New Common Uniqueness Results on Generalized Normed Space
JAYASHREE PATIL1, BASEL HARDAN2*, AHMED A. HAMOUD3, KIRTIWANT P. GHADLE4
AND ALAA A. ABDALLAH5
1Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, Maharshtra,
INDIA.
2,5Department of Mathematics, Abyan University, Abyan, YEMEN.
3Department of Mathematics, Taiz University, Taiz P.O. Box 6803, YEMEN.
4Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad,
Maharashtra, INDIA.
Abstract: - In this study, by evaluating two mappings that do not both exhibit direct continuity features, fresh
results were found supporting the uniqueness of the solutions in generalized spaces.
Key-Words: - Contractive mapping, Fixed point, Banach space, Uniqueness.
Received: December 27, 2022. Revised: September 11, 2023. Accepted: October 14, 2023. Published: November 14, 2023.
1 Introduction
One of the key areas in the
development of functional analysis is fixed
point theory. Additionally, it has been
successfully applied in chemistry, biology,
economics, computer science, engineering, and
other scientific fields. Studying the fixed point
and confirming the uniqueness of the solution is
a very old study. Banach [1] is the first
developer of this study and presented it in its
basic form by presenting the Banach
contraction principle, which is considered the
raw material for all subsequent developments
that appeared in this field. Banach proved his
principle in a perfect one-dimensional space,
and then more exciting results followed after
that based on the development of space or the
development of contraction, see ([5],[7],[14-
18]). In our study, we dealt with the
development in space and the contraction as
well.
[2] introduced the idea of 2-Banach
spaces, and these spaces have since been
investigated by demonstrating the existence of
fixed points of contractive mappings. The fixed
point theory of mappings has been developed in
such spaces as it has in other spaces. Iseki [8]
also achieved the fundamental findings
regarding the fixed points of mappings in 2-
Banach spaces for the first time. After Iseki's
works fixed point results in these spaces are
found in [9].
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[3], offered a fascinating n-norm
theory on linear spaces then numerous authors,
including Kim et al. [10], Malceski [12], Misiak
[13], and Gunawan [4], have developed linear
n-normed spaces systematically. In a linear n-
Banach space current research on the functional
analysis parts we're referring to [6].
2 Main Results
A new form of the common fixed points results in
generalized Banach space will be discussed in this
section.
Definition 2.1. is a continuous mapping on an
Banach space at if every sequence
, satisfied
implies that
,
for all
Theorem 2.1. Let T be a continuous mapping
and be a contractive self-mapping on an
Banach space and let
such that
, (2.1)
for every in , satisfied
,
(2.2)
for all and , then and T
have a unique common fixed point in
Proof. Consider: ,
. Now
,
for all and
Hence,
,
implies .
Thus for any two positive integers with
and for all we get
.
Treating the case in a similar, we obtain
that is a Cauchy sequence in , and
since is complete, there exists a point
such that
Since is a contractive mapping thus it is a
continuous,
Hence, is a common fixed point of and .
Then the existence of (2.2) has been proven.
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Now for any suppose that
be another fixed point of and then
, where . Hence, for all
.
This implies Uniqueness has been
proven.
Theorem 2.2. Let and are contractive
mappings on an Banach spaces, suppose
for there exists a function:
such that
,
for all . Then and have a unique
common fixed point in .
Proof. For , take
and
Now ,
. Also, for all
and
Therefore
.
Thus,
Hence,
,
.
Thus for any two positive integers and
with and for all
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,
Treating the case in a similar we
conclude that is a Cauchy sequence in
, similarly we can show that is a
Cauchy sequence in , and since is
complete, there exists a common fixed point in
.
Let
and
that is for all
and
.
As and are continuous,
and
.
Therefore,
.
This implies
Similarly it can be show that .
Now for all ,
.
Hence, z is a common fixed point of and
.
Let be an another common fixed
point of and , then .
So,
.
Hence, is the unique common fixed point
of and in .
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3 Conclusion
The above results can be generalized to obtain a
conclusion whose proof is left to the reader
Theorem 3.1 Let be a family of
continuous self-mapping of an Banach
spaces , suppose that for any
there exists a function
, such that for
all
.
Then there exists a unique satisfying
for all
4 Recommendation
The nth spaces are possible, wide, sizes spaces,
so it's really interesting to develop this study to
search at specific conditions and contractions
for the existence of double or triple fixed points.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
B.H. and A.H.; methodology, B.H. and A.A.;
validation, B.H., A.H. and A.A.; formal analysis,
B.H.; resources, B.H. and A.A.; data curation,
B.H.; writing-original draft preparation, B.H.;
writing-review and editing, A.H., B.A. and A.A.;
supervision, A.H., J.P. and K.G. All authors have
read and agreed to the published version of the
manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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