Abstract
In the present paper, we prove a common coupled fixed point theorem in the setting of a generalized metric space in the sense of Mustafa and Sims. Our results improve and extend the corresponding results of Shatanawi. We also present an application to integral equations.
Keywords:
Gmetric space; common coupled coincidence fixed point; common fixed point; integral equation1 Introduction and preliminaries
The study of fixed points of mappings satisfying certain contractive conditions has been in the center of rigorous research activity. For a survey of common fixed point theory in metric and cone metric spaces, we refer the reader to [19]. In 2006, Bhaskar and Lakshmikantham [10] initiated the study of a coupled fixed point in ordered metric spaces and applied their results to prove the existence and uniqueness of solutions for a periodic boundary value problem. For more works in coupled and coincidence point theorems, we refer the reader to [1113].
Some authors generalized the concept of metric spaces in different ways. Mustafa and Sims [14] introduced the notion of Gmetric space, in which the real number is assigned to every triplet of an arbitrary set as a generalization of the notion of metric spaces. Based on the notion of Gmetric spaces, many authors (for example, [1533]) obtained some fixed point and common fixed point theorems for mappings satisfying various contractive conditions. Fixed point problems have also been considered in partially ordered Gmetric spaces [3439].
The purpose of this paper is to obtain some common coupled coincidence point theorems in Gmetric spaces satisfying some contractive conditions.
The following definitions and results will be needed in the sequel.
Definition 1.1[14]
Let X be a nonempty set, and let be a function satisfying the following axioms:
(G4) (symmetry in all three variables);
(G5) for all (rectangle inequality),
then the function G is called a generalized metric, or more specifically, a Gmetric on X, and the pair is called a Gmetric space.
Definition 1.2[14]
Let be a Gmetric 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 Gconvergent to x.
Thus, if in a Gmetric space , then for any , there exists such that for all .
Proposition 1.3[14]
Letbe aGmetric space, then the following are equivalent:
Definition 1.4[14]
Let be a Gmetric space. A sequence is called GCauchy sequence if for each , there exists a positive integer such that for all ; i.e., if as .
Definition 1.5[14]
A Gmetric space is said to be Gcomplete if every GCauchy sequence in is Gconvergent in X.
Proposition 1.6[14]
Letbe aGmetric space, then the following are equivalent:
(2) For every, there existssuch thatfor all.
Proposition 1.7[14]
Letbe aGmetric space. Then the functionis jointly continuous in all three of its variables.
Definition 1.8[14]
Let and be Gmetric space, and let be a function. Then f is said to be Gcontinuous at a point if and only if for every , there is such that and implies that . A function f is Gcontinuous at X if and only if it is Gcontinuous at all .
Proposition 1.9[14]
LetandbeGmetric spaces, then a functionisGcontinuous at a pointif and only if it isGsequentially continuous atx; that is, wheneverisGconvergent tox, isGconvergent to.
Proposition 1.10[14]
Letbe aGmetric space. Then for anyx, y, z, ainX, it follows that
Definition 1.11[10]
An element is called a coupled fixed point of a mapping if and .
Definition 1.12[11]
An element is called a coupled coincidence point of the mappings and if and .
Definition 1.13[11]
Let X be a nonempty set. Then we say that the mappings and are commutative if .
2 Main results
We start our work by proving the following crucial lemma.
Lemma 2.1Letbe aGmetric space. Letandbe four mappings such that
for all, where, and. Suppose thatis a common coupled coincidence point of the mappings pair, and. Then
Proof Since is a common coupled coincidence point of the mappings pair , and , we have and . Assume that . Then by (2.1), we get
Also by (2.1), we have
Therefore,
which is a contradiction. So, , and hence,
□
Theorem 2.1Letbe aGmetric space. Letandbe four mappings such that
for all, where, and. Suppose that, , andgsatisfy the following conditions:
(ii) gXisGcomplete;
(iii) gisGcontinuous and commutes with, , .
Then there exist uniquesuch that
Proof Let . Since , , , we can choose such that , , , , and . Combining this process, we can construct two sequences and in X such that
If , then , where , . If , then , where , . If , then , where , . On the other hand, if , then , where , . If , then , where , . If , then , where , . Without loss of generality, we can assume that and , for all .
