**Turkish Journal of Analysis and Number Theory**

##
(α_{o}-λ_{o})-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results

Department of Mathematics University of Peshawar, Peshawar 25000, Pakistan
Abstract | |

1. | Introduction and Preliminaries |

2. | Multiplicative (α_{o}, λ_{o})-contraction and Fixed Point Results |

References |

### Abstract

In this manuscript we introduce new type of contraction mapping in the framework of multiplicative metric space and some fixed point results. Also some example for the support of our constructed results.

**Keywords:** complete multiplicative metric space, multiplicative contraction mapping, multiplicative (α_{o}-λ_{o})-contraction, fixed point

Received February 09, 2016; Revised June 05, 2016; Accepted June 13, 2016

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

### Cite this article:

- Bakht Zada. (α
_{o}-λ_{o})-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results.*Turkish Journal of Analysis and Number Theory*. Vol. 4, No. 3, 2016, pp 67-73. http://pubs.sciepub.com/tjant/4/3/3

- Zada, Bakht. "(α
_{o}-λ_{o})-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results."*Turkish Journal of Analysis and Number Theory*4.3 (2016): 67-73.

- Zada, B. (2016). (α
_{o}-λ_{o})-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results.*Turkish Journal of Analysis and Number Theory*,*4*(3), 67-73.

- Zada, Bakht. "(α
_{o}-λ_{o})-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results."*Turkish Journal of Analysis and Number Theory*4, no. 3 (2016): 67-73.

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### 1. Introduction and Preliminaries

The Banach-contraction principal was introduced by Banach ^{[1]}. It is one of the important results for metric fixed point theory and also vast applicability in mathematical analysis, like used to establish the existence of solution of integral equation. After Banach contraction mapping, a new type of contraction mapping was introduced by Kannan ^{[5, 6]}, which is known as Kannan-contraction. Many researcher work on the generalization and fixed point theory of Kannan-contraction mapping like in ^{[4, 8, 9, 11]}. Like Kannan, Chatterjea ^{[3]} also introduced a similar contractive condition and fixed point theorems in metric space. After that, in 2008, a new concept of multiplicative distance was introduced by Bashirov ^{[2]}.

**Definition ****1.1** *Let** ** **be a non-empty set, then multiplicative metric is a** **mapping** ** **satisfying the following conditions*:

(1) * **for all** *,

(2) * **if and only if** *,

(3) ,

(4) * **for all *.

*The pair** ** **is known as multiplicative metric space*.

Ozavsar and Cevikel ^{[10]} studied multiplicative metric space and its topological properties, they also introduce the concepts of Banach-contraction, Kannan-contraction and Chatterjea-contraction mappings in the framework of multiplicative metric space and proved fixed point results on complete multiplicative metric space.

**Definition 1.2** ^{[10]} *Let** ** **be a multiplicative metric space then the** **mapping** ** **is multiplicative Banach-contraction if*

(1.1) |

*for all** ** **where** *

**Definition 1.3** ^{[10]} *Let ** **be a multiplicative metric space then the mapping** ** **is multiplicative Kannan-contraction if*

(1.2) |

*for all** ** **where*

**Definition 1.4** ^{[10]} *Let** ** **be a multiplicative metric space then the** **mapping** ** **is multiplicative Chatterjea-contraction if*

(1.3) |

*for all** ** **where** *

The concept of α_{o}-admissible mapping was introduced by B. Samet, C. Vetro and P. Vetro ^{[7]}:

**Definition 1.5 ***Suppose** **, **and let** ** **be a mapping.** **Then** ** **is said to be** **-admissible mapping if**:*

*for all** ** **for which*

### 2. Multiplicative (α_{o}, λ_{o})-contraction and Fixed Point Results

Now we will introduce (α_{o}, λ_{o})-contraction mapping in the framework of multiplicative metric space.

Let be the class of functions for which for all where , and is self-mapping.

**Definition 2.1** *Suppose** ** **be a multiplicative metric space and let a mapping** ** **then** ** **is said to be multiplicative** **-Banach-contraction if there exists** ** **and ** **s**uch that*

(2.1) |

*for all** ** **where** *

**Remark 2.2** *When** ** **for all** ** **and** ** **for all** ** **where** ** **then multiplicative** **-contraction mapping reduces to multiplicative Banach-contraction mapping*.

