Keywords
complete b-metric space, Altering distance function, α-admissible maping
complete b-metric space, Altering distance function, α-admissible maping
In this revised version of our article, we have thoroughly incorporated the reviewers' suggestions to enhance clarity and depth. Notably, we have provided a more detailed and explicit definition of contractive mappings, including the novel α−ψ−ϕ contractive mappings, which facilitates a better understanding of the theoretical framework presented. Additionally, we have clarified the role of the function ϕ in our main results, thereby strengthening the theoretical foundation of our findings.
The introduction has been expanded to include pertinent references, which contextualize our work within the broader landscape of fixed-point theory and highlight the significance of our contributions. We have also restructured Theorem 3.1 to provide a clearer and more unambiguous statement, ensuring that our main results are readily accessible to the reader.
Furthermore, we have included a comparative analysis between Theorem 3.1 and Theorem 3.2, elucidating their differences and the implications of each theorem within the framework of b-metric spaces. This version also features a discussion on the advantages of α−ψ−ϕ contractive mappings over traditional α−ψ contractive mappings, underscoring the practical applications of our findings in various mathematical and interdisciplinary fields.
Lastly, we have cited relevant literature that supports our claims and justifies the theoretical constructs used in our work. Overall, these revisions aim to provide a comprehensive and cohesive understanding of our novel contributions to fixed-point theory, addressing the feedback received while enhancing the article's overall quality.
See the authors' detailed response to the review by Tanakit Thianwan
See the authors' detailed response to the review by Konsalraj Julietraja and Agilan P
Banach's principle of contraction theory, as elucidated in Ref. 1, has served as a cornerstone for numerous research endeavors, facilitating expansions across various journals. The ubiquity of this theorem underscores its significance in extending the frontiers of mathematical discourse. Such expansions often entail imposing contraction restrictions and additional constraints on the ambient spaces, thereby giving rise to a diverse array of structures, including b-metric spaces.2,3 The notion of b-metric spaces, an extension of conventional metric spaces, has been instrumental in enriching the landscape of mathematical investigations.3–6 This versatile framework has found application in a myriad of contexts, contributing to discussions on topological structures and yielding insights into the interplay between fixed and common fixed points.7–9
In this context, our research aims to address the pressing need for a deeper understanding of contractive mapping within the framework of b-metric spaces.10 The fundamental question we seek to explore is how the principles of contraction theory can be extended to encompass this novel mapping concept. This investigation is motivated by the observation that while traditional contraction mappings have been extensively studied, the exploration of more generalized forms, such as contractive mappings, remains relatively uncharted territory. Our work not only seeks to fill this gap in the literature but also aims to contribute to the advancement of mathematical theory by introducing and elucidating the properties of contractive mappings. Through a meticulous synthesis of internet sources, scholarly publications, and pertinent literature, we establish the theoretical framework for this novel concept.
The novelty of our approach lies in the integration of contractive mappings into the framework of b-metric spaces, thus expanding the scope of both contraction theory and metric space theory. This novel synthesis opens up new avenues for exploration and offers fresh insights into the dynamics of fixed point theorems in non-standard spaces. In pursuit of our objectives, we present a comprehensive analysis, bolstered by corollaries, illustrative examples, and rigorous proofs, in support of our main result. By elucidating the intricacies of contractive mapping within the context of b-metric spaces, we aim to provide a solid foundation for further research and to inspire new avenues of inquiry in this vibrant field of mathematical exploration.
