#### Abstract

We establish the existence and uniqueness of coupled common fixed point for symmetric -contractive mappings in the framework of ordered *G*-metric spaces. Present work extends, generalize, and enrich the recent results of Choudhury and Maity (2011), Nashine (2012), and Mohiuddine and Alotaibi (2012), thereby, weakening the involved contractive conditions. Our theoretical results are accompanied by suitable examples and an application to integral equations.

#### 1. Introduction and Preliminaries

The structure of *G*-metric spaces introduced by Mustafa and Sims [1] is a generalization of metric spaces. The theory of fixed points in this generalized structure was initiated by Mustafa et al. [2], in which Banach contraction principle was established in *G*-metric spaces. After that different authors proved several fixed point results in this space. References [3–9] are some examples of these works.

*Definition 1 (see [1]). *Let be nonempty set, and 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 is called a -metric on and the pair is a called a -metric space.

*Definition 2 (see [1]). *Let be a -metric space, and let be a sequence of points of . A point is said to be the limit of the sequence if , and then we say that the sequence is -convergent to .

Thus, if in *G*-metric space then, for any , there exists a positive integer such that for all .

In [1], the authors have shown that the -metric induces a Hausdorff topology, and the convergence described in the above definition is relative to this topology. This is a Hausdorff topology, so a sequence can converge at most to one point.

*Definition 3 (see [1]). *Let be a -metric space. A sequence is called *G*-Cauchy if, for every , there is a positive integer such that for all ; that is, as .

Lemma 4 (see [1]). *If is a G-metric space, then the following are equivalent:*(1)

*is -convergent to ,*(2)

*as ,*(3)

*as ,*(4)

*as .*

Lemma 5 (see [1]). *If is a -metric space, then the following are equivalent:*(1)*the sequence is G-Cauchy;*(2)

*for every , there exists a positive integer such that for all .*

Lemma 6 (see [1]). *If is a -metric space, then for all .*

Lemma 7. *If is a -metric space, then for all .*

*Definition 8 (see [1]). *Let , be two *G*-metric spaces. Then a function is *G*-continuous at a point if and only if it is *G*-sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Lemma 9 (see [1]). *Let be a -metric space; then the function is jointly continuous in all three of its variables.*

*Definition 10 (see [1]). *A -metric space is said to be -complete (or a complete *G*-metric space) if every -Cauchy sequence in is convergent in .

*Definition 11 (see [10]). *Let be a -metric space. A mapping is said to be continuous if for any two *G*-convergent sequences and converging to and , respectively, is *G*-convergent to .

Recently, fixed point theorems under different contractive conditions in metric spaces endowed with the partial ordering have been studied by various authors. Works noted in [11–25] are some examples in this direction. Bhaskar and Lakshmikantham [11] introduced the notion of coupled fixed points and proved some coupled fixed point theorems for a mapping having mixed monotone property. The work [11] was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem.

Lakshmikantham and iri [12] extended the notion of mixed monotone property by introducing the notion of mixed -monotone property in partially ordered metric spaces.

*Definition 12 (see [11]). *Let be a partially ordered set and . The mapping is said to have the mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any ,

*Definition 13 (see [12]). *Let be a partially ordered set , and . We say that the mapping has the mixed -monotone property if is monotone -nondecreasing in its first argument and monotone -nonincreasing in its second argument; that is, for any ,

*Definition 14 (see [11]). *An element is called a coupled fixed point of the mapping if and .

*Definition 15 (see [12]). *An element is called a coupled coincidence point of the mappings and if and .

*Definition 16 (see [12]). *An element is called a coupled common fixed point of the mappings and if and .

*Definition 17 (see [12]). *The mappings and are called commutative if
for all .

Let be a partially ordered set, and let be a -metric on such that is a complete -metric space.

Choudhury and Maity [10] established some coupled fixed point theorems for the mixed monotone mapping under a contractive condition of the form where .

Various authors extended and generalized the results of Choudhury and Maity [10] under different contractive conditions in *G*-metric spaces. For more works, one can see [26–34]. Nashine [31] generalized and extended the contractive condition (4) and thereby obtained the coupled coincidence points for a pair of commuting mappings under the following contraction:
where .

Mohiuddine and Alotaibi [33] further generalized the contraction (4) by considering the following more general contractive condition: where are functions satisfying some appropriate conditions mentioned in [33].

Interestingly, for , , with , condition (6) reduces to (4).

On the other hand, Karapinar et al. [32] improved various results present in the literature of coupled fixed point theory of *G*-metric spaces by considering a generalized -contraction. Assigning the value to with for , Karapinar et. al. [32] in an alternative way generalized the contraction (4) for a pair of commutative mappings as follows:
where .

