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
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 then . Since is a 0-Cauchy
sequence, thus uniqueness of limit implies that does not
converge to . Taking and in Equation 1 we have Where, Thus taking in the above
inequality we have — a contradiction. Therefore Thus is a fixed
point of . Now if then taking in Equation
1 we have, where Therefore from the above
inequality we have — a contradiction. Therefore Thus is a fixed
point of . Therefore and has a common
fixed point. Uniqueness: To prove the
uniqueness let us assume on the contrary that there exists two common fixed
points and of and . Then . Then taking and in Equation 1 we have Where, Therefore from the above inequality we have — a contradiction. Thus 4. APPLICATIONS The following results are
the direct applications of the above theorem; Theorem 4.1 Suppose be a 0-complete metric like space and be two functions such that whenever there exists such that Equation 5 Then and will have a
unique common fixed point in X. Proof. Considering in Equation
1 we can have the following results. Theorem 4.2 Suppose be a 0-complete metric like space and be two functions such that whenever there exists such that Equation 6 Then and will have a
unique common fixed point in X. Proof. Considering in equation Equation 1 we can have the following results. 5. AUTHORS CONTRIBUTION Conceptualization: M.Mitra,
J.D. Bhutia. Methodology: M.Mitra, J.D.
Bhutia, K. Tiwary. Formal Analysis: M.Mitra,
J.D. Bhutia, K. Tiwary. Investigation: M.Mitra,
J.D. Bhutia; Supervision: K. Tiwary. 6. CONSENT OF STATEMENT All authors have read and
agreed to publish this manuscript.
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