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Coupled fixed point theorems for generalized contractive mappings in partially ordered G-metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 172 (2012)
Abstract
In this paper, we establish some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings having the mixed monotone property in partially ordered G-metric spaces. The results on fixed point theorems are generalizations of the recent results of Choudhury and Maity (Math. Comput. Model. 54:73-79, 2011) and Luong and Thuan (Math. Comput. Model. 55:1601-1609, 2012).
1 Introduction and preliminaries
One of the simplest and the most useful results in the fixed point theory is the Banach-Caccioppoli contraction [1] mapping principle, a power tool in analysis. This principle has been generalized in different directions in different spaces by mathematicians over the years (see [2–10] and references mentioned therein). On the other hand, fixed point theory has received much attention in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [11] and they presented applications of their results to matrix equations. Subsequently, Nieto and RodrÃguez-López [12] extended the results in [11] for non-decreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions (see also [13–19]).
In recent times, fixed point theory has developed rapidly in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Some of these works are noted in [13, 17, 20, 21]. Bhaskar and Lakshmikantham [21] introduced the concept of a coupled fixed point and the mixed monotone property. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. After the publication of this work, several coupled fixed point and coincidence point results have appeared in the recent literature. Works noted in [22–25] are some examples of these works.
Mustafa and Sims [26, 27] introduced a new structure of generalized metric spaces, which are called G-metric spaces, as a generalization of metric spaces to develop and introduce a new fixed point theory for various mappings in this new structure. Later, several fixed point theorems in G-metric spaces were obtained by [28–34].
To fix the context in which we are placing our results, recall the following notions. Throughout this article, denotes a partially ordered set with the partial order ⪯. By , we mean but . A mapping is said to be non-decreasing (non-increasing) if for all , implies (, respectively).
The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham [21].
Definition 1.1 [21]
Let be a partial ordered set. A mapping is said to be have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,
and
The following concepts were introduced in [35].
Definition 1.2 [35]
Let be a partial ordered set and and be two mappings. We say that F has the mixed g-monotone property if is g-monotone non-decreasing in x and it is g-monotone non-increasing in y, that is, for any
and
Let and be mappings. An element is said to be:
-
(i)
a coupled fixed point of a mapping F if
-
(ii)
a coupled coincidence point of mapping F and g if
-
(iii)
a coupled common fixed point of mappings F and g if
Consistent with Mustafa and Sims [26, 27], the following definitions and results will be needed in the sequel.
Definition 1.4 (G-metric space [27])
Let X be a non-empty set. Let be a function satisfying the following properties:
(G1) if ;
(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 1.5 [27]
Let X be a G-metric space, and let be a sequence of points of X, a point is said to be the limit of a sequence if as and sequence is said to be G-convergent to x.
From this definition, we obtain that if in a G-metric space X, then for any , there exists a positive integer N such that for all .
It has been shown in [27] that the G-metric induces a Hausdorff topology and the convergence described in the above definition is relative to this topology. So, a sequence can converge, at the most, to one point.
Definition 1.6 [27]
Let X 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 .
We next state the following lemmas.
Lemma 1.7 [27]
If X is a G-metric space, then the following are equivalent:
-
(1)
is G-convergent to x.
-
(2)
as .
-
(3)
as .
-
(4)
as .
Lemma 1.8 [27]
If X is a G-metric space, then the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
for every , there exists a positive integer N such that , for all .
Lemma 1.9 [27]
If X is a G-metric space, then for all .
Lemma 1.10 If X is a G-metric space, then for all .
Definition 1.11 [27]
Let , be two generalized metric spaces. A mapping 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 [27]
A G-metric space X is called a symmetric G-metric space if
for all .
Definition 1.13 [27]
A G-metric space X is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in X is convergent in X.
Definition 1.14 Let X be a G-metric space. A mapping is said to be continuous if for any two G-convergent sequences and converging to x and y, respectively, is G-convergent to .
Definition 1.15 Let X be a non-empty set and and two mappings. We say F and g are commutative (or that F and g commute) if
Recently, Choudhury and Maity [36] studied necessary conditions for the existence of a coupled fixed point in partially ordered G-metric spaces. They obtained the following interesting result.
