**Turkish Journal of Analysis and Number Theory**

## An Extended Coupled Coincidence Point Theorem

**Esra Yolacan**^{1}, **Mehmet Kir**^{2,}, **Hukmi Kiziltunc**^{3}

^{1}Republic of Turkey Ministry of National Education, Mathematics Teacher, 60000 Tokat, Turkey

^{2}Department of Civil Engineering, Faculty of Engineering, Şırnak University, 73000, Turkey

^{3}Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey

### Abstract

In this paper, we prove some coupled coincidence point theorem for a pair {F,G} of mappings F,G:C^{2}→C without mixed G-monotone property of F. Our results improve and generalize results given by Karapinar et al. (Arab J Math (2012) 1: 329-339) and Jachymski (Nonlinear Anal. 74, 768-774 (2011)). The theoretic results are also accompanied with suitable example.

**Keywords:** coupled coincidence point, generalized compatibility, ordered set

Received December 29, 2015; Revised February 06, 2016; Accepted February 14, 2016

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Esra Yolacan, Mehmet Kir, Hukmi Kiziltunc. An Extended Coupled Coincidence Point Theorem.
*Turkish Journal of Analysis and Number Theory*. Vol. 4, No. 1, 2016, pp 23-30. http://pubs.sciepub.com/tjant/4/1/5

- Yolacan, Esra, Mehmet Kir, and Hukmi Kiziltunc. "An Extended Coupled Coincidence Point Theorem."
*Turkish Journal of Analysis and Number Theory*4.1 (2016): 23-30.

- Yolacan, E. , Kir, M. , & Kiziltunc, H. (2016). An Extended Coupled Coincidence Point Theorem.
*Turkish Journal of Analysis and Number Theory*,*4*(1), 23-30.

- Yolacan, Esra, Mehmet Kir, and Hukmi Kiziltunc. "An Extended Coupled Coincidence Point Theorem."
*Turkish Journal of Analysis and Number Theory*4, no. 1 (2016): 23-30.

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### 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.

**De****fi****nition 1.** ^{[1]} *An element** ** **is said to be a coupled **fi**xed point of the mapping** ** **if** ** **and** *

**De****fi****nition 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.

**De****fi****nition 3**. ^{[2]} *An element** ** **is said to be a coupled coincidence point of a mapping** ** **and** ** **if** ** **and*

**De****fi****nition 4. **^{[2]} Let be a partially ordered set and and . We say F has the mixed g-monotone property if for any

*and*

**De****fi****nition 5**. ^{[2]} *Let C be a nonempty set and** ** **and** ** **We say F and g are commutative if** ** **for all** *

**De****fi****nition 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:

(*φ*1) *φ* is continuous,

(*φ*2) *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 eithe**r*

*(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.

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

**De****fi****nition 8.** ^{[3]} *An element** ** **is said to be a coupled coincidence point of a mappings** ** **if** ** **and** *

**De****fi****nition 9.** ^{[3]} *Let** ** **We say that pair** ** **is generalized compatible if*

*whenever** ** **and** ** **are sequences in C such that*

**De****fi****nition 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 (*φ*2), 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*.

**De****fi****nition 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 (*φ*2), 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 de**fi**ned 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** ** **satis**fi**es the g**eneralized compatibility hypoth**esis. Let** ** and** ** **be two sequences in C such that*

*Then we** must have** ** **and one can easly prove that*

*Let** ** **be de**fi**ned 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.

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