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Research Article

Open Access Peer-reviewed

Nisha Sharma, Lakshmi Narayan Mishra^{ }, Vishnu Narayan Mishra

Received November 18, 2017; Revised February 22, 2018; Accepted April 13, 2018

In this paper, we prove some common fixed point theorems for various types of compatible mappings in the setting of complete multiplicative metric spaces.

The study of expansive mappings is an interesting research area in the fixed point theory. It was grown in 1984 from the paper of Wang *et **al**.** *^{ 7} by introducing the concept of expanding mappings in the complete metric space. Daffer and Kaneko ^{ 10} used two self mappings to generalize the result of Wang in a complete metric space.

**Definition ****1.1. ***Let ** **be a nonempty set. A multiplicative** **metric is a mapping** **satisfying the following** conditions*:

i) and iff

ii)

iii) (multiplicative triangle inequality).

Then mapping together with, i.e. is a multiplicative metric space.

**Example 1.2. **(^{ 9}) Let be defined as where and . Then is a multiplicative metric and is a multiplicative metric space. We may call it usual multiplicative metric spaces.

**Example 1.3. **(^{ 9}) Let be a metric space. Define a mapping on by

where and . Then is a multiplicative metric and is known as the discrete multiplicative metric space.

In 2012, Ozavsar and Cevikel ^{ 5} gave the concept of multiplicative contraction mappings and proved some fixed point theorems of such mappings in a multiplicative metric space.

Also they proved the Banach Contraction Principle in the setting of multiplicative metric spaces as follows:

*“**Let f be a multiplicative contraction mapping of a complete multiplicative metric space ** **into itself. Then f has a unique fixed point*.”

In 1984, Wang et al. ^{ 7} and in 1993, Rhoades ^{ 8} proved some fixed point theorems for expansion mappings, which corresponds to some contractive mappings in metric spaces.

**Definition 1.6****. **Let *f* be a mapping of a multiplicative metric space into itself. Then *f *is said to be an *expansive mapping *if there exists a constant such that

Recently Kang et al. ^{ 6} introduced the notion of compatible mappings and its variants as follows:

**Definition 1.7****.** Let *f *and * *be two mappings of a multiplicative metric space into itself. Then *f *and are called

(1) *Compatible *if

where is a sequence in such that

for some

(2) *Compatible of type (A)* if

where is a sequence in such that

for some .

(3) *Compatible of type (B)* if

and

where is a sequence in such that

for some

(4) *Compatible of type (C)* if

and

where is a sequence in such that

for some

(5) *Compatible of type (P)* if

where is a sequence in such that

for some

Next is a result which is useful for our main results.

**Proposition 1.8****.**** ***Let f and ** be compatible mappings of a multiplicative metric space ** into itself. Suppose that **for some ** **Then ** if f is continuous at t*.

In 1993, Rhoades ^{ 8} proved the following fixed point theorem for expansive mappings in metric spaces as follows:

**Definition 2.1. ***Let f and ** be compatible mappings of a complete metric space into itself satisfying the condition *

*where **and assume that ** and f is continuous. Then f and **have a unique common fixed point*.

**Theorem 2.2. **Let *f* be a mapping of a multiplicative metric space into itself. Then *f *is said to be *multiplicative contraction* if a real constant such that

Now we generalized Theorem 2.2 in the string of setting of multiplicative metric spaces in the following way:

**Theorem 2.3****. ***Let f and ** be compatible mappings of a complete multiplicative metric space into itself satisfying the condition*

(2.1) |

* **where ** **and assume that ** and f is continuous. Then f and **have a unique common fixed point*.

*Proof. *Let Since such that In general, such that

(2.2) |

From (2.1), consider

(2.3) |

Again,

(2.4) |

Therefore using (2.3) and (2.4), we have

and so on.

In general, we have

(2.5) |

where or

Now for with consider

As It follows that the sequence is a multiplicative Cauchy sequence. Since is complete, we have

(2.6) |

Since *f *and are compatible and *f *is continuous, by Proposition 1.15,

(2.7) |

Consider

Letting we get

(2.8) |

Now we show that is fixed point of *f *and *. *Again considering

Letting we get

Hence is fixed point of *f *and *. *

Uniqueness follows easily from (2.1). This completes the proof.

Next we prove a common fixed point theorem for compatible mappings of type (B) as follows:

**Theorem 2.3****. ** *Let f and ** be compatible mappings of type (B) of a complete multiplicative metric space into itself satisfying the condition (2.1) and assume that ** and f is continuous. Then f and ** have a unique common fixed point.*

**Proof***. *From the proof of Theorem 2.2, is a multiplicative Cauchy sequence. Since is complete, there exists a point such that

Since *f *is continuous, we have

Since *f* and *g* are compatible of type (B), so

Letting we have

Consider

Again consider

[using (2.1)]

Letting we have

Since

Next we have to show that is a fixed point of *f *and , that easily follows from the proof of Theorem 2.2.

Uniqueness follows from (2.1).This completes the proof.

Now we prove a common fixed point theorem for compatible mappings of type (C) as follows:

**Theorem 2.4****. ***Let f and **be compatible mappings of type (C) of a complete multiplicative metric space into itself satisfying the conditions (2.1) and assume that ** and f is continuous. Then f and **have a unique common fixed point*.

