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Coupled common fixed point theorems for a pair of commuting mappings in partially ordered G-metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 64 (2013)
Abstract
In this paper, we introduce a new contractive condition for a pair of commuting mappings in partially ordered G-metric spaces. Some new coupled coincidence point and coupled common fixed point theorems are obtained. An example is given to support the usability of our results. The results presented in this paper generalize and improve the corresponding results of Nashine and Shatanawi from partially ordered metric spaces to partially ordered G-metric spaces.
MSC:47H10, 54H25, 54E50.
1 Introduction and preliminaries
In 2004, Ran and Reurings [1] showed the existence of fixed points of nonlinear contraction mappings in metric spaces endowed with a partial ordering and presented applications of their results to matrix equations. Subsequently, Nieto and Rodríguez-López [2, 3] extended the corresponding results in [1]. They proved the existence theorems of a unique fixed point for mappings in partially ordered sets and gave some applications to the existence of a unique solution for a first-order ordinary differential equation with periodic boundary conditions.
In 2005, the well-known Tarski theorem [4] was used in [5] to study the existence of solutions for fuzzy equations and to prove existence theorems for fuzzy differential equations.
It is well known that fixed point theory in partially ordered metric spaces as one of the most important tools of nonlinear analysis has been widely applied to matrix equations (for more details, see [1] and the references therein), ordinary differential equations (for more details, see [2, 3, 6] and [7] and the references therein), fuzzy differential equations (for more details, see [5] and the references therein), integral equations (for more details, see [8] and [9] and the references therein) and intermediate value theorems (for more details, see [10] and the references therein). For more results on partially ordered metric spaces, one refers to the results in [11–21].
In 2006, Bhaskar and Lakshmikantham [22] introduced the notion of a coupled fixed point and proved some interesting coupled fixed point theorems for the mappings satisfying a mixed monotone property; while Lakshmikantham and Ćirić [23] introduced the concept of a mixed g-monotone mapping and proved coupled coincidence and coupled common fixed point theorems that extended the theorems due to Bhaskar and Lakshmikantham [22]. Subsequently, many authors obtained many coupled coincidence and coupled fixed point theorems in ordered metric spaces; see [24–29] and the references therein.
In 2006, Mustafa and Sims [30] introduced the notion of generalized metric spaces or simply G-metric spaces as a generalization of the concept of a metric space. Based on the concept of a G-metric space, many authors obtained many fixed point and common fixed point theorems for the mappings satisfying different contractive conditions; see [31–42] for more details. Fixed point problems also have been considered in partially ordered G-metric spaces; see [43–55] and the references therein for more details.
Inspired and motivated by the facts said above, in this paper we establish coupled coincidence point and coupled common fixed point theorems for a pair of commuting mappings in partially ordered G-metric spaces. An example is given to support the usability of our results. The results presented in this paper generalize and improve the corresponding results of Nashine and Shatanawi [29] from partially ordered metric spaces to partially ordered G-metric spaces.
First, we present some known definitions and propositions.
Definition 1.1 [30]
Let X be a nonempty set, and let be a function satisfying the following axioms:
-
(G1)
if ;
-
(G2)
for all with ;
-
(G3)
for all with z≠ y;
-
(G4)
(symmetry in all three variables);
-
(G5)
for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X and the pair is called a G-metric space.
It is known that the function on a G-metric space X is jointly continuous in all three of its variables, and if and only if ; see [30] and the references therein for more details.
Definition 1.2 [30]
Let be a G-metric space, and let be a sequence of points in X. A point x in X 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 such that for all .
Proposition 1.1 [30]
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as ;
-
(4)
as .
Definition 1.3 [30]
Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for each , there exists a positive integer such that for all ; that is, as .
Definition 1.4 [30]
A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X.
Proposition 1.2 [30]
Let be a G-metric space. Then the following are equivalent:
-
(1)
The sequence is G-Cauchy;
-
(2)
For every , there exists such that for all .
Definition 1.5 [30]
Let and be two G-metric spaces. Then is G-continuous at if and only if it is G-sequentially continuous at x; that is, whenever is G-convergent to x, is G-convergent to .
Proposition 1.3 [30]
Let be a G-metric space. Then, for any , we have .
Definition 1.6 [43]
Let 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 1.7 [22]
An element is called a coupled fixed point of the mapping if and .
Definition 1.8 [23]
An element is called a coupled coincidence point of the mappings and if and .
Definition 1.9 [23]
Let X be a nonempty set. Then we say that the mappings and are commutative if .
