Common Fixed Point Results of Suzuki-type Rational Z-contractions
PAIWAN WONGSASINCHAI, CHATUPHOL KHAOFONG
Department of Mathematics, Faculty of Science and Technology
Rambhai Barni Rajabhat University, Chanthaburi 22000, THAILAND
Division of Mathematics, Faculty of Science and Technology,
Rajamangala University of Technology Krungthep (RMUTK),
2 Nang Linchi Rd., Sathorn, Bangkok 10120, THAILAND
Abstract: -In this paper, we combine the -admissible mappings and the simulation function in this paper to
create the generalized version of Suzuki - type rational Z-contraction mapping. This notion is also employed in
the setting of metric spaces to get some common fixed point theorems. Appropriate examples are also provided
to validate the results acquired.
Key-Words: Suzuki - type rational, -admissible mappings, Z-contraction mapping, metric spaces
Received: October 5, 2022. Revised: December 8, 2022. Accepted: January 6, 2023. Published: February 2, 2023.
1 Introduction
Samet et al., [1], proposed --contractive type map-
ping and -admissible mappings. Karapinar and
Samet, [2], take the concept further by introducing
generalized --contractive type mapping broaden
the Banach contraction principle, Khojastesh et al.,
[3], presented simulation function and the notion of
Z-contraction with respect to simulation function. Ar-
goubi et al., [4], extend the results of Joonaghany et
al., [5]. In this paper, we introduce Suzuki - type ra-
tional Z-contraction.
For more results in rational type contractions and
Z-contractions, we refer to the papers in, [6], [7], [8],
[9], [10], [11], [12], [13], [14], [15], and references
therein.
2 Preliminaries
Throughout this article, the term refers to the set of
all nonnegative integers. Furthermore, represents
real numbers, and
Samet et al., [1], defined the class of - acceptable
mappings in 2012.
Definition 2.1. [1] A mapping is called
-admissible if for all we have
implies
where is a given function.
Definition 2.2. [1] Let be a nonempty set,
and The two
mappings is called a pair of -admissible
mappings, if
and implies
and and
and for all
Khojasteh et al., [3], introduced the simulation
function class in 2015. Furthermore, Argoubi et al.,
[4], modified the simulation function definition and
defined it as follows.
Definition 2.3. [4] A simulation function is a function
that satisfies the following
conditions
(i) for all
(i) if and are sequences in such
that
then
Joonaghany et al., [5], proposed a new concept
of the -simulation function, and with it, the Z-
contraction in the standard metric space. The con-
cept of the Z-contraction encompasses several dis-
tinct types of contraction, including the Z-contraction
defined in, [3].
Denote that is continuous
and nondecreasing, and
Definition 2.4. [5] We say that is a
-simulation function, if there exists such that
() for all
()if and are the sequences in
such that
then
Let Zis a collection of all -simulation func-
tions. Take note that “simulation” becomes “simu-
lation function” in sentence, [3].
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Example 2.5. [5] Let
(i) for all
where
(ii) for all
where so that and
for each
(ii) for all
where is a mapping
such that, for each
It is clear that Z
Lemma 2.6. [16] Let be a metric space, and
let be a sequence in such that
If numbers is not a Cauchy sequence. Then,
there exists an and monotone increasing se-
quences of natural and such that
and and
(i)
(ii)
(iii)
(iv)
Motivated by the all above results, we develop the
concept of Suzuki-type rational Z-contraction and
demonstrate several typical fixed point results in met-
ric spaces. We also provide an example that supports
our primary theorem.
3 Main Result
Now we state our main results.
