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Fixed Point Theorems for Expansive Type Mappings in Multiplicative Metric Spaces

Nisha Sharma, Lakshmi Narayan Mishra*, Vishnu Narayan Mishra

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, Tamil Nadu, India

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

Keywords: multiplicative metric space, compatible mapping, expansive mappings

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

Citation: 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, 6 (2). 52-56. January 2018. doi:10.12691/tjant-6-2-4

1. Introduction and Preliminaries

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.

2. Main Results

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

[picture] | (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

[picture] | (2.2)

From (2.1), consider

[picture] | (2.3)

Again,

[picture] | (2.4)

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

and so on.

In general, we have

[picture] | (2.5)

where or

Now for with consider

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

[picture] | (2.6)

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

[picture] | (2.7)

Consider

Letting we get

[picture] | (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.

References

[1]   | A.E. Bashirov, E.M. Kurplnara, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl. 337 (2008), 36-48.
[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.
[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.
[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.
[5]   | M. Ozavsar, A.C. Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces, arXiv:1205.5131v1 [math.GM], 2012.
[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.
[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.
[8]   | B.E. Rhoades, An expansion mapping theorem, Jnanabha, 23 (1993), 151-152.
[9]   | M. Sarwar, R. Badshah-e, Some unique fixed point theorems in multiplicative metric space, arXiv:1410.3384v2 [math.GM], 2014.
[10]   | P.Z. Daffer, H. Kaneko, on expansive mappings, Math. Japonica, 37(1992), 733-735.