ON THE EXISTENCE OF UNIQUE COMMON FIXED POINT OF TWO MAPPINGS IN A METRIC-LIKE SPACE Mainak Mitra
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) 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,
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.
In 1994, S.G. Matthew introduced a
generalization of metric space and referred it as a Partial Metric Space.
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.
In 2012, A.A. Harandi introduced a new
generalization of metric space called a Metric-Like space
1) . 2) . 3) . 4) . then the ordered pair will be called
a Metric-Like Space or a Dislocated Metric Space.
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 .
Then — a
contradiction The above fact leads to the
following definition,
• 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.
The point is called a
In 2012, D. Wardowski introduced the idea of -contraction in the following manner;
1. is strictly
increasing. 2. . 3. such that . The followings are some
examples of such functions
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 .
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
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
where is a
non-decreasing function such that 1) . 2) . 3) . Then has a fixed
point.
where is a
non-decreasing continuous function such that . Then has a unique
fixed point.
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) , Fabijano et al.(2020).3. MAIN RESULT We now introduce our main
result;
for some . Then and will have a
unique common fixed point in X.
i.e. in general Further let us denote by Now let us consider the
following cases
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.
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 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 |