**Journal of Mathematical Sciences and Applications**

## Generalized (ψ,φ)-weak Contractions In 0-complete Partial Metric Spaces

**Mehmet Ali Akturk**^{1,}, **Esra Yolacan**^{2}

^{1}Istanbul University, Faculty of Engineering, Department of Engineering Sciences, Avcilar Campus-34320, Istanbul, Turkey

^{2}Republic of Turkey Ministry of National Education, Mathematics Teacher, 60000 Tokat, Turkey

### Abstract

In this paper, we prove some common fixed point theorems in 0-complete partial metric spaces. Our results extend and generalize many existing results in the literature. Some examples are included which show that the generalization is proper.

**Keywords:** partial metric space, weak contraction, fixed point

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

### Cite this article:

- Mehmet Ali Akturk, Esra Yolacan. Generalized (ψ,φ)-weak Contractions In 0-complete Partial Metric Spaces.
*Journal of Mathematical Sciences and Applications*. Vol. 4, No. 1, 2016, pp 14-19. https://pubs.sciepub.com/jmsa/4/1/3

- Akturk, Mehmet Ali, and Esra Yolacan. "Generalized (ψ,φ)-weak Contractions In 0-complete Partial Metric Spaces."
*Journal of Mathematical Sciences and Applications*4.1 (2016): 14-19.

- Akturk, M. A. , & Yolacan, E. (2016). Generalized (ψ,φ)-weak Contractions In 0-complete Partial Metric Spaces.
*Journal of Mathematical Sciences and Applications*,*4*(1), 14-19.

- Akturk, Mehmet Ali, and Esra Yolacan. "Generalized (ψ,φ)-weak Contractions In 0-complete Partial Metric Spaces."
*Journal of Mathematical Sciences and Applications*4, no. 1 (2016): 14-19.

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### 1. Introduction and Preliminaries

Partial metric spaces were introduced by Matthews in ^{[9]} as a part of the study of denotational semantics of dataow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation ^{[10, 11, 12, 13, 14]}.

**De****fi****nition 1.** ^{[9]} *A partial metric on a nonempty set** ** **is a function** ** **such that for all** ** *

*(pms1)** *

*(pms2)** *

*(pms3)** *

*(pms4)** *

*The pair** ** **is called a partial metric space*.

If is a partial metric on then the function given by is a metric on Each partial metric p on inroduces a topology on which has as a base the family of open balls for all and

**De****fi****nition 2**. ^{[9]} Let be a partial metric space, and let be any sequence in and Then

*(a) a sequence** ** **is convergent to x with respect to** ** **if** **;*

*(b) a sequence** ** **is a Cauchy sequence in** ** **if** ** **existsand is **fi**nite;*

*(c) ** **is called complete if for every Cauchy sequence** ** in** ** **there exists** ** **such that** *

In 2010, Romaguera proved in [4-Theorem 2.3] that a partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point. Since then several papers have dealt with fixed point theory for single-valued and multi-valued operators in 0-complete partial metric space (see [1-8]^{[1]} and references therein).

**De****fi****nition 3**. ^{[4]} *Let** ** **be a partial metric space*.* A sequence** ** **in X is called a** *0*-**Cauchy sequence if** ** **The space** ** is said to be** ** *0*-**complete if every *0*-**Cauchy sequence in** ** **converges with respect to** ** **to a point** ** **such that** *

**Remark 1.** ^{[15, 16]} *Let** ** **be a partial metric space*. *If** ** **as** ** **then** ** **as** ** **for all** ** *

**Lemma 1.** ^{[2]} *Let** ** **be a partial metric space and let** ** **be a sequence in** ** **such that*

(1.1) |

*If** ** **is not *0-*Cauchy sequence in** **, then there exists** ** **and two sequences** ** **and** **of positive integers such that** ** **and the following sequences tend to** ** **as** *

(1.2) |

**Definition 4**. ^{[17]} *Let f and g be self maps of a set X*. *If** ** **for some** ** **then x is called a coincidence point of f and g*, *and w is called a point of coincidence of f and g*. *The pair f, g of self maps is weakly compatible if they commute at their coincidence points*.

**Proposition 1.** ^{[17]} *Let f and g be weakly compatible self maps of a set X. If f and g have a unique point of coincidence w = fx = gx, then w is the unique common **fi**xed point of f and g*.

### 2. Main Results

Denote by the set of functions satisfying the following conditions:

is continuous nondecreasing;

for all and

Denote by the set of functions satisfying the following conditions:

is a lower semi-continuous functions;

for all and

**Theorem 1.** *Let** ** **be a *0-*complete partial metric spaces. Suppose mappings** ** **satisfy*

(2.1) |

*where** ** **and ** **and*

(2.2) |

*for all** ** **If the range of g contains the range of f and** f (X) or g (X) is a closed subset of X*,* then f and g have a unique point of coincidence in X. Moreover*,* if f and g are weakly compatible*,* then f and g have a unique common fixed point z and** *

*Proof.* First, we prove that f and g have a unique point of coincidence (if it exists). If with and with , we assume c1 6= c2. Using (2.1) and (2.2), we have

which is a contradiction. Thus , that is, Thus, the point of coincidence of *f *and *g* is unique (if it exists).

We construct a sequence as follows:

Let Choose a point such that This can be done, as the range of *g* contains the range of *f*. Continuing in the same way, having chosen we get such that (say). Therefore, we get the sequence such that for all Consider the two possible cases:

(i) for some

In this case is a point of coincidence and then the proof is finished.

