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# A new common coupled fixed point theorem in generalized metric space and applications to integral equations

Feng Gu* and Yun Yin

Author Affiliations

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, China

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Fixed Point Theory and Applications 2013, 2013:266  doi:10.1186/1687-1812-2013-266

 Received: 8 June 2013 Accepted: 19 September 2013 Published: 7 November 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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:
G-metric space; common coupled coincidence fixed point; common fixed point; integral equation

### 1 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 [1-9]. 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 [11-13].

Some authors generalized the concept of metric spaces in different ways. Mustafa and Sims [14] introduced the notion of G-metric 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 G-metric spaces, many authors (for example, [15-33]) 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 G-metric spaces [34-39].

The purpose of this paper is to obtain some common coupled coincidence point theorems in G-metric 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:

(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 generalized metric, or more specifically, a G-metric on X, and the pair is called a G-metric space.

Definition 1.2[14]

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 1.3[14]

Letbe aG-metric space, then the following are equivalent:

(1) isG-convergent tox.

(2) as.

(3) as.

(4) as.

Definition 1.4[14]

Let be a G-metric space. A sequence is called G-Cauchy sequence if for each , there exists a positive integer such that for all ; i.e., if as .

Definition 1.5[14]

A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X.

Proposition 1.6[14]

Letbe aG-metric space, then the following are equivalent:

(1) The sequenceisG-Cauchy.

(2) For every, there existssuch thatfor all.

Proposition 1.7[14]

Letbe aG-metric space. Then the functionis jointly continuous in all three of its variables.

Definition 1.8[14]

Let and be G-metric space, and let be a function. Then f is said to be G-continuous at a point if and only if for every , there is such that and implies that . A function f is G-continuous at X if and only if it is G-continuous at all .

Proposition 1.9[14]

LetandbeG-metric spaces, then a functionisG-continuous at a pointif and only if it isG-sequentially continuous atx; that is, wheneverisG-convergent tox, isG-convergent to.

Proposition 1.10[14]

Letbe aG-metric space. Then for anyx, y, z, ainX, it follows that

(i) if, then;

(ii) ;

(iii) ;

(iv) ;

(v) ;

(vi) .

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 aG-metric space. Letandbe four mappings such that

(2.1)

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,

Since , we get

which is a contradiction. So, , and hence,

□

Theorem 2.1Letbe aG-metric space. Letandbe four mappings such that

(2.2)

for all, where, and. Suppose that, , andgsatisfy the following conditions:

(i) , , ;

(ii) gXisG-complete;

(iii) gisG-continuous 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

(2.3)

Similarly, we have

(2.4)

By combining (2.3) and (2.4), we get

(2.5)

In the same way, we can show that

(2.6)

and

(2.7)

It follows from (2.5), (2.6) and (2.7) that for all , we have

(2.8)

Where . From (G3), we have and . Hence, by the (G3) and (2.8), we get

(2.9)

Therefore, for all , , by (G5) and (2.9), we have

(2.10)

Which implies that

Thus, and are all G-Cauchy in gX. Since gX is G-complete, we get that and are G-convergent to some and , respectively. Since g is G-continuous, we have is G-convergent to gx and is G-convergent to gy. That is,

(2.11)

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

(2.12)

Since the sequences , and are all a subsequence of , then they are all G-convergent to x. Similarly, we may show that , and are all G-convergent 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 aG-metric space. Letandbe mappings such that

(2.13)

for all, where, and. Suppose that, , andgsatisfy the following conditions:

(1) , , ;

(2) gXisG-complete;

(3) gisG-continuous 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.

Example 2.1 Let . Define by

for all . Then is a complete G-metric 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 aG-metric space. Letandbe four mappings such that

(2.14)

for all, where, and. Suppose that, , andgsatisfy the following conditions:

(i) , , ;

(ii) gXisG-complete;

(iii) gisG-continuous and commutes with, , .

Then there exist uniquesuch that

If we take in Corollary 2.2, then the following corollary is obtained.

Corollary 2.3Letbe aG-metric space. Letandbe four mappings such that

(2.15)

for all, where, and. Suppose thatFandgsatisfy the following conditions:

(i) ;

(ii) gXisG-complete;

(iii) gisG-continuous and commutes withF.

Then there exist uniquesuch that

Now, we give an example to support Corollary 2.3.

Example 2.2 Let . Define by

for all . Then is a complete G-metric 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 aG-metric space. Letandbe mappings such that

(2.16)

for all, where, and. Suppose thatFandgsatisfy the following conditions:

(1) ;

(2) gXisG-complete;

(3) gisG-continuous and commutes withF.

Then there exist uniquesuch that.

Now, we introduce an example to support Corollary 2.4.

Example 2.3 Let . Define by

for all . Then is a complete G-metric space. Define a map

by

for all . Also, define by for .

Clearly, we can get , and g is G-continuous 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 G-complete metric space.

Consider the following integral equations:

(3.1)

Next, we will analyze (3.1) under the following conditions:

(i) is continuous.

(ii) () are continuous functions.

(iii) There exist constants () such that

for all and .

(iv) , where

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

(3.2)

By conditions (iii),

and

Taking these inequalities into (3.2), we obtain

(3.3)

Using the Cauchy-Schwartz inequality in (3.3), we get

(3.4)

Similarly, we can obtain the following estimate

(3.5)

(3.6)

Substituting (3.4), (3.5) and (3.6) into (3.3), we obtain that

(3.7)

Taking for all , and

then inequality (3.7) becomes

(3.8)

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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