and , then F has a coupled fixed point in X.
Proof If F satisfies (3.12), then F satisfies (3.11). □
Now, we present an example to illustrate Theorem 3.1.
Example 3.8 Let be endowed with the usual ordering, and let -metric on X be given by , where .
Define and as
and for all .
Define by , , where .
Let be such that and . Now, we have
Obviously, all the conditions of Theorem 3.1 are satisfied. Moreover, is a coupled coincidence point of F and g.
4 Applications
In this section, we obtain some coupled coincidence point theorems for mappings satisfying some contractive conditions of integral type in an ordered complete -metric space.
Denote by Λ the set of all functions verifying the following conditions:
-
(I)
μ is a positive Lebesgue integrable mapping on each compact subset of .
-
(II)
For all , .
Corollary 4.1 Replace the contractive condition (3.1) of Theorem 3.1 by the following condition:
There exists such that
If the other conditions of Theorem 3.1 hold, then F and g have a coupled coincidence point.
Proof Consider the function . Then (4.1) becomes
Taking and and applying Theorem 3.1, we obtain the proof. □
Corollary 4.2 Substitute the contractive condition (3.1) of Theorem 3.1 by the following condition:
There exists such that
Then F and g have a coupled coincidence point if the other conditions of Theorem 3.1 hold.
Proof Again, as in Corollary 4.1, define the function . Then (4.2) changes to
Now, if we define and and apply Theorem 3.1, then the proof is obtained. □
As in [46], let be fixed. Let be a family of N functions which belong to Λ. For all , we define
We have the following result.
Corollary 4.3 Replace the inequality (3.1) of Theorem 3.1 by the following condition:
Assume further that the other conditions of Theorem 3.1 are also satisfied. Then F and g have a coupled coincidence point.
Proof Consider and . Then the above inequality becomes
Now, applying Theorem 3.1, we obtain the desired result. □
Another consequence of our theorems is the following result.
Corollary 4.4 Replace the contractive condition (3.1) of Theorem 3.1 by the following condition:
There exist such that
Let the other conditions of Theorem 3.1 be satisfied. Then F and g have a coupled coincidence point.
5 Conclusions
We saw that the results of Cho et al. [25] and the results of Choudhury and Maity [23] also hold in the context of -metric spaces with some simple changes in the contractive conditions. The most difference between the concepts of G-metric and -metric is that the -metric function is not necessarily continuous in all its three variables (see, Example 2.15). On the other hand, by a simple but essential lemma (Lemma 2.16), we can prove many fixed point results in this new structure.
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Mustafa, Z., Roshan, J.R. & Parvaneh, V. Coupled coincidence point results for -weakly contractive mappings in partially ordered -metric spaces. Fixed Point Theory Appl 2013, 206 (2013). https://doi.org/10.1186/1687-1812-2013-206
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DOI: https://doi.org/10.1186/1687-1812-2013-206