Definition 3.1. Let be a metric space. Let
be two mappings. we call the pair
is Suzuki - type rational Z-contraction if for
all and such that
implies
(1)
where Z,
and
which
and
Theorem 3.2. Let be a complete metric space,
and let be two mappings and
Suppose that the following condi-
tions are satisfied
(i) is pair of - admissible mappings
(ii) there exists such that
and
(iii) the pair is Suzuki - type rational Z-
contraction
(iv) either, and are continuous or for every se-
quence in such that and
for all and
we have and
Then and have a unique common fixed point in
Proof. By condition (ii), there exists such that
. Define the sequence in by
letting such that
continuing in this manner, we obtain
and
From is a pair of -admissible, we have
and
continuing this process, we get
for all
In the same way, we get
for all
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If for some then is a
common fixed point for or . Consequently, we
assume that for all
Because
from (1), we have
and
So,
Because is strictly increasing, we have
(2)
where
(3)
and
(4)
which
(5)
and
(6)
From (4), (5) and (6), we obtain
If then by (2)
becomes
which is a contradiction. Thus we conclude that
(7)
By (2), we get
As a result, we can conclude that the sequence
is nonnegative and nonincreasing.
Therefore, there exists such that
We assert that Assume, on the other hand, that
(8)
For each we have
from (1), we have
where
and hence
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By condition of Definition 2.4, we have
which is a contradiction. Thus we conclude that
(9)
Now we will demonstrate that is a Cauchy
sequence. Assume, on the other hand, that
is not a Cauchy sequence. Then, there exists an
and monotone increasing sequences of nat-
ural numbers and such that and
and
(i)
(ii)
(iii)
(iv)
As a result of the definition of , we have
(10)
which
(11)
and
(12)
From (10), (11) and (12), we obtain
(13)
and hence
By condition of Definition 2.4, we have
(14)
In contrast, we assert that for sufficiently large
if then
(15)
When we let as in (15), we get the
contradiction. Therefore,
and from (1), we have
where
Therefore,
(16)
which contradicts (14). This contradiction proves that
is a Cauchy sequence, and since is complete,
there exists such that as
We assert that is a fixed point shared by and
Because and are continuous, we can conclude
that
and
Hence, that is, is a common fixed
point of and From (iv), we have for every se-
quence in such that and
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for all and as
this implies and as
Now we show that Assume
Now we assert that, for each at least
one of the following statements is true.
or
Assume, on the other hand,
and
For some we have
which is a contradiction, and thus the claim is true.
From (1), we have
implies
So,
Because is strictly increasing, we have
(17)
where
(18)
and
(19)
which
(20)
and
(21)
Letting in (19), we obtain
From (17), we have
(22)
where
Letting in (22), we obtain
which is a contradiction. Therefore, In the
same way, we can find that Therefore, the
pair has a common fixed point
We claim and have a unique common fixed
points Therefore
and Therefore,
and from (1), we have
Because is strictly increasing,
(23)
where
and
(24)
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which
(25)
and (26)
From (24), (25) and (26), we obtain
(27)
From (23) and (27), we have
which is a contradiction. Therefore, and have a
unique common fixed point.
Corollary 3.3. Let be a complete metric space,
and let be a mapping and
Assume that the following conditions
are satisfied
(i) if for all ,
implies
(28)
where Z,
and
(ii) is admissible mapping
(iii) there exists such that
(iv) either, and are continuous or for every se-
quence in such that and
for all and
we have and
Then has a unique fixed point in
Proof. The proof follows from Theorem 3.2 by taking
.
Example 3.4. Let and let
be defined by
if
if
We define by
and
for all . Let and are continuous self-
mappings on and are two
mappings defined by
if
otherwise
and
if
otherwise
We now define by
for all and
. Now
implies
where Zand
Therefore, for and the pair
is a Suzuki - type rational Zcontraction. In either
case and then pair
is a Suzuki - type rational Zcontraction.
As a result, the presumptions of Theorem 3.2 are
all met, and and have a common fixed point in
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Acknowledgment
Paiwan Wongsasinchai (paiwan.w@rbru.ac.th)
was financially supported by the Research and
Development Institute of Rambhai Barni Rajab-
hat University. Finally, Chatuphol Khaofong
(Chatuphol.k289@hotmail.com) was financially
supported by Rajamangala University of Technology
Krungthep (RMUTK).
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_US
Paiwan Wongsasinchai (paiwan.w@rbru.ac.th)
was financially supported by the Research and
Development Institute of Rambhai Barni Rajab-
hat University. Finally, Chatuphol Khaofong
(Chatuphol.k289@hotmail.com) was financially
supported by Rajamangala University of Technology
Krungthep (RMUTK).