Definition 1.10 [22]
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any , we have
and
Definition 1.11 [23]
Let be a partially ordered set and and . The mapping F is said to have the mixed g-monotone property if F is monotone g-nondecreasing in its first argument and is monotone g-nonincreasing in its second argument; that is, for any , we have
and
2 Main results
Theorem 2.1 Let be a partially ordered set and be a G-metric space. Let and be mappings such that F has the mixed g-monotone property on X. Suppose there exist non-negative real numbers α, β, γ and L with such that
for all with and . Further suppose and is a complete subspace of X. Also suppose that X satisfies the following properties:
-
(i)
if a non-decreasing sequence in X converges to , then for all ;
-
(ii)
if a non-increasing sequence in X converges to , then for all .
If there exist two elements such that and , then F and g have a coupled coincidence point. That is, there exists such that and .
Proof In view of the condition of the theorem, we see that there exist such that and . Since , we can choose such that and . Again since , we can choose such that and . Continuing in this way, we construct two sequences and in X such that
Now we prove that
We will use the mathematical induction. Let . Since and , in view of and , we have and ; that is, (2.3) hold for . We assume that (2.3) hold for some . As F has the mixed g-monotone property, and , from (2.2) we get that
In the same way, we obtain that
By combining (2.4) and (2.5), we obtain and . Thus, by the mathematical induction, we conclude that (2.3) hold for all .
We check easily that
and
It follows from (2.1) and (2.2) that
Hence, we obtain
Similarly, using (2.1) and (2.2), we can prove that
Suppose that and , then . From (2.7) and (2.8), we have
which implies that
and
Next, we will prove that and are G-Cauchy sequences in .
In fact, for each , , from (G5) we have
and
Therefore, combining inequalities (2.10), (2.11) and (2.9), we have
This implies that and are G-Cauchy sequences in . Since is a complete subspace of X, there exist such that and as . Since is a non-decreasing sequence and is a non-increasing sequence, by the assumption conditions (i) and (ii), we have and for all .
It follows from (2.1) and (2.2) that
Taking the limit as in the above inequality, we obtain . Hence , and so .
Similarly, one can show that . Thus we proved that F and g have a coupled coincidence point. This concludes the proof. □
Theorem 2.2 Let be a partially ordered set and be a complete G-metric space. Let and be mappings such that F has the mixed g-monotone property on X. Suppose there exist non-negative real numbers α, β, γ and L with such that
for all with and . Further suppose and g is continuous non-decreasing 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 in X converges to , then for all ;
-
(ii)
if a non-increasing sequence in X converges to , then for all .
-
(i)
If there exist two elements such that and , then F and g have a coupled coincidence point. That is, there exists such that and .
Proof Following the proof of Theorem 2.1, we will get two G-Cauchy sequences and in X such that is a non-decreasing sequence and is a non-increasing sequence in X. Since is a complete G-metric space, there is such that and .
Since g is continuous, we have and as .
First, suppose that (a) holds, that is, F is continuous. Then
On the other hand, since g commutes with F, we have
and
Therefore, by the uniqueness of limit, we get and .
Second, suppose that (b) holds. Since is a non-decreasing sequence such that , is a non-increasing sequence such that and g is a non-decreasing function, by (i) and (ii) we get that and hold for all .
By (2.14), we have
Letting , we get , which implies that , so that .
Similarly, we can show that . Thus we proved that F and g have a coupled coincidence point. □
Remark 2.1 Theorems 2.1 and 2.2 generalize and extend the corresponding results in [[29], Theorems 2.1 and 2.2] from partially ordered metric spaces to partially ordered G-metric spaces.
In Theorem 2.2, if we take (I is an identity mapping), then we have the following corollary.
Corollary 2.1 Let be a partially ordered set and be a complete G-metric space. Let be a mapping such that F has the mixed monotone property on X. Suppose there exist non-negative real numbers α, β, γ and L with such that
for all with and , and also suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence in X converges to , then for all ;
-
(ii)
if a non-increasing sequence in X converges to , then for all .
-
(i)
If there exist two elements such that and , then F and g have a coupled fixed point. That is, there exists such that and .
Also, if we take in Theorem 2.2, then we get the following.
Corollary 2.2 Let be a partially ordered set and be a complete G-metric space. Let and be mappings such that F has the mixed g-monotone property on X. Suppose there exists a non-negative real number L such that
for all with and . Further suppose and g is continuous non-decreasing 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 in X converges to , then for all ;
-
(ii)
if a non-increasing sequence in X converges to , then for all .
-
(i)
If there exist two elements such that and , then F and g have a coupled coincidence point. That is, there exists such that and .
Corollary 2.3 Let be a partially ordered set and be a complete G-metric space. Let and be mappings such that F has the mixed g-monotone property on X. Suppose there exist non-negative real numbers α, β, γ and L with such that
for all with and . Further suppose and g is continuous non-decreasing 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 in X converges to , then for all ;
-
(ii)
if a non-increasing sequence in X converges to , then for all .
-
(i)
If there exist two elements such that and , then F and g have a coupled coincidence point. That is, there exists such that and .
