(αo-λo)-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results
Department of Mathematics University of Peshawar, Peshawar 25000, PakistanAbstract | |
1. | Introduction and Preliminaries |
2. | Multiplicative (αo, λo)-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 (αo-λo)-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. (αo-λo)-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. "(αo-λo)-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). (αo-λo)-Contractive Mapping in Multiplicative Metric Space and Fixed Point Results. Turkish Journal of Analysis and Number Theory, 4(3), 67-73.
- Zada, Bakht. "(αo-λo)-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 αo-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 (αo, λo)-contraction and Fixed Point Results
Now we will introduce (αo, λo)-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.
References
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[3] | Chatterjea, Fixed point theorems. Acad. Bulgare Sci. 25, 727-730, (1972). | ||
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