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ON THE EXISTENCE OF UNIQUE COMMON FIXED POINT OF TWO MAPPINGS IN A METRIC-LIKE SPACE
Mainak Mitra 1
, Jigmi Dorjee
Bhutia 2, Kalishankar Tiwary 1
1 Department of Mathematics, Raiganj University, West Bengal, India
2 Department of Mathematics, Kalimpong College, Kalimpong,
West Bengal, India
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ABSTRACT |
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The main objective of this article is to introduce a fixed-point result involving two mappings satisfying contraction in a metric-like space. The article has been designed in the following manner. In the first section the authors have mentioned some definitions and commonly used notations. In the second section, they have mentioned some fixed-point results. In the third section the authors have introduced their main result and in the last section, using these results, the authors have obtained some conditions that assure the existence of a common fixed point of a pair of mappings. |
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Received 17 October 2022 Accepted 19 November 2022 Published 09 December 2022 Corresponding Author Mainak
Mitra, mr.mainakmitra@outlook.com DOI 10.29121/ijetmr.v9.i12.2022.1256-+
Funding: This research
received no specific grant from any funding agency in the public, commercial,
or not-for-profit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CC-BY, authors retain the copyright, allowing anyone to download,
reuse, re-print, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author.
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Keywords: Metric-Like
Space, 0-Complete Metric-Like Space, Common Fixed Point, F-Contraction |
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1. INTRODUCTION
In 1922, Banach (1922) introced his result, known as Banach’s Fixed Point
Theorem. After that several mathematicians worked on this result and made some
successful attemps to generalize his idea in other ways. Recently in 2012, as a
generalization of Banach’s Contraction Condition, Wardowski (2012) introduced a new type of contraction condition,
called an 
-contraction. After this introduction, several
mathemticians have widely used this idea to introduce some interesting results
of fixed point. Beside these generalizations, some other mathematicians tried
to generalize the notion of a Metric Space. Partial Metric Space Matthews (1994), Metric-Like Space Amini-Harandi (2012)are some notable generalizatons of the idea of a
Metric Space. During these years several results on fixed point have been
introduced on these spaces.
In our paper we’ll consider the Metric-Like Space and the -contraction
to investigate the existence of a unique common fixed point for a pair of
mapping. Foe that we’ll mention some definitions and results firts.
The idea of a Metric Space was introduced by M. Fréchet in the year 1906. Through this idea of Metric Space, he tried to define the distance between two points of an arbitrary set in an abstract manner as follows,
Definition 1.1 (Metric Space) Suppose 
and 
be a mapping such that
1) 
.
2) 
.
3) 
.
4) 
.
then the mapping 
will be called
a Metric or a Distance Function on 
and the ordered
pair 
will be called
a Metric Space.
Example 1.2 The set 
of all real numbers forms metric space with
respect to metric 
, defined by 
.
Example 1.3 The set 
of all complex numbers forms metric space with
respect to metric ,defined by 
, where 
In 1994, S.G. Matthew introduced a
generalization of metric space and referred it as a Partial Metric Space.
Definition 1.4 (Partial Metric Space) Matthews (1994)
Suppose and 
be a mapping such that
1) 
; 
.
2) 
.
3) 
; .
4) 
; 
.
then the mapping 
will be called
a partial metric on and the ordered
pair 
will be called
a Partial Metric Space.
Example 1.5 In the set of real numbers forms a partial metric space
with respect to the mapping 
.
Example 1.6 In the set of real numbers forms a partial metric space
with respect to the mapping 
.
In 2012, A.A. Harandi introduced a new
generalization of metric space called a Metric-Like space
Definition 1.7 (Metric-Like Space) Amini-Harandi (2012)
Suppose and 
be a mapping such that
1) 
.
2) 
.
3) 
.
4) 
.
then the ordered pair will be called
a Metric-Like Space or a Dislocated Metric Space.
Example 1.8 In the set of real numbers forms a metric-like space with
respect to the mapping 
.
Example 1.9 In the set of real numbers forms a metric-like space with
respect to the mapping 
.
Example 1.10 The set 
of all real-valued continuous functions
defined on a compact interval 
forms a metric-like space with respect to te
mapping 
.
Remark 1.11 Clearly every Metric Space is a Partial Metric Space
as well as a Metric-Like Space. But 1.6, 1.8 shows that the converse is not
true.
Remark 1.12 Every Partial Metric Space is a Metric-Like Space but
1.9 shows that the converse is not true.
Definition 1.13 (Convergence of a Sequence) Amini-Harandi (2012) In a metric-like space 
suppose 
be a sequence. Then is said to converge to some limit 
if 
Definition 1.14 (Cauchy Sequence and Completeness) Amini-Harandi (2012)
1.
In a metric-like space a sequence is said to be a
Cauchy Sequence if 
exists
finitely.
2. A
metric-like space is said to be a
complete metric like space if for every Cauchy Sequence in there exists 
such that 
.
Remark 1.15 In a metric-like space the limit of a sequence may not
be unique. For example in the space 
where 
consider the sequence 
. Then for any real number 
, 
.