Proof By noting that if α, β and γ are non-negative real numbers, from (2.18) we have
From Theorem 2.2, we see that F and g have a coupled coincidence point ; that is, and . □
Remark 2.2 Corollaries 2.1, 2.2 and 2.3 generalize and extend the corresponding results in [[29], Corollaries 2.1, 2.2 and 2.3] from partially ordered metric spaces to partially ordered G-metric spaces.
Now we will prove the existence and uniqueness theorem of a coupled common fixed point. Note that if is a partially ordered set, then we endow the product space with the following partial order:
Theorem 2.3 In addition to the hypotheses of Theorem 2.1, suppose that and for every , there exists a such that is comparable to and . Then F and g have a unique coupled common fixed point; that is, there exists a unique such that and .
Proof From Theorem 2.1, the set of coupled coincidence points of F and g is nonempty. Suppose and are coupled coincidence points of F and g, that is, , , and , then
By assumption, there exists such that is comparable with and . Put , and choose so that and . Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences and such that
Further, set , , and . In the same way, we define the sequences , , and . It is easy to show that
Since
and
are comparable, we see that and . It is easy to show that and are comparable, that is, and for all . Thus from (2.1), we have
Since and , we have
Therefore, we have
Similarly, using (2.1), and , we can prove that
From (2.21) and (2.22), we have
Taking the limit as , we get
which implies that
Similarly, we show that
By the rectangle inequality (G5), Proposition 1.3, (2.23) and (2.24), we obtain
and
It implies that and . Thus we have (2.20) holds. This implies that .
Since and , by the commutativity of F and g, we have
Denote , . Then from (2.25) we have
Thus is a coupled coincidence point of F and g. Then from (2.25) with and it follows and , that is,
From (2.26) and (2.27), we get
Therefore, is a coupled common fixed point of F and g. To prove the uniqueness, assume that is another coupled common fixed point, then by (2.25) we have and . This completes the proof of Theorem 2.3. □
In Theorem 2.3, if we take (I is an identity mapping), then we have the following corollary.
Corollary 2.4 In addition to the hypotheses of Corollary 2.1, suppose that , and for every , there exists a such that is comparable to and . Then F has a unique coupled fixed point; that is, there exists a unique such that and .
Theorem 2.4 In addition to the hypotheses of Theorem 2.1, if and are comparable and , then F and g have a coupled coincidence point such that .
Proof By Theorem 2.1 we construct two sequences and in X such that and , where is a coincidence point of F and g. Suppose , then it is an easy matter to show that for all . Thus, by (2.1) we have
By taking the limit as , we get . Hence .
A similar argument can be used if . This completes the proof of Theorem 2.4. □
In Theorem 2.4, if we take (I is an identity mapping), then we have the following corollary.
Corollary 2.5 In addition to the hypotheses of Theorem 2.1, if and are comparable and , then F has a coupled coincidence point of the form .
Remark 2.3 Theorems 2.3 and 2.4 generalize and extend the corresponding results in [[29], Theorems 2.3 and 2.4] from partially ordered metric spaces to partially ordered G-metric spaces.
Remark 2.4 Corollaries 2.4 and 2.5 generalize and extend the corresponding results in [[29], Corollaries 2.4 and 2.5] from partially ordered metric spaces to partially ordered G-metric spaces.
Now, we introduce an example to support Theorem 2.1.
Example 2.1 Let , then is a partially ordered set with a natural ordering of real numbers. Let for all . Set F and g as
Then
-
(1)
is a complete G-metric space;
-
(2)
is complete;
-
(3)
;
-
(4)
X satisfies (i) and (ii) of Theorem 2.1;
-
(5)
F has the mixed g-monotone property;
-
(6)
For any , F and g satisfy
(2.28)
for all with and . Thus by Theorem 2.1, F and g have a coupled coincidence point. Moreover, is a coupled fixed point of F.
Proof The proofs of (1)-(5) are clear. The proof of (6) is divided into the following cases.
Case 1. If . In this case, we have , and so
Hence, we get
Case 2. . We divide the study in two sub-cases:
(a) If , then , so . Therefore, we get
Hence, we have
(b) If , hence ; the case where is obvious because we get . If , we have . Therefore
In all the above cases, inequality (2.1) of Theorem 2.1 is satisfied for and any . All the required hypotheses of Theorem 2.1 are satisfied. Clearly, F and g have a coupled coincidence point. Moreover, is a coupled fixed point of F. □
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Acknowledgements
The authors are grateful to the editor and the reviewer for suggestions which improved the contents of the article. This work is supported by the National Natural Science Foundation of China (11071169, 11271105) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).
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Gu, F., Zhou, S. Coupled common fixed point theorems for a pair of commuting mappings in partially ordered G-metric spaces. Fixed Point Theory Appl 2013, 64 (2013). https://doi.org/10.1186/1687-1812-2013-64
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DOI: https://doi.org/10.1186/1687-1812-2013-64