This research aims to extend traditional contraction theory by introducing contractive mappings within the framework of b-metric spaces, providing a more generalized approach to fixed point theory. While contraction mappings have been extensively studied in conventional metric spaces, the generalization to b-metric spaces is significant because it allows for a broader range of applications, especially in real-world contexts where strict metric conditions, such as symmetry or the triangle inequality, may not hold exactly. This flexibility makes b-metric spaces highly relevant for practical applications in areas like optimization, dynamic systems, and computer science. Our work not only fills an existing gap in the literature by exploring these more generalized contractive conditions but also demonstrates how this framework can expand the applicability of contraction theory to more complex, real-world problems. Through rigorous analysis, we establish the theoretical foundation for contractive mappings, setting the stage for further research and practical applications. In addition to advancing the theoretical framework of contractive mappings within b-metric spaces, this research holds potential for practical applications in fields like data science and machine learning.8,9 The generalization to b-metric spaces is particularly useful for analyzing algorithms in these domains, where ideal conditions such as exact distances or symmetric relationships often do not hold. For instance, clustering algorithms and optimization problems in machine learning frequently operate in non-Euclidean spaces, making the flexibility of b-metric spaces and generalized contraction mappings highly relevant.10,11 By providing a more robust mathematical foundation for these non-standard spaces, our work can support more effective modeling of real-world systems, such as large-scale data sets or complex networks, ultimately contributing to improved algorithm performance and deeper insights in these fields.
This section serves as a foundational platform for our subsequent analysis, comprising key definitions, illustrative examples, essential notations, and fundamental theorems. These elements collectively provide the necessary groundwork and contextual understanding required to establish and prove our main result.
Let be a non-empty set and be a given real number. A function is a b-metric on the given below principles are apply;
Then, b-metric space a pair of with coefficient s.
Let X be any non empty set. Let be a self mapping on X and : × X be mapping.
We say that is α-admissible if for all
Consider Let and and such that and . Then is α-admissible.
Consider X as any set that is not empty. If and : X are mappings, then is triangular α-admissible mapping if
Let , and are mappings and
We define and
Then, is triangular α-admissible.
A function is referred to as an altering distance function if the given below principles are apply;
where, is the set of all altering distance functions.
Let : ) ) given by;
then is an altering distance function.
Assuming that a b-metric space ( ) with s equal to or greater than one and are given, then sequence with for every and is called a picard sequence.
Suppose that a b-metric space with s equal to or greater than one and are given. If a picard sequence of initial point fulfills;
Whenever and Then is a Cauchy sequence.
A mapping is called a contractive mapping if there exists a constant such that for all the following inequality holds:
Let be a b-metric space with , and let be a function defined on pairs of points in . A mapping is referred to as an contractive mapping if there exist functions such that for all with , the following condition is satisfied:
This definition extends the classical notion of contractive mappings by incorporating the functions and that allow for more flexible contraction conditions, particularly in the context of b-metric spaces.
Let be a complete b-metric space with . Consider a mapping and a function . Assume the following conditions hold:
(i) is an contractive mapping.
(ii) is a triangular -admissible mapping
(iii) There exists an initial point such that .
(iv) is continuous.
Then, there exists a point such that , indicating that is a fixed point of the mapping .
Let be given as (iii), i.e. )
From the definition (2.6) and for every If for and ,then is a fixed point of
Next, Let is not equal with , then is also not equal to zero for every n belongs to
Due to the triangular -admissible mapping in (ii);
; for each .
; for each
by continuing in this process we get;
for each .
From definition (3.1), obtain;
Thus, from (3.2) we have;
Since, is continuous and increasing, we obtain;
Since, the sequence is limited from the lower. Therefore r ≥ 0 implies
We want to show r = 0.
Suppose that r > 0.
Now, of the two side of (3.4), gives:
Since, > 0, ≥ 1 and < ,
This implies and < 1
, a contradiction.
Hence, (3.5). So is a Cauchy sequence by (3.4) and Using proposition (2.1).
Let’s now demonstrate the fixed point ;-
X) implies from a complete b-metric space
So, using definition (3.1), with , gives;
Taking the limit as of the two side of (3.6), gives:
By the limit’s uniqueness and the Complete b-metric space it gives;
Hence, the fixed point of is .
Assume that is a complete b-metric space with s greater than or equal to one, and that and ) are given.
