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 such 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 defined as follows:

for all where is defined by

(2.2)

we define the mapping and as follows:

(2.3)

and

(2.4)

for all where is defined 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. T0 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.

(A1) If for all fixed point

(A2) there exist such that and for all

then has a unique fixed 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.

Definition 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 fixed point

(B2) there exist such that and for all

then has a unique fixed 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 fixed point.

(C1) If for all fixed point

(C2) there exist such that and for all

then has a unique fixed 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 Definition 2.1, in Definition 2.5 and2.6 with we obtain multiplicative Banach-contraction mapping, multiplicative Kannan-contraction mapping and multiplicative Chatterjea-contraction mapping respectively.

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