1. Introduction and Preliminaries

In the sequel, let be a non-empty set. Throughout this paper, we use indifferently the notation to denote the product space will denote a partial order on and will be a metric on Also, with respect to abbreviated as w.r.t.

Definition 1. ^[1] An element is said to be a coupled fixed point of the mapping if and

Definition 2. ^[1] Let be a partially ordered set and be a mapping. We say that F has the mixed monotone property if is monotone nondecreasing in a and is monotone non-increasing in b; that is, for any

and

Lakshmikantham and Ćirić ^[2] introduced the concept of mixed g-monotone mapping.

Definition 3. ^[2] An element is said to be a coupled coincidence point of a mapping and if and

Definition 4. ^[2] Let be a partially ordered set and and . We say F has the mixed g-monotone property if for any

and

Definition 5. ^[2] Let C be a nonempty set and and We say F and g are commutative if for all

Definition 6. ^[4] Let be a metric space, be a mapping and g be a self mapping on C. A hybrid pair F, g is compatible if

and

whenever and are sequences in C such that

with

Denote by the set of functions satisfying:

(φ1) φ is continuous,

(φ2) φ (t) < t for all t > 0 and φ (t) = 0 if nd only if t = 0.

Using the concept of compatibility, Karapinar et al. ^[5] proved the following theorem.

Theorem 1. ^[5] Let be a partially ordered set, and suppose there is a metric on C such that is a complete metric space. Let and be two mappings having the g-mixed monotone property on C such that there exists two elements with

Suppose there exists and such that

for all with and Suppose g is continuous and compatible with Also suppose either

(a) F is continuous or;

(b) C has the following properties:

(1) if a non-decreasing sequence then for all n;

(2) if a non-increasing sequence then for all n:

Then there exists such that and that is, F and g have a coupled coincidence point in C.

Hussain et al. ^[3] introduced the concept of G-increasing mappings and concept of generalized compatibility for the pair Also, they introduced some coupled coincidence point results.

Definition 7. ^[3] Suppose that are two mappings. F is said to be G-increasing w.r.t if for all with we have

Definition 8. ^[3] An element is said to be a coupled coincidence point of a mappings if and

Definition 9. ^[3] Let We say that pair is generalized compatible if

whenever and are sequences in C such that

Definition 10. ^[3] Let be two maps. We say that the pair is commuting if for all

Remark 1. ^[3] A commuting pair is a generalized compatible but not conversely in general.

In this paper, we prove the existence of a coupled coincidence point theorem for a pair of mapping with contraction (2.1) in complete metric space without mixed G-monotone property of F. Therefore provided coupled fixed point results need not the mixed monotone property of F. Our results represent new version of results given by Karapinar et al. ^[5] and Jachymski ^[6]. The theoretic results are also accompanied with suitable example.

2. Main Results

Theorem 2. Let be a partially ordered set, and suppose there is a metric on C such that is a complete metric space. Assume that are two generalized compatible mappings such that F is G-increasing w.r.t G is continuous and has the mixed monotone property, and there exists two elements with

Suppose there exists non-negative real numbers and such that

(2.1)

for all with and Suppose that for any there exists such that

(2.2)

Also suppose that either

(a) F is continuous or;

(b) C has the following properties:

(1) if a non-decreasing sequence then for all n,

(2) if a non-increasing sequence then for all n.

Then F and G have a coupled coincidence point in C.

Proof. Let be such that and By (2.2), there exists such that and Continuing this process, we construct sequences and in C such that

(2.3)

Since F is G-increasing w.r.t and using the mathematical induction, we have

(2.4)

Since and from (2.1) and (2.3), we have

(2.5)

Similarly, we have

(2.6)

By (2.5) and (2.6), we obtain

(2.7)

Owing to (φ2), by (2.7), we have

Set

then sequence is non-increasing. Hence, there is some such that . We claim that Suppose, to the contrary, that then by (2.7) and using the property of we have

which is a contradiction. Therefore = 0, i.e.,

(2.8)

Now, we show that is Cauchy sequence in endowed with the metric defined by

(2.9)

for all If is not a Cauchy sequence in . Then there exists for which we can find two sequences of positive integers (m(k)) and (n (k)) such that for all positive integer k with n (k) > m(k) > k, we have

(2.10)

From (2.9), we get

(2.11)

and

(2.12)

From (2.12) and using triangle inequality, we have

(2.13)

and

(2.14)

From (2.11), (2.13), (2.14), we have

(2.15)

Letting in (2.15) and by (2.8), we obtain

(2.16)

