ON THE EXISTENCE OF UNIQUE COMMON FIXED POINT OF TWO MAPPINGS IN A METRIC-LIKE SPACE Mainak Mitra 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 In our paper we’ll consider the Metric-Like Space and the 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 1) 2) 3) 4) then the mapping Example 1.2 The set Example 1.3 The set 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 1) 2) 3) 4) then the mapping Example 1.5 In the set Example 1.6 In the set 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 1) 2) 3) 4) then the ordered pair Example 1.8 In the set Example 1.9 In the set Example 1.10 The set 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 Definition 1.14 (Cauchy Sequence and Completeness) Amini-Harandi (2012) 1.
In a metric-like space 2. A
metric-like space Remark 1.15 In a metric-like space the limit of a sequence may not
be unique. For example in the space 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 Then 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 metricclike space 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 The point Definition 1.20 (Weakly Compatible Mapping) Jungck (1996)
Suppose In 2012, D. Wardowski introduced the idea of Definition 1.21 ( 1. 2. 3. The followings are some
examples of such functions Example 1.22 Example 1.23 Example 1.24
then Notations: In our paper we’ll commonly
use the folloing notations 1) 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 1) 2) 3) If a sequence 4) If 5) Suppose will converge to Theorem 2.2 Amini-Harandi (2012)
Suppose where 1) 2) 3) Then Theorem 2.3 Amini-Harandi (2012)
Suppose where Theorem 2.4 Karapinar and Salimi
(2013) Suppose where 1) 2) 3) If the range of 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 for some Proof. Suppose i.e. in general Further let us denote by Now let us consider the
following cases Case 1: Suppose If Therefore Now if Where, Therefore from Equation
2 we have — a contradiction. Hence Therefore if Case 2: Now let us assume that Then for Where,
If Equation 4 —a contradiction. Thus Similarly, condsidering Therefore Thus . Now if — a contracdiction.
Therefore Now we claim that converges to Now if Where, = Taking — a contradiction as If This proves that Now if Since Where, Thus taking — a contradiction. Therefore Thus Now if where Therefore from the above
inequality we have — a contradiction. Therefore Thus Therefore Uniqueness: To prove the
uniqueness let us assume on the contrary that there exists two common fixed
points 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 Then Proof. Considering Theorem 4.2 Suppose Then Proof. Considering 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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