**Example 2.3*** ** **is multiplicative metric space, where** ** **and** ** **be de**fi**ned as follows:*

*for all** ** **where** ** **is de**fi**ned by*

(2.2) |

*we de**fi**ne the mapping** ** **and** ** **as follows*:

(2.3) |

*and*

(2.4) |

*for all** ** **where** ** **is de**fi**ned by** ** ** **is** **multiplicative** **-Banach-contraction mapping*.

**NOTE.** In the above example is not multiplicative Banach-contraction mapping: that is, for and we have

for all

So this mapping is said to be extension of multiplicative Banach-contraction mapping.

Now we prove some fixed point results for Multiplicative -Banach-contraction mapping.

**Theorem 2.4** *Let** ** **be a complete multiplicative metric space and** **assume that** ** **be** **-Banach-contraction mapping satisfying** **the conditions*:

1. *there exist** ** **such that *;

2. *T**0 **is** **-admissible*;

3. *one of the conditions holds*;

(a) *is continuous*;

(b) if a sequence such that for all and as then

*Then** ** **has a **fi**xed point*.

*(A1) If ** **for all** fi**xed point *

*(A2) there exist ** **such that ** **and ** **for all *

*then** ** **has a unique **fi**xed point.*

**Proof.** Assume such that Define the sequence such that for all

(2.5) |

Assume that

Since is -admissible and and similarly by induction, we get

Applying Inequality (2.1) with and we have

Again, using inequality (2.1) with and we have

By continuing this process, we get

(2.6) |

As we get is multiplicative Cauchy sequence in

From the completeness of there exist such that as

We suppose that is continuous from condition(3a), so

Now, we suppose that condition(3b) holds: As is multiplicative Cauchy sequence. So, there exist such that as

From condition(3b), we have

And

As for all Therefore, we have

Assume that in the above inequality, we get that is which shows that is fixed point of

To show uniqueness of let is another fixed point of if condition(A1) holds, then the fixed point is unique from (2.1). Now we have to show that condition(A2) holds. From(A2), we have such that

(2.7) |

As is -admissible, from (2.7), we have

(2.8) |

So

Taking in the above inequality, we have

and similarly

By uniqueness of limit, we have which shows the uniqueness of fixed point.

**De****fi****nition 2.5*** **Suppose** ** **be a multiplicative metric space and let a** **mapping** ** then** ** **is said to be multiplicative** **-Kannan-contraction if there exists** ** **and** ** **such that*

(2.9) |

*for all** ** **where*

**Definition 2.6** *Suppose** ** **be a multiplicative metric space and let** **a mapping** ** then** ** **is said to be multiplicative** **-Chatterjea-contraction if there exists** ** **and** ** **such that*

(2.10) |

*for all** ** where*

**Theorem 2.7** *Let** ** **be a complete multiplicative metric space and** **assume that** ** **be** **-Kannan-contraction mapping satisfying** **the conditions*:

1. *there exist** ** **such that** **;*

2.* ** **is **-admissible;*

3.* one of the conditions holds;*

*(a)** ** **is continuous;*

*(b) if a sequence** ** **such that** ** **for all** ** **and** ** **as** ** **then** *

Then has a fixed point.

*(B1) If** ** **for all **fi**xed point** *

*(B2) there exist** ** **such that** ** **and** ** **for all** *

*then ** **has a unique **fi**xed point*.

**Proof.** Assume such that Define the sequence such that for all

(2.11) |

Assume that

Since is -admissible and and similarly by induction, we get

Applying Inequality (2.9) with and we have

and so

Suppose such that we have

Taking we get and is multiplicative Cauchy sequence in

From the completeness of there exist such that as

We suppose that is continuous from condition(3a), so

Now, we suppose that condition(3b) holds: As is multiplicative Cauchy sequence. So, there exist such that as

From condition(3b), we have

And

As for all Therefore, we have

Assume that in the above inequality, we get that is which shows that is fixed point of

To show uniqueness of let is another fixed point of if condition(B1) holds, then the fixed point is unique from (2.9). Now we have to show that condition(B2) holds. From(B2), we have such that

(2.12) |

As is -admissible, from (2.12), we have

(2.13) |

So

Taking in the above inequality, we have

and similarly

By uniqueness of limit, we have which shows the uniqueness of fixed point.