*Proof.* From the proof of Theorem 2.2, is a multiplicative Cauchy sequence. Since is complete, there exists a point such that

Since *f *is continuous, we have

Since *f *and are compatible of type (C), so

Letting we have

Consider

Again consider

Letting we have

The rest of the proof follows easily from Theorem 2.2.

Now we prove a common fixed point theorem for compatible mappings of type (P) as follows:

**Theorem 2.5****. ***Let f and ** be compatible mappings of type (P) of a complete metric space into itself satisfying the condition (2.1) and assume that ** and f is continuous. Then f and **have a unique common fixed point*.

**Proof.*** *From the proof of Theorem 2.2, is a multiplicative Cauchy sequence. Since is complete, there exists such that

Since *f *and *g *are compatible of type (P) and *f *is continuous, we have

Consider

The rest of the proof follows easily from Theorem 2.2.

[1] | A.E. Bashirov, E.M. Kurplnara, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl. 337 (2008), 36-48. | ||

In article | View Article | ||

[2] | M. Abbas, B. Ali, Y.I. Suleiman, Common fixed points of locally contractive mappings in multiplicative metric spaces with application, Int. J. Math. Sci., 2015(2015), Article ID 218683, 7 pages. | ||

In article | View Article | ||

[3] | S.M. Kang, P. Kumar, S. Kumar, common fixed points for compatible mappings of types in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 1755-1767. | ||

In article | View Article | ||

[4] | X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric spaces, Fixed Point Theory Appl., 48 (2014), 9 pages. | ||

In article | View Article | ||

[5] | M. Ozavsar, A.C. Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces, arXiv:1205.5131v1 [math.GM], 2012. | ||

In article | View Article | ||

[6] | S.M. Kang, P. Kumar, S. Kumar, P. Nagpal, S.K. Garg, Common fixed points for compatible mappings and its variants in multiplicative metric spaces, Int. J. Pure Appl. Math., 102 (2015), 383-406. | ||

In article | View Article | ||

[7] | S.Z. Wang, B.Y, Li, Z.M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Japon., 29 (1984), 631-636. | ||

In article | |||

[8] | B.E. Rhoades, An expansion mapping theorem, Jnanabha, 23 (1993), 151-152. | ||

In article | |||

[9] | M. Sarwar, R. Badshah-e, Some unique fixed point theorems in multiplicative metric space, arXiv:1410.3384v2 [math.GM], 2014. | ||

In article | View Article | ||

[10] | P.Z. Daffer, H. Kaneko, on expansive mappings, Math. Japonica, 37(1992), 733-735. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2018 Nisha Sharma, Lakshmi Narayan Mishra and Vishnu Narayan Mishra

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

Nisha Sharma, Lakshmi Narayan Mishra, Vishnu Narayan Mishra. Fixed Point Theorems for Expansive Type Mappings in Multiplicative Metric Spaces. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 2, 2018, pp 52-56. https://pubs.sciepub.com/tjant/6/2/4

Sharma, Nisha, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Fixed Point Theorems for Expansive Type Mappings in Multiplicative Metric Spaces." *Turkish Journal of Analysis and Number Theory* 6.2 (2018): 52-56.

Sharma, N. , Mishra, L. N. , & Mishra, V. N. (2018). Fixed Point Theorems for Expansive Type Mappings in Multiplicative Metric Spaces. *Turkish Journal of Analysis and Number Theory*, *6*(2), 52-56.

Sharma, Nisha, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Fixed Point Theorems for Expansive Type Mappings in Multiplicative Metric Spaces." *Turkish Journal of Analysis and Number Theory* 6, no. 2 (2018): 52-56.

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[1] | A.E. Bashirov, E.M. Kurplnara, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl. 337 (2008), 36-48. | ||

In article | View Article | ||

[2] | M. Abbas, B. Ali, Y.I. Suleiman, Common fixed points of locally contractive mappings in multiplicative metric spaces with application, Int. J. Math. Sci., 2015(2015), Article ID 218683, 7 pages. | ||

In article | View Article | ||

[3] | S.M. Kang, P. Kumar, S. Kumar, common fixed points for compatible mappings of types in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 1755-1767. | ||

In article | View Article | ||

[4] | X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric spaces, Fixed Point Theory Appl., 48 (2014), 9 pages. | ||

In article | View Article | ||

[5] | M. Ozavsar, A.C. Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces, arXiv:1205.5131v1 [math.GM], 2012. | ||

In article | View Article | ||

[6] | S.M. Kang, P. Kumar, S. Kumar, P. Nagpal, S.K. Garg, Common fixed points for compatible mappings and its variants in multiplicative metric spaces, Int. J. Pure Appl. Math., 102 (2015), 383-406. | ||

In article | View Article | ||

[7] | S.Z. Wang, B.Y, Li, Z.M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Japon., 29 (1984), 631-636. | ||

In article | |||

[8] | B.E. Rhoades, An expansion mapping theorem, Jnanabha, 23 (1993), 151-152. | ||

In article | |||

[9] | M. Sarwar, R. Badshah-e, Some unique fixed point theorems in multiplicative metric space, arXiv:1410.3384v2 [math.GM], 2014. | ||

In article | View Article | ||

[10] | P.Z. Daffer, H. Kaneko, on expansive mappings, Math. Japonica, 37(1992), 733-735. | ||

In article | |||