Let’s the given below principles are apply:-
(i) is - - Contractive mapping;
(ii) is triangular -admissible mapping;
(iii) implies ) ;
(iv) if in X implies ) is equal to or greater than one for every and as , then there exist a subsequence of so that ) for every .
Then, a fixed point of appears.
Proof: Proceed analogously to the proof of first theorem and use (iv) ) for a subsequence of and we get;
Taking the limit as on two side of (3.7), gives:-
By uniqueness of limit and Complete b-metric space we have;
Hence, the fixed point of is
Remark: Theorem 3.1 provides a direct route to establishing the existence of a fixed point through initial conditions, Theorem 3.2 builds on this by examining the implications of sequential convergence and the relationships between subsequent points in the sequence, thus offering a more nuanced perspective on the fixed point's existence within the framework of contractive mappings in b-metric spaces.
Let and ) are the predetermined mappings, while is a complete metric space;
Let’s the given below principles are apply:-
(i) is - - Contractive mapping;
(ii) is triangular -admissible mapping;
(iii) let implies ) ; and
(iv) if in X implies ) for every and as
For some subsequence of so that ) for every .
Then, a fixed point of appears
If s is one, From Theorems 3.1 and 3.2 One finds this consequence.
If X = {0,3,4}, is given by is a complete b-metric space with s = 2, and also and are given as follows;
First, show that is α-admissible mapping.
If = 0, ,
is α-admissible mapping.
Secondly, If = 0, , the given equation holds.
is triangular α-admissible.
Lastly, check that is Contractive Mapping;
18 < 219
is Contractive Mapping.
Remark: In examining the distinctions between α−ψ contractive mappings and α−ψ−ϕ contractive mappings, it becomes evident that the latter provides a more flexible and nuanced approach to addressing fixed-point problems in b-metric spaces. While α−ψ contractive mappings focus primarily on the contraction properties defined by the functions α and ψ, the inclusion of the additional function ϕ in α−ψ−ϕ contractive mappings introduce a layer of complexity that allows for broader applications. This extension facilitates the accommodation of various contraction behaviors and interactions between the mappings, enhancing the potential for convergence. Furthermore, the α−ψ−ϕ framework enables the exploration of new classes of mappings and their properties, thus opening new avenues for research and applications in fields such as fixed-point theory, iterative methods, and mathematical modeling. By leveraging these advantages, researchers can derive more robust results and insights into the behavior of mappings in more complex environments, making α−ψ−ϕ contractive mappings a significant advancement over their α−ψ counterparts.
In conclusion, this research has introduced the concept of contractive mapping within b-metric spaces, expanding upon contraction theory principles. Through rigorous analysis and illustrative examples, we've highlighted its significance and applicability. Future directions include exploring stability in iterative methods, investigating connections with other mathematical areas, and extending the analysis to broader classes of spaces and mappings. Overall, this exploration of contractive mappings opens new avenues for research with broad potential implications.
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Is the work clearly and accurately presented and does it cite the current literature?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Are the conclusions drawn adequately supported by the results?
Yes
Competing Interests: No competing interests were disclosed.
Reviewer Expertise: Jungck Type Contractions, Fixed point theory and analysis.
Competing Interests: No competing interests were disclosed.
Reviewer Expertise: Fixed point theory
Is the work clearly and accurately presented and does it cite the current literature?
Partly
Is the study design appropriate and is the work technically sound?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
No
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Are the conclusions drawn adequately supported by the results?
Partly
Competing Interests: No competing interests were disclosed.
Reviewer Expertise: Applied Mathematics, Fixed point theory, Chemical Graph Theory
Is the work clearly and accurately presented and does it cite the current literature?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Yes
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
References
1. Baewnoi K, Yambangwai D, Thianwan T: A novel algorithm with an inertial technique for fixed points of nonexpansive mappings and zeros of accretive operators in Banach spaces. AIMS Mathematics. 2024; 9 (3): 6424-6444 Publisher Full TextCompeting Interests: No competing interests were disclosed.
Reviewer Expertise: Fixed point theory
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