**Theorem 2.8** *Let ** be a complete multiplicative metric space and as-sume that** ** **be** **-Chatterjea-contraction mapping satisfying the conditions*:

*1. there exist** ** **such that** **;*

*2. ** **is** ** **-admissible;*

*3. one of the conditions holds;*

*(a) ** **is continuous;*

*(b) if a sequence** ** **such that** ** **for all** ** **and** ** **as** ** **then** *

*Then** ** **has a **fi**xed point.*

*(C1) If** ** **for all **fi**xed point** *

*(C2) there exist** ** **such that** ** **and** ** for all** *

*then** ** **has a unique **fi**xed point.*

**Proof.** Assume such that Define the sequence such that for all

(2.14) |

Assume that

Since is -admissible and and similarly by induction, we get

Applying Inequality (2.10) with and we have

and so

Suppose such that we have

Taking we get and is multiplicative cauchy sequence in

From the completeness of there exist such that as

We suppose that is continuous from condition(3a), so

Now, we suppose that condition(3b) holds: As is multiplicative Cauchy sequence. So, there exist such that as

From condition(3b), we have

And

As for all Therefore, we have

Assume that in the above inequality, we get that is which shows that is fixed point of

To show uniqueness of let is another fixed point of if condition(C1) holds, then the fixed point is unique from (2.9). Now we have to show that condition(C2) holds. From(C2), we have such that

(2.15) |

As is -admissible, from (2.15), we have

(2.16) |

So

Taking in the above inequality, we have

and similarly

By uniqueness of limit, we have which shows the uniqueness of fixed point.

**Remark 2.9** *The multiplicative** **-Banach-contraction mapping, multiplicative** **-Kannan-contraction mapping and multiplicative** **-Chatterjea-contraction mapping is the generalization of multiplicative Banach-contraction mapping, multiplicative Kannan-contraction mapping and multiplicative Chatterjea-contraction mapping respectively, i.e by simply **putting** ** **in De**fi**nition 2.1,** ** **in De**fi**nition 2.5 and2.6 with** ** **we obtain multiplicative Banach-contraction mapping,** **multiplicative Kannan-contraction mapping and multiplicative Chatterjea-contraction** **mapping respectively*.

### References

[1] | Banach, Sur les operations dans les ensembles abstrait et leur application aux equations integrales. Fundam. Math.3, 133-181, (1922). | ||

In article | |||

[2] | Bashirov, Kurpunar, Ozyapici, Multiplicative calculus and its applications. Math. Anal. Appl. 337, 36-48, (2008). | ||

In article | View Article | ||

[3] | Chatterjea, Fixed point theorems. Acad. Bulgare Sci. 25, 727-730, (1972). | ||

In article | |||

[4] | Ghosh, A generalization of contraction principle. Int. J. Math. Math. Sci. 4(1), 201-207, (1981). | ||

In article | View Article | ||

[5] | Kannan, Some results on fixed points. Calcutta Math. Soc. 60, 71-76, (1968). | ||

In article | |||

[6] | Kannan, Some results on fixed points. II. Math. Mon. 76, 405-408, (1969). | ||

In article | View Article | ||

[7] | B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings,. Nonlinear Analysis: Theory, Methods and Applications, vol. 75, no. 4, pp. 2154-2165, (2012). | ||

In article | |||

[8] | Shioji, N, Suzuki, T, Takahashi, Contractive mappings, Kannan mappings and metric completeness. Proc. Am. Math.Soc. 126, 3117-3124, (1998). | ||

In article | View Article | ||

[9] | Subrahmanyam, Completeness and fixed-points. Monatshefte Math. 80, 325-330, (1975). | ||

In article | View Article | ||

[10] | Ozavsar, Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces. arXiv:1205.5131v1 [math.GM] (2012). | ||

In article | |||

[11] | Zamfirescu, Fixed point theorems in metric spaces. Arch. Math. 23, 292-298, (1972). | ||

In article | View Article | ||