Theorem 1.16 [36]
Let be a partially ordered set such that X is a complete G-metric space and be a mapping having the mixed monotone property on X. Suppose there exists such that
for all for which and , where either or . If there exists such that
and either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n,
then F has a coupled fixed point.
Let Θ denote the class of all functions satisfying the following condition:
for all with .
Remark 1.17 If the function satisfies (1.4), then, for any with either or , . Indeed, suppose that , we have . Taking and for all , we have, by (1.4), that
Example 1.18 The following are some examples of φ, for all ,
-
(1)
for ;
-
(2)
for ;
-
(3)
for some .
Using basically these concepts, Luong and Thuan [37] proved the following coupled fixed point theorem for nonlinear contractive mappings having the mixed monotone property in partially ordered G-metric spaces.
Theorem 1.19 [[37], Theorem 2.1]
Let be a partially ordered set and suppose that there exists a G-metric G on X such that is a complete G-metric space. Let be a mapping having the mixed monotone property on X. Suppose that there exists such that
for all and . Suppose that either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n.
If there exist such that and , then F has a coupled fixed point in X.
Starting from the results in Choudhury and Maity [36] and Luong and Thuan [37], our main aim in this paper is to obtain more general coincidence point theorems and coupled common fixed point theorems for mixed monotone operators satisfying a contractive condition which is significantly more general that the corresponding conditions (1.3) and (1.5) in [36] and [37], respectively, thus extending many other related results in literature. We also provide an illustrative example in support of our results.
2 Coupled coincidence points
The first main result in this paper is the following coincidence point theorem which generalizes [[36], Theorem 3.1] and [[37], Theorem 2.1].
Theorem 2.1 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a mapping and be a mapping having the mixed g-monotone property on X. Suppose that there exists such that
for all for which and where
If there exists such that
and suppose , g is continuous and commutes with F, and also suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n,
then F and g have a coupled coincidence point, that is, there exists such that and .
Proof Let such that and . Since , we can choose such that and . Again since , we can choose such that and . Continuing this process, we can construct sequences and in X such that
Next, we show that
Since and , therefore, (2.3) holds for . Next, suppose that (2.3) holds for some fixed , that is,
Since F has the mixed g-monotone property, from (2.4) and (1.1), we have
for all , and from (2.4) and (1.2), we have
for all . If we take and in (2.5), then we obtain
If we take and in (2.6), then
Now, from (2.7) and (2.8), we have
Therefore, by the mathematical induction, we conclude that (2.3) holds for all . Since and for all so from (2.1), we have
Setting
and
we have, by (2.10), that
As for all , we have
Then the sequence is decreasing. Therefore, there exists such that
Now, we show that . Suppose, to contrary, that . From (2.12), the sequences and have convergent subsequences and , respectively. Assume that
and
which gives that . From (2.11), we have
Then taking the limit as in the above inequality, we obtain
which is a contradiction. Thus ; that is,
Next, we show that and are G-Cauchy sequences. On the contrary, assume that at least one of or is not a G-Cauchy sequence. By Lemma 1.8, there is an for which we can find subsequences , of and , of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (2.16). Then
By Lemma 1.10, we have
and
In view of (2.16)-(2.19), we have
Then letting in the last inequality and using (2.15), we have
By Lemma 1.9 and Lemma 1.10, we have
and
It follows from (2.21) and (2.22) that
Since , we get
and also, from (2.1),
From (2.23) and (2.24), we have
This implies that
From (2.20), the sequences and have subsequences converging to, say, and , respectively, and . By passing to subsequences, we may assume that
Taking in (2.25) and using (2.15), we have
which is a contradiction. Therefore, and are G-Cauchy sequences. By G-completeness of X, there exists such that
This together with the continuity of g implies that
Now, suppose that assumption (a) holds. From (2.2) and the commutativity of F and g, we obtain
Similarly, we have
Hence, is a coupled coincidence point of F and g.
Finally, suppose that assumption (b) holds. Since is non-decreasing satisfying and is non-increasing satisfying , we have
Using the rectangle inequality and (2.1), we get
Letting in the above inequality, we obtain that
which gives that ; that is, and . Therefore, is a coupled coincidence point of F and g. The proof is complete. □
Setting in Theorem 2.1, we obtain the following new result:
Theorem 2.2 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a mapping having the mixed monotone property on X. Suppose that there exists such that
for all and . Suppose that either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n.