From triangle inequality

(2.17)

and

(2.18)

From (2.11), (2.17) and (2.18), we have

(2.19)

Again, from the triangle inequality,

(2.20)

and

(2.21)

Thus,

(2.22)

Letting in (2.19) and by (2.8), (2.22), (2.16), we get

(2.23)

In view of

and from (2.1) and (2.3), we have

(2.24)

Similarly, we have

(2.25)

Using (2.24) and (2.25), we get

(2.26)

Taking the limit as in (2.26), and from (2.8), (2.16), (2.23) and (2), we obtain

which is a contradiction. Therefore, is Cauchy sequence in which implies that and are Cauchy sequence in Since is a complete metric space, there exists such that

(2.27)

Since the pair satisfies the generalized compatibility, by (2.27), we have

(2.28)

Suppose the assumption (a) holds. For all , we get

Taking the limit as in (2.27), by (2.28), and since F and G are continuous, we have

(2.29)

Similarly, we show that

(2.30)

Hence is a coupled coincidence point of F and G.

Next, suppose the assumption (b) holds. From (2.4) and (2.27), we obtain is non-decreasing sequence, as and is non-increasing sequence, as Therefore, we get

(2.31)

Since the pair satisfies the generalized compatibility and G is continuous, from (2.28), we obtain

(2.32)

and

(2.33)

Next, we have

Since G has the mixed monotone property, it follows from (2.31) that

and From (2.1), (2.32) and (2.33), we obtain

Then we get Similarly,

By Remark 1, we have the following Corrollary.

Corollary 1. Under the assumption of Theorem 2, suppose that are two commuting mappings such that F is G-increasing w.r.t , G is continuous and has the mixed monotone property, then F and G have a coupled coincidence point in C.

Definition 11. ^[3] Let be a partially ordered set and and We say F is g-increasing w.r.t if for any

and

The consequence of the main results of Karapinar et al. ^[5] (Theorem 1) without g-mixed monotone property of F is given in the following corollary.

Corollary 2. Let and be two mappings such that is g-increasing w.r.t Under the assumption of Theorem 1, suppose that the pair is compatible, then and g have a coupled coincidence point in C.

Corollary 3. Let and be two mappings such that is g-increasing w.r.t Under the assumption of Theorem 1, suppose that the pair is compatible, then and g have a coupled coincidence point in C.

Corollary 4. Taking L = 0 in (2.1), then Corrollary 2 and 3 provides the conclusion of the main results of Jachymski ^[6].

Now, we shall prove the uniqueness of coupled fixed point. Note that if is a partially ordered set, then we endow the product with the following partial order relation:

where is one-one.

Theorem 3. In addition to the hypotheses of Theorem 2, suppose that for every there exists another which is comparable to and . Then F and G have a unique coupled coincidence point.

Proof. Owing to Theorem 2, 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

By assumption, there exists such that is comparable to and . We define sequences as follows

Since is comparable with we assume that which implies and We assume that for some We derive that

Since F is G increasing, we have implies and implies Then, we have

and

Hence we obtain

(2.34)

By (2.1) and (2.34), we have

(2.35)

Similarly, we have

(2.36)

From (2.35) and (2.36), we get

(2.37)

Owing to (φ2), by (2.37), we have

Set

then sequence is decreasing. Hence, there is some such that

We claim that Suppose, to the contrary, that Taking the limit as in (2.37) and using the property of , we have

which is a contradiction. Therefore , i.e.,

This implies that

(2.38)

Similarly, we get

(2.39)

From (2.38) and (2.39), we have and

Next, we discuss an example to support Theorem 2.

Example 1. Let with the usual metric for all We consider the following order relation on C

Let be defined by

and

Clearly, G is continuous and has the mixed monotone property. Moreover, F is G-increasing.

Now, we prove that for any there exists such that and It is easy to see the following cases.

Case 1: If then we have and

Case 2: If then we have and

Case 3: If a < b, then we have and

Now, we show that the pair satisfies the generalized compatibility hypothesis. Let and be two sequences in C such that

Then we must have and one can easly prove that

Let be defined by for all Now, we verify the contraction (2.1) for all with and We have the following cases.

Case 1: or we have Thus, (2.1) holds.

Case 2: , we have

Thus, (2.1) holds.

Case 3: we have

Thus, (2.1) holds.

Therefore, all the conditions of Theorem 2 are satisfied and is a coupled coincidence point of F and G.

Acknowledgement

The authors would like to thank the editor and referees for their valuable comments and suggestions.

References

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