By (2.2) and (G3), we have
Similarly, we have
By combining (2.3) and (2.4), we get
In the same way, we can show that
and
It follows from (2.5), (2.6) and (2.7) that for all , we have
Where . From (G3), we have and . Hence, by the (G3) and (2.8), we get
Therefore, for all , , by (G5) and (2.9), we have
Which implies that
Thus, and are all GCauchy in gX. Since gX is Gcomplete, we get that and are Gconvergent to some and , respectively. Since g is Gcontinuous, we have is Gconvergent to gx and is Gconvergent to gy. That is,
Also, since g commutes with , and , respectively, we have
Thus, from condition (2.2), we have
Letting , using (2.11) and the fact that G is continuous on its variables, we get that
Hence, . Similarly, we may show that . Also for the same reason, we may show that , , and . Therefore, is a common coupled coincidence point of the pair , and . By Lemma 2.1, we obtain
Since the sequences , and are all a subsequence of , then they are all Gconvergent to x. Similarly, we may show that , and are all Gconvergent to y. From (2.2), we have
Letting , and using the fact that G is continuous on its variables, we get that
Similarly, we may show that
Thus, using the Proposition 1.10(iii), we have
Since , so the last inequality happens only if and . Hence, and . From (2.12), we have , thus, we get
To prove the uniqueness, let with such that
Again using condition (2.2) and Proposition 1.10(iii), we have
Since , we get , which is a contradiction. Thus, , , and g have a unique common fixed point. □
Remark 2.1 Theorem 2.1 extends and improves Theorem 3.2 of Shatanawi [26].
The following corollary can be obtained from Theorem 2.1 immediately.
Corollary 2.1Letbe aGmetric space. Letandbe mappings such that
for all, where, and. Suppose that, , andgsatisfy the following conditions:
(2) gXisGcomplete;
(3) gisGcontinuous and commutes with, , .
Then there exist uniquesuch that
Remark 2.2 If and , then Corollary 2.1 is reduced to Theorem 3.2 of Shatanawi [26].
Now, we give an example to support Corollary 2.1.
for all . Then is a complete Gmetric space. Define a map
by
for all . Also, define by for . Then . Through calculation, we have
Then the mappings , , and g are satisfying condition (2.13) of Corollary 2.1 with . So that all the conditions of Corollary 2.1 are satisfied. By Corollary 2.4, , , and g have a unique common fixed point. Moreover, 0 is the unique common fixed point for all of the mappings , , and g.
If , then Theorem 2.1 is reduced to the following.
Corollary 2.2Letbe aGmetric space. Letandbe four mappings such that
for all, where, and. Suppose that, , andgsatisfy the following conditions:
(ii) gXisGcomplete;
(iii) gisGcontinuous and commutes with, , .
Then there exist uniquesuch that
If we take in Corollary 2.2, then the following corollary is obtained.
Corollary 2.3Letbe aGmetric space. Letandbe four mappings such that
for all, where, and. Suppose thatFandgsatisfy the following conditions:
(ii) gXisGcomplete;
(iii) gisGcontinuous and commutes withF.
Then there exist uniquesuch that
Now, we give an example to support Corollary 2.3.
for all . Then is a complete Gmetric space. Define a map by
for all . Also, define by for . Then . Through calculation, we have
Then the mappings , , and g are satisfying condition (2.15) of Corollary 2.3 with . So that all the conditions of Corollary 2.3 are satisfied. By Corollary 2.3, F and g have a unique common fixed point. Moreover, 0 is the unique common fixed point for all of the mappings F and g.
If we take in Theorem 2.1, then the following corollary is obtained.
Corollary 2.4Letbe aGmetric space. Letandbe mappings such that
for all, where, and. Suppose thatFandgsatisfy the following conditions:
(2) gXisGcomplete;
(3) gisGcontinuous and commutes withF.
Then there exist uniquesuch that.
Now, we introduce an example to support Corollary 2.4.
for all . Then is a complete Gmetric space. Define a map
by
for all . Also, define by for .
Clearly, we can get , and g is Gcontinuous and commutes with F.
By the definition of the mappings of F and g, for all , we have
Then the mappings F and g are satisfying condition (2.16) of Corollary 2.4 with , . So that all the conditions of Corollary 2.4 are satisfied. By Corollary 2.4, F and g have a unique common fixed point. Here is the unique common fixed point of mappings F and g; that is, .
3 Application to integral equations
Throughout this section, we assume that is the set of all continuous functions defined on . Define by
for all . Then is a Gcomplete metric space.
Consider the following integral equations:
Next, we will analyze (3.1) under the following conditions:
(ii) () are continuous functions.
(iii) There exist constants () such that
The aim of this section is to give an existence theorem for a solution of the above integral equations by using the obtained result given by Theorem 2.1.
Theorem 3.1Under conditions (i)(iv), integral equation (3.1) has a unique common solution in.
Proof First, we consider (). By virtue of our assumptions, is well defined (this means that for then ()). Then we can get
By conditions (iii),
and
Taking these inequalities into (3.2), we obtain
Using the CauchySchwartz inequality in (3.3), we get
Similarly, we can obtain the following estimate
Substituting (3.4), (3.5) and (3.6) into (3.3), we obtain that
then inequality (3.7) becomes
By condition (iv), we know that
This proves that the operator () and satisfy contractive condition (2.2) appearing in Theorem 2.1 with . Therefore, , , have a unique common coupled fixed point, that is, , and so, is the unique solution of equation (3.1). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript.
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 (11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030) and the Physical Experiment Center of Hangzhou Normal University.
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