Present work extend and generalize several results present in the literature of fixed point theory of -metric spaces. Our result directly derive a result of Karapinar et. al. [32]. We give suitable examples to show how our results generalize and enrich the well-known results of Choudhury et al. [10], Nashine [31], and Mohiuddine and Alotaibi [33] by significantly weakening the involved contractive conditions. The effectiveness of the present work is shown by suitable examples and an application to the integral equations.

#### 2. Main Results

Before proving our results, we need the following.

Denote by the class of all functions with the following properties: is continuous and nondecreasing; for all ; for all .

We note that and imply if and only if .

Denote by the class of all functions with the following properties: for all ;.Some examples of are (where ), and , and examples of are (where ), .

Let be a -metric space, and let , be two mappings. We say that and are symmetric -contractive mappings on if there exist and such that for all .

Our first result is the following.

Theorem 18. *Let be a partially ordered set, and suppose that there exists a -metric on such that is a complete -metric space. Let and g be symmetric -contractive mappings on with and (or and ) such that has the mixed -monotone property. Assume that ; both the mappings and commutes and are continuous.**If there exist two elements with and (or and , then there exist such that and ; that is, and have a coupled coincidence point in .*

*Proof. *Without loss of generality, assume that there exist such that , . Since , we can choose such that . Again we can choose such that , .

Continuing this process, we can construct sequences and in such that
We will prove, for all , that
Since , , and , , we have ; that is, (10) holds for .

Suppose that (10) holds for some ; that is, , . As has the mixed -monotone property, using (9), we have
Then by mathematical induction, it follows that (10) holds for all .

If for some , we have , and then and ; that is, and have a coupled coincidence point. So now onwards, we suppose that for all ; that is, we suppose that either or .

Using (8)–(10), we have
Since is nonnegative, using (12), we get
By monotonicity of , we get
Let ; then is a monotone decreasing sequence. Therefore, there exists some 0 such that
We claim that .

On the contrary, suppose that .

Taking limit as on both sides of (12) and using the properties of and , we have
Thus, ; that is,
Next, we shall show that and are -Cauchy sequences.

If possible, suppose that at least one of and is not a -Cauchy sequence. Then there exists 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 (18). Then,
Using (18), (19) and Lemma 7, we get
Letting and using (17), we have
Using Lemma 6 and Lemma 7, we get
Similarly, we can obtain
Now, for all , using (22)-(23) in (18), we get
By monotonicity of and property , we have
Also, since , and , using (8) and (9), we have
Combining (25) and (26), we obtain that
On letting , using (17), (21) and continuity of , we get
Therefore both and are *G*-Cauchy sequences in . Now, since the *G*-metric space is *G*-complete, there exist in such that the sequences and are, respectively, -convergent to and , and then by Lemma 4, we have
Using the -continuity of , and Definition 8, we get
Since and , hence using commutativity of and we obtain
Since the mapping is *G*-continuous and the sequences and are, respectively, -convergent to and , hence using Definition 11, the sequence is -convergent to . By uniqueness of limit and using (30), and (32) we get . Similarly, we can show that . Hence, is a coupled coincidence point of and .

In the next theorem, we omit the continuity hypotheses of the mapping along with the commutativity of mappings and . We need the following definition.

*Definition 19. *Let be a partially ordered set, and suppose that there exists a -metric on . We say that is regular if the following conditions hold:(i)if a nondecreasing sequence such that , then for all ,(ii)if a nonincreasing sequence such that , then for all .

Theorem 20. *Let be a partially ordered set, and suppose that there exists a -metric on . Let and be symmetric -contractive mappings on with and (or and ) such that has the mixed -monotone property. Assume that is regular. Suppose that is -complete and . Suppose that there exist with and (or and ); then and have a coupled coincidence point in ; that is, there exist such that and .*

*Proof. *Proceeding exactly as in Theorem 18, we have that and are *G*-Cauchy sequences in the complete *G*-metric space . Then there exist such that and ; that is,
Since is nondecreasing and is nonincreasing, using the regularity of , we have and for all . Using (8), we get
On letting and using (34) and the properties of and , we obtain that
which yields
Hence we can obtain that
On the other hand, by condition (G5), we have
Letting in (39) and using (34)–(38), we have
Thus and . Therefore, we proved that is a coupled coincidence point of and .

Next we give an example in support of Theorem 18 that shows that Theorem 18 is more general than Theorem 3.1 in [31], since the contractive condition (8) is more general than (5).

*Example 21. *Let . Then is a partially ordered set with the natural ordering of real numbers. Let be defined by
Then is a complete -metric space.

Define by , and by , .