Remark 1.16 In a metric-like space a convergent sequence may not
be a Cauchy sequence.
Remark 1.17 In a complete metric like space if a sequence is a Cauchy sequence such that 
then its limit will be unique. For this,
suppose on the contrary that the limit of the sequence is not unique. Then
there will exist 
with the following properties
Then 
— a
contradiction
The above fact leads to the
following definition,
Definition 1.18 (0-Cauchy Sequence and 0-Complete Space) Shukla et al. (2013)
• In a metric like space a sequence will be called
a 0-Cauchy sequence if 
.
• A metricclike space is said to be a
0-Complete metric like space if every 0-Cauchy sequence in converges to
some point such that 
.
Clearly every 0-Cauchy sequence ia a Cauchy
sequence and a complete metric like space is a 0-complete metric like space.
Definition 1.19 (Coincidence Point and Point of Coincidence) Jungck (1996) Suppose and 
be two functions. A point 
is said to be a point of coincidence of 
and 
if there exists such that
The point is called a coincidence
point of and .
Definition 1.20 (Weakly Compatible Mapping) Jungck (1996)
Suppose and be two functions. Then and are said to be weakly compatible if they
commutes at their coincidence points i.e. if 
, for some then 
.
In 2012, D. Wardowski introduced the idea of -contraction in the following manner;
Definition 1.21 (-Contraction) Wardowski (2012)
Suppose, 
be a mapping such that
1. is strictly
increasing.
2. 
.
3. 
such that 
.
The followings are some
examples of such functions
Example 1.22 
Example 1.23 
Example 1.24
.
Now suppose be a metric-like
space and 
be a mapping
such that whenever 
, 
such that
then 
is said to be
an -contraction defined on .
Notations:
In our paper we’ll commonly
use the folloing notations
1) 
satisfies the
three conditons mentioned in Definition 1.21}
2) For a mapping 
,
3) For mappings 
,
4) For mappings ,
2. SOME RESULTS
Lemma 2.1 Karapinar and Salimi
(2013)
In a metric-like space the following results hold
1) 
.
2) 
.
3) If a sequence converges to such that , then for all 
converges to 
.
4) If be a sequence
in such that 
as 
, then 
as .
5) Suppose be a sequence
in such that 
. If 
then there
exists 
and two
subsequences 
and 
of such that the
following sequences
will converge to 
as 
Theorem 2.2 Amini-Harandi (2012)
Suppose be a complete metric-like space and be a map such that
where 
is a
non-decreasing function such that
1) 
.
2) 
.
3) 
.
Then has a fixed
point.
Theorem 2.3 Amini-Harandi (2012)
Suppose be a complete metric-like space and be a map such that
where 
is a
non-decreasing continuous function such that 
. Then has a unique
fixed point.
Theorem 2.4 Karapinar and Salimi
(2013) Suppose be a complete metric-like space and be a map such that
where is a
non-decreasing function such that
1) .
2) .
3) .
If the range of contains the
range of and 
or 
is a closed
subset of X, then and will have a
unique point of coincidence in X. Moreover if the mappings are weakly
compatible, then they will have a unique common fixed point 
such that 
.
For more fixed point
results in a metric-like space we refer Amini-Harandi (2012),
Karapinar and Salimi
(2013), Shukla et al. (2013), Fabijano et al.(2020).
3. MAIN RESULT
We now introduce our main
result;
Theorem 3.1 Suppose be a 0-complete metric like space and be two functions such that whenever 
there exists 
such that
for some 
. Then and will have a
unique common fixed point in X.
Proof. Suppose 
and define a
sequence as follows
i.e. in general
Further let us denote by
Now let us consider the
following cases
Case 1: Suppose 
for some 
, then 
.
If 
then,
Therefore 
is a fixed
point of .
Now if 
, then we have Equation
1 we have
Where,
Therefore from Equation
2 we have
— a contradiction.
Hence 
i.e. 
is the fixed
point of and
consequently the common fixed point of and . If 
, then proceeding in the similar way we can prove that
and will have a
common fixed point.
Therefore if for some then and will have a
common fixed point.
Case 2: Now let us assume that 
.
Then for 
and 
we have from Equation
1
Where,
If 
, then Equation
3 will imply
Equation 4
—a contradiction.
Thus
Similarly, condsidering 
and in Equation
1 we can prove that
Therefore
Thus 
is a monotonic
decreasing sequence. Since it is bounded below by 
, it is convergent. Suppose the
.
Now if 
, then taking limit as on both sides
of (??) we have
— a contracdiction.
Therefore
Now we claim that is a 0-Cauchy
Sequence i.e.
. For this on
the contrary, let us assume that is not a
0-Cauchy Sequence. Then from Lemma 2.1 we have, 
and two
subsequences 
of such that the
following sequences
converges to .
Now if 
and 
are both even
then taking 
and 
we have
Where,
=
Taking in the above
inequality we have
— a contradiction as .
If and are both odd or
one is even and other is odd then choosing suitable terms as and 
as above we
will arrive at a contradiction.
This proves that is a 0-Cauchy
Sequence. Since is a 0-complete
metric like space, thus is converges to
with 
.
Now if