(ii) for every

From (2.1) and (2.2), using properties of functions and , we obtain

which implies that

Then, we have

If then Furthermore, it implies that

which is a contradiction. Therefore, we have

(2.3) |

It follows from (2.3) that the sequence is nonincreasing. Therefore,

Letting in inequality

we obtain and Thus

(2.4) |

We next prove that is a 0-Cauchy sequence in the space . It is sufficient to show that is a 0-Cauchy sequence. Suppose the opposite. Then using Lemma 1, we see that there exist and two sequences and of positive integers and sequences

(2.5) |

all tend to when Using (2.1) and (2.2), we get that

(2.6) |

Using (2.4) and (2.5), we obtain

Letting in (2.6), we get that which is a contradiction if

This show that is a 0-Cauchy sequence in the space and is a 0-Cauchy sequence in the space

If is closed in then there exist such that and

Now, putting and in (2.1) and (2.2) we have

(2.7) |

Letting in (2.7) and by Remark 1, we obtain

This implies that is, Hence,* f *and *g* have a unique point of coincidence. By Proposition 1, *f* and *g* have a unique common fixed point.

When is closed set in the proof similar.

**Corollary 1**. *Let** ** **be a *0*-**complete partial metric spaces*.* Suppose mapping** * *satisfy*

(2.8) |

*where ** **and** ** **and*

(2.9) |

*for all** ** **Then f has a unique **fi**xed point** ** **and *

*Proof*. Taking (the identity mapping of ), along the lines of the proof of Theorem 1, we get the desired results. In view of the analogy, we skip the details of the proof.

**Corollary 2**. *Let** ** **be a *0-*complete partial metric spaces*. *Suppose mappings** ** **satisfy*

(2.10) |

*where ** **and*

(2.11) |

*for all** ** **If the range of g contains the range of f and f (X) or g (X) is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point z and** *

*Proof*. To prove the above corollary it suffices to take in Theorem 1.

**Corollary 3**. *Let (X; p) be a *0*-**complete partial metric spaces. Suppose mapping f : X ! X satisfy*

(2.12) |

*where ** **and*

(2.13) |

*for all** ** **Then** ** **has a unique **fi**xed point** ** **and** *

*Proof*. Taking in Corollary 2, we have desired results.

**Corollary 4.** ^{[2]} *Let** ** **be a *0-*complete partial metric spaces. Suppose mappings** ** **satisfy*

(2.14) |

*where* * **and*

(2.15) |

*for** all** ** **If the range of g contains the range of f and** ** **or** ** **is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point z and** ** *

*Proof.* To prove the above corollary it suffices to take in Corollary 2.

**Corollary 5**. *Let** ** **be a *0-*complete partial metric spaces. Suppose mapping** ** **satisfy*

(2.16) |

*where** ** **and*

(2.17) |

*for all** ** **Then f has a unique** fi**xed point** ** **and ** *

**Corollary 6.** ^{[18]} *Let** ** **be a *0-*complete partial metric spaces. Suppose mapping** ** **and there exist nonnegative constants **b**i **satisfying** ** **such** **that, for each** ** *

(2.18) |

*Then** f has a unique **fi**xed point** ** **and** *

Corollary 6 is a simple consequence of Corollary 5.

**Corollary 7.*** Let** ** be a *0-*complete partial metric spaces. Suppose mapping** ** **satisfy*

(2.19) |

*fo**r each** ** **and** *.* Then f has a unique **fi**xed point ** **and** *

*Proof*. It follows from Corollary 6.

**Conclusion 1.** *1. Our theorems and corolaries which include the corresponding results announced in Boyd and Wong *^{[19]}* (1969), Rhoades *^{[20]}* (1977) as special cases fundamentally improve and generalize the results of Ahmad et al. *^{[2]}* (2012) and Radenovi**ć** *^{[18]}* (2013)*.

*2. Taking** ** **and ** **in Corollary 6, we obtain extension of Kannan Theorem on a 0-complete partial metric spaces.*

*3. Taking** ** **and** ** **in Corollary 6, we obtain extension of Reich Theorem on a *0-*complete partial metric spaces.*

*4. Taking** ** **and** ** **in Corollary 6, we obtain extension of Chatterjea Theorem on a *0-*complete partial metric spaces.*

Now, we give a example which illustrate Theorem 1.

**Example 1.** *Let** **, **and le** ** **be de**fi**ned by** ** **for all** ** **Then,** ** **is a 0-complete partial metric space. De**fi**ne** *

*Take** ** **and** ** **for eac**h *

*We distinguish **fi**ve cases:*

**Case 1:***If (x = *0* and y = *0*) or (x *=* *0 *and y = *1*) or (x = 0 and y = *2*) or (x = *1* and y = *1*) or (x = *1* and y *=* *2*) or (x = *2* and y = *2*)*, *we have*

*where*

**Case 2:*** If x = *0* and y = *3*, we have*

*and*

*Hence,*

* Case 3*:

*If x*= 1

*and y*= 3,

*we have*

*and*

*Thus,*

**Case 4: ***If x *= 2 *and y* = 3,* we have*

*and*

*Thus,*

**Case 5:*** If x *= 3 *and y *= 3, *we have*

*and*

*Thus,*

*It is obvious that all the condition of Theorem 1 is satisfied. Therefore, we apply Theorem 1 and f and g have a unique common **fi**xed point, i.e. *0.

The following is a example which illustrate our results and that the generalizations are proper.

**Example 2.*** Let** ** **and let** ** **be de**fi**ned by** ** **for all** ** **Then, ** **is a 0-complete partial metric space, but it is not complete partial metric space. De**fi**ne** ** **by*

*Then all the conditions of Theorem 1 are satis**fi**ed with** ** **and** ** **and f and g have a unique common **fi**xed point, i.e. *0.

**Acknowledgement 1**. The authors wish to thank the editor and referees for their helpful comments and suggestions.

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