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.

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

(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.

References