The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions (cf. [1–31]). Recently, Bhaskar and Lakshmikantham [8], Nieto and Rodríguez-López [28, 29], Ran and Reurings [30], and Agarwal et al. [1] presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [8] noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book [22].

Recently, Al-Homidan et al. [2] introduced the concept of a -function defined on a quasi-metric space which generalizes the notions of a -function and a -distance and establishes the existence of the solution of equilibrium problem (see also [3–7]). The aim of this paper is to extend the results of Lakshmikantham and Ćirić [24] for a mixed monotone nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a -function . We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to Lakshmikantham and Ćirić [24] and many others.

Recall that if is a partially ordered set and such that for implies , then a mapping is said to be nondecreasing. Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham [8] introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 1.1 (Bhaskar and Lakshmikantham [8]).

Let be a partially ordered set and . The mapping is said to have the mixed monotone property if is nondecreasing monotone in its first argument and is nonincreasing monotone in its second argument, that is, for any

Definition 1.2 (Bhaskar and Lakshmikantham [8]).

An element is called a coupled fixed point of the mapping if

The main theoretical result of Lakshmikantham and Ćirić in [24] is the following coupled fixed point theorem.

Theorem 1.3 (Lakshmikantham and Ćirić [24, Theorem ]).

Let be a partially ordered set, and suppose, there is a metric on such that is a complete metric space. Assume there is a function with and for each , and also suppose that and such that has the mixed -monotone property and

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following property:

(i)if  a  nondecreasing  sequence ,then   for all

(ii)if  a  nonincreasing  sequence ,then   for all

If there exists such that

then there exist such that

that is, and have a coupled coincidence.

Definition 1.4.

Let be a nonempty set. A real-valued function is said to be quasi-metric on if

for all

if and only if

for all .

The pair is called a quasi-metric space.

Definition 1.5.

Let be a quasi-metric space. A mapping is called a -function on if the following conditions are satisfied:

for all

if and is a sequence in such that it converges to a point (with respect to the quasi-metric) and for some then ;

for any , there exists such that , and implies that

Remark 1.6 (see [2]).

If is a metric space, and in addition to the following condition is also satisfied:

for any sequence in with and if there exists a sequence in such that then

then a -function is called a -function, introduced by Lin and Du [27]. It has been shown in [27]that every -distance or -function, introduced and studied by Kada et al. [21], is a -function. In fact, if we consider as a metric space and replace by the following condition:

for any , the function is lower semicontinuous,

then a -function is called a -distance on . Several examples of -distance are given in [21]. It is easy to see that if is lower semicontinuous, then holds. Hence, it is obvious that every -function is a -function and every -function is a -function, but the converse assertions do not hold.

Example 1.7 (see [2]).

(a) Let . Define by

and by

Then one can easily see that is a quasi-metric and is a -function on , but is neither a -function nor a -function.

(b) Let Define by

and by

Then is a -function on However, is neither a -function nor a -function, because is not a metric space.

The following lemma lists some properties of a -function on which are similar to that of a -function (see [21]).

Lemma 1.8 (see [2]).

Let be a -function on Let and be sequences in , and let and be such that they converge to and Then, the following hold:

(1)if and for all , then . In particular, if and , then ;

(2)if and for all , then converges to ;

(3)if for all with , then is a Cauchy sequence;

(4)if for all , then is a Cauchy sequence;

(5)if are -functions on , then is also a -function on .

2. Main Results

Analogous with Definition 1.1, Lakshmikantham and Ćirić [24] introduced the following concept of a mixed -monotone mapping.

Definition 2.1 (Lakshmikantham and Ćirić [24]).

Let be a partially ordered set, and and We say has the mixed -monotone property if is nondecreasing -monotone in its first argument and is nondecreasing -monotone in its second argument, that is, for any

Note that if is the identity mapping, then Definition 2.1 reduces to Definition 1.1.

Definition 2.2 (see [24]).

An element is called a coupled coincidence point of a mapping and if

Definition 2.3 (see [24]).

Let be a nonempty set and and one says and are commutative if

for all

Following theorem is the main result of this paper.

Theorem 2.4.

Let be a partially ordered complete quasi-metric space with a -function on . Assume that the function is such that

Further, suppose that and are such that has the mixed -monotone property and

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following property:

(i)if  a  nondecreasing  sequence  ,  then  for all

(ii)if  a  nonincreasing  sequence  ,  then   for all

If there exists such that

then there exist such that

that is, and have a coupled coincidence.

Proof.

Choose to be such that and Since we can choose such that and Again from , we can choose such that and Continuing this process, we can construct sequences and in such that

We will show that

We will use the mathematical induction. Let Since and and as and we have and Thus, (2.9) and (2.10) hold for Suppose now that (2.9) and (2.10) hold for some fixed Then, since and and as has the mixed -monotone property, from (2.8) and (2.9),

and from (2.8) and (2.10),

Now from (2.11) and (2.12), we get

Thus, by the mathematical induction, we conclude that (2.9) and (2.10) hold for all . Therefore,

Denote

We show that

Since and from (2.11) and (2.5), we have

Similarly, from (2.11) and (2.5), as and

Adding (2.17) and (2.18), we obtain (2.16). Since for it follows, from (2.16), that

and so, by squeezing, we get

Thus,

Now, we prove that and are Cauchy sequences. For and since for each we have

This means that for ,

Therefore, by Lemma 1.8, and are Cauchy sequences. Since is complete, there exists such that

and (2.24) combined with the continuity of yields

From (2.11) and commutativity of and

We now show that and

Case 1.