Clearly , and are continuous, has the mixed -monotone property, and the pair is commutative and satisfies condition (8) but does not satisfy the condition (5). Assume, to the contrary, that there exists some such that (5) holds. Then, we must have
for all and . Taking , in the last inequality and setting , we obtain
which implies that , a contradiction since . Hence and do not satisfy (5).

Indeed, for and , we have
By summing up the above six inequalities, we get exactly (8) with , . Also, , are the two points in such that and . Now , , , and satisfy all the conditions of Theorem 18; by Theorem 18, we obtain that and have a coupled coincidence point , but Theorem 3.1 in [31] cannot be applied to and in this example.

Now, putting (the identity map of ) in the previous results, we obtain the following.

Corollary 22. *Let be a partially ordered set, and let be a -metric on . Let be a mapping satisfying (8) (with ); that is,
**
for all with and (or and ). Assume that is complete and has the mixed monotone property. Also suppose that either*(a)* is continuous, or*(b)* is regular.**If there exist two elements with and (or and ), then there exist such that and ; that is, has a coupled fixed point in .**The following example shows that the contractive condition (45) is more general than the contractive conditions (4) and (6).*

*Example 23. *Let . Then is a partially ordered set with the natural ordering of real numbers. Let be defined by
Then is a complete -metric space.

Define by , .

Then is continuous and satisfies the mixed monotone property. We note that satisfies condition (45) but does not satisfy the conditions (4) and (6). Indeed, assume that there exists , such that (4) holds. Then, we must have
by which, for , we get
which for implies , a contradiction, since . Hence does not satisfy (4).

Next, we prove that (6) is not satisfied, either. Assume, to the contrary, that there exist functions and satisfying appropriate conditions as in [33] such that (6) holds. This means that
for all and . Taking , in the previous inequality and setting , we obtain
Since satisfy the subadditive property, we have , and therefore, we deduce that, for all , ; that is, , which contradicts the definition of .

This shows that does not satisfy (6).

Finally, we prove that (45) holds. Indeed, for and , we have
By summing up the above six inequalities, we get exactly (45) with , . Also, , are the two points in such that and .

By Corollary 22, we obtain that has a coupled fixed point , but Theorems 3.1 and 3.2 in [10] and Theorem 3.1 in [33] cannot be applied to in this example.

Corollary 24. *Let be a partially ordered set, and let be a G-metric on . Let be a mapping having mixed monotone property on . Suppose there exists such that
*

*for all with and (or and ). Assume that is complete, and also suppose that either*(i)

*is continuous, or*(ii)

*is regular.*

*If there exist two elements with and (or and ), then there exist such that and ; that is, has a coupled fixed point in .*

*Proof. *Note that if , then for all , . Now divide (52) by 4 and take , ; then condition (52) reduces to (8) with and ; and hence by Theorem 18 and Theorem 20, we obtain Corollary 24.

The following result provides us the recent result of Karapinar et. al. [32, Corollary 2.5].

Theorem 25 (see [32]). *Let be a partially ordered set, and suppose that there exists a -metric on such that is a complete -metric space. Let and be two mappings such that has the mixed -monotone property on and . Suppose that there exists a real number such that
**
for all with and (or and ). Also suppose that either*(i)* and are continuous, is complete, and commutes with , or*(ii)* is complete and is regular.**If there exist two elements with ) and (or and ), then there exist such that and .*

*Proof. *Define , and , . Then (53) holds. Hence the result follows from Theorem 18 or Theorem 20.

*Remark 26. *The choice of functions and in Example 21 shows that Theorem 25 is more general than Theorem 3.1 in [31], since the contractive condition (53) is more general than (5). Indeed, the contractive condition (5) does not hold for the choice of functions and , but (53) holds exactly for with and and yields as the coupled coincidence point of and .

Now, putting (the identity map of ) in Theorem 25, we obtain the following.

Corollary 27. *Let be a partially ordered set, and let be a -metric on such that is a complete G-metric space. Let be a mapping having mixed monotone property on . Suppose there exists a real number such that
*

*for all with and (or and ). Suppose that either*(i)

*is continuous, or*(ii)

*is regular.*

*If there exist two elements with and (or and ), then there exist such that and ; that is, has a coupled fixed point in .*

*Remark 28. *The choice of function in Example 23 shows that Corollary 27 is more general than Theorem 3.1 and Theorem 3.2 in [10], since the contractive condition (54) is more general than (4). Indeed the contractive condition (4) does not hold for the choice of function , but (54) holds exactly for with , .

Next we prove the existence and uniqueness of the coupled common fixed point for our main result.

Theorem 29. *In addition to the hypotheses of Theorem 18, suppose that for every , there exists a such that is comparable to and . Then and have a unique coupled common fixed point; that is, there exists a unique *