If there exists such that
then F has a coupled fixed point in X.
Remark 2.3 Theorem 2.2 is more general than [[37], Theorem 2.1] since the contractive condition (2.29) is weaker than (1.5), a fact which is clearly illustrated by the following example.
Example 2.4 Let be a set endowed with order . Let the mapping be defined by
for all . Then G is a G-metric on X. Define the mapping by
Then the following properties hold:
-
(1)
F is mixed monotone;
-
(2)
F satisfies condition (2.29) but F does not satisfy condition (1.5).
Indeed, we first show that F does not satisfy condition (1.5). Assume to the contrary, that there exists , such that (1.5) holds. This means
Setting and , by Remark 1.17, we get
which gives a contradiction. Hence, F does not satisfy condition (1.5). Now, we prove that (2.29) holds. Indeed, we have
By the above, we get exactly (2.29) with .
By Theorem 2.1, we also obtain the following new result for the coupled coincidence point theorem for mixed g-monotone operators F satisfying a contractive condition.
Theorem 2.5 Let be a partially ordered set and suppose that there exists a G-metric G on X such that is a complete G-metric space. Let , so that F is a mapping having the mixed g-monotone property on X. Suppose that there exists such that
for all and . Suppose that either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n.
If there exists such that
then F and g have a coupled coincidence point.
Let Ψ denote the class of all functions satisfying
Corollary 2.6 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a mapping and be a mapping having the mixed g-monotone property on X. Suppose that there exists such that
for all for which and where
If there exists such that
and either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n,
then F and g have a coupled coincidence point.
Proof In Theorem 2.1, taking for all , we get the desired results. □
Corollary 2.7 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a mapping and be a mapping having the mixed g-monotone property on X. Suppose that there exists such that
for all for which and where
If there exists such that
and either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-decreasing sequence is such that , then for all n,
then F and g have a coupled coincidence point.
Proof In Theorem 2.1, taking for all , we get the desired results. □
3 Coupled common fixed point
Now, we shall prove the existence and uniqueness theorem of a coupled common fixed point. If is a partially ordered set, we endow the product set with the partial order relation:
Theorem 3.1 In addition to the hypotheses of Theorem 2.1, suppose that for all , there exists such that is comparable with and . Then F and g have a unique coupled common fixed point.
Proof From Theorem 2.1, the set of coupled coincidences is non-empty. Assume that and are coupled coincidence points of F and g. We shall show that
By assumption, there exists such that is comparable with and . Putting , and choosing such that
Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences and in X by
Since and are comparable, without restriction to the generality, we can assume that
and
This actually means that
and
Using that F is a mixed g-monotone mapping, we can inductively show that
and
Thus, from (2.1), we get
which implies that
that is, the sequence is decreasing. Therefore, there exists such that
We shall show that . Suppose, to the contrary, that . Therefore, and have subsequences converging to , , respectively, with
Taking the limit up to subsequences as in (3.2), we have
which is a contradiction. Thus, ; that is,
which implies that
Similarly, one can show that
Therefore, from (3.3), (3.4) and the uniqueness of the limit, we get and . So, (3.1) holds. Since and , by commutativity of F and g, we have
Denote and , then by (3.5), we get
Thus, is a coincidence point. Then from (3.1) with and , we have and , that is,
From (3.6) and (3.7), we get
Then, is a coupled common fixed point of F and g. To prove the uniqueness, assume that is another coupled common fixed point. Then by (3.1), we have and . The proof is complete. □
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Acknowledgements
The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. Finally, the first author is supported by the ‘Centre of Excellence in Mathematics’ under the Commission on Higher Education, Ministry of Education, Thailand.
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Wangkeeree, R., Bantaojai, T. Coupled fixed point theorems for generalized contractive mappings in partially ordered G-metric spaces. Fixed Point Theory Appl 2012, 172 (2012). https://doi.org/10.1186/1687-1812-2012-172
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DOI: https://doi.org/10.1186/1687-1812-2012-172