Suppose that the assumption (a) holds. Taking the limit as in (2.26), and using the continuity of , we get

Thus,

Case 2.

Suppose that the assumption (b) holds. Let . Now, since is continuous, is nondecreasing with for all , and is nonincreasing with for all , so is nondecreasing, that is,

with , for all , and is nonincreasing, that is,

with , for all .

Let

Then replacing by and by in (2.16), we get such that We show that

In , replacing by and by , we get

that is, for

or for ,

Let , and Then, since , and by axiom of the -function, we get

Therefore, by the triangle inequality and ( ), we have for

Case 3.

This implies that

Case 4.

Also, we have

or

That is, for ,

Hence, by Lemma 1.8, and Thus, and have a coupled coincidence point.

The following example illustrates Theorem 2.4.

Example 2.5.

Let with the usual partial order Define by

and by

Then is a quasi-metric and is a -function on Thus, is a partially ordered complete quasi-metric space with a -function on Let for Define by

and by , where Then, has the mixed -monotone property with

and , are both continuous on their domains and . Let be such that and There are four possibilities for (2.5) to hold. We first compute expression on the left of (2.5) for these cases:

(i) and ,

(ii) and

(iii) and

(iv) and

On the other hand, (in all the above four cases), we have

Thus, satisfies the contraction condition (2.5) of Theorem 2.4. Now, suppose that be, respectively, nondecreasing and nonincreasing sequences such that and , then by Theorem 2.4, and for all

Let Then, this point satisfies the relations

Therefore, by Theorem 2.4, there exists such that and

Corollary 2.6.

Let be a partially ordered complete quasi-metric space with a -function on . Suppose and are such that has the mixed -monotone property and assume that there exists such that

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following properties:

(i)if  a  nondecreasing  sequence , then  for all

(ii)if  a  nonincreasing  sequence , then  for all .

If there exists such that

then there exist such that

that is, and have a coupled coincidence.

Proof.

Taking in Theorem 2.4, we obtain Corollary 2.6.

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 with the following partial order:

From Theorem 2.4, it follows that the set of coupled coincidences is nonempty.

Theorem 2.7.

The hypothesis of Theorem 2.4 holds. 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 exist a unique such that

Proof.

By Theorem, 2.1   . Let . We show that if and , then

By assumption there is such that is comparable with and Put , and choose so that and Then, as in the proof of Theorem 2.4, we can inductively define sequences and such that

Further, set , , , , and, as above, define the sequences and Then it is easy to show that

for all Since and are comparable; therefore and It is easy to show that and are comparable, that is, and for all From (2.5) and properties of , we have

where From this, it follows that, for each ,

Similarly, one can prove that

where Thus by Lemma 1.8, and . Since and , by commutativity of and , we have

Denote Then from (2.61),

Thus, is a coupled coincidence point. Then, from (2.55), with and , it follows that and ; that is,

From (2.62) and (2.63),

Therefore, is a coupled common fixed point of and To prove the uniqueness, assume that is another coupled common fixed point. Then, by (2.55), we have and

Corollary 2.8.

Let be a partially ordered complete quasi-metric space with a -function on . Assume that the function is such that for each Let and let be a mapping having the mixed monotone property on and

Also suppose that either

(a) is continuous or

(b) has the following properties:

(i)if a nondecreasing  sequence  , then for all

(ii)if  a  non-increasing  sequence  , then  for all

If there exists such that

then, there exist such that

Furthermore, if are comparable, then that is,

Proof.

Following the proof of Theorem 2.4 with (the identity mapping on ), we get

We show that Let us suppose that We will show that are comparable for all that is,

where , Suppose that (2.69) holds for some fixed Then, by mixed monotone property of

and (2.69) follows. Now from (2.69), (2.65), and properties of we have

where Similarly, we get

where . Hence, by Lemma 1.8, that is,

Corollary 2.9.

Let be a partially ordered complete quasi-metric space with a -function on . Let be a mapping having the mixed monotone property on . Assume that there exists a such that

Also, suppose that either

(a) is continuous or

(b) has the following properties:

(i)if  a  nondecreasing  sequence  , then  for all

(ii)if  a  nonincreasing  sequence  , then  for all

If there exists such that

then, there exist such that

Furthermore, if are comparable, then that is,

Proof.

Taking in Corollary 2.8, we obtain Corollary 2.9.

Remark 2.10.

As an application of fixed point results, the existence of a solution to the equilibrium problem was considered in [2–7]. It would be interesting to solve Ekeland-type variational principle, Ky Fan type best approximation problem and equilibrium problem utilizing recent results on coupled fixed points and coupled coincidence points.