Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces

Tamene Raji; Nasir Ali; Gudeta Hanchalu; Fikadu Tesgera Tolasa; Berhanu Seboka

doi:10.12688/f1000research.150979.1

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Research Article

Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces

[version 1; peer review: 2 approved with reservations]

Tamene Raji¹, Nasir Ali², Gudeta Hanchalu³, Fikadu Tesgera Tolasa ⁴, Berhanu Seboka⁵

Tamene Raji¹, Nasir Ali², [...] Gudeta Hanchalu³, Fikadu Tesgera Tolasa ⁴, Berhanu Seboka⁵

PUBLISHED 03 Jun 2024

Author details Author details

¹ Mathematics, Dambi Dollo University, Dembi Dolo, Oromia, Ethiopia
² Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari, 61100, Pakistan
³ Dambi Dollo University, Dembi Dolo, Oromia, Ethiopia
⁴ Dambi Dollo University, Dembi Dolo, Oromia, Ethiopia
⁵ Mathematics, Ambo University, Ambo, Oromia, Ethiopia

Tamene Raji
Roles: Conceptualization

Nasir Ali
Roles: Data Curation, Formal Analysis

Gudeta Hanchalu
Roles: Investigation, Methodology, Writing – Original Draft Preparation

Fikadu Tesgera Tolasa
Roles: Supervision, Validation

Berhanu Seboka
Roles: Resources

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background

This paper explores α-ψ-ϕ contractive mappings, extending the field of self-map and fixed-point theorems.

Methods

We analyze α-ψ-ϕ contractive mappings using rigorous mathematical proofs and logical deductions.

Results

A key main result is established, supported by intuitive corollaries and practical examples, highlighting the applicability of our findings.

Conclusions

Our work provides a fresh perspective on contractive mappings, simplifying complex mathematical concepts and enriching the literature on fixed-point theorems.

Keywords

complete b-metric space, Altering distance function, α-admissible maping

Corresponding author: Fikadu Tesgera Tolasa

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2024 Raji T et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Raji T, Ali N, Hanchalu G et al. Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces [version 1; peer review: 2 approved with reservations]. F1000Research 2024, 13:566 (https://doi.org/10.12688/f1000research.150979.1) First published: 03 Jun 2024, 13:566 (https://doi.org/10.12688/f1000research.150979.1) Latest published: 16 Dec 2024, 13:566 (https://doi.org/10.12688/f1000research.150979.2)

1. Introduction

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 α-ψ-ϕ 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 α-ψ-ϕ 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 α-ψ-ϕ 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 α-ψ-ϕ 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 α-ψ-ϕ 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.

2. Preliminaries

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.

Definition 2.1.2,3

Let $X$ be a non-empty set and $s \geq 1$ be a given real number. A function $d : X \times X \to [0, \infty)$ is a b-metric on $X$ $if$ the given below principles are apply;

(i) $d (x, y) = 0$ iff $x = y$ , $\forall$ $x, y \in X;$
(ii) $d (x, y) = d (y, x)$ , $\forall$ $x, y \in X;$
(iii) $d (x, z) \leq s [d (x, y) + d (y, z)],$ $\forall$ $x, y, z \in X .$

Then, b-metric space a pair of $(X, d)$ with coefficient s.

Definition 2.2.11

Let X be any non empty set. Let $f : X \to X$ be a self mapping on X and $α$ : $X$ × X $\to [0, \infty)$ be mapping.

We say that $f$ is α-admissible if $α (x, y) \geq 1 \Rightarrow α (fx, fy) \geq 1$ for all $x, y \in X .$

Example 2.1.7

Consider $X = {0, 1, 2, 3},$ Let $f : X \to X$ and $α : X \times X$ $\to [0, \infty)$ and $d (x, y) = | x - y |$ such that $f_{0} = 0, f_{1} = 2, f_{2} = 1, f_{3} = 3,$ and ${\begin{cases} α (x, y) = 1, if (x, y) \in {(0, 1), (0, 2), (1, 1), (2, 2), (1, 2), (2, 1), (1, 3), (2, 3)} \\ α (x, y) = 0, other wise \end{cases}$ . Then $f$ is α-admissible.

Definition 2.3.12

Consider X as any set that is not empty. If $f : X \to X$ and $α$ : $X \times$ X $\to [0, \infty)$ are mappings, then $f$ is triangular α-admissible mapping if

(i) $f$ is α-admissible mapping;
(ii) ${\begin{matrix} α (x, y) \geq 1 \\ α (y, z) \geq 1 \end{matrix} \Rightarrow α (x, z) \geq 1$ , for all $x, y, z \in X$ .

Example 2.2.13

Let $X = [0, \infty)$ , $f : X \to X$ and $α : X \times X$ $\to [0, \infty)$ are mappings and $d (x, y) = {| x - y |}^{2}$

We define $fx = {\begin{matrix} \frac{x^{2}}{2}, x \in [0, 1] \\ x + 1, x > 1 \end{matrix}$ and $α (x, y) = {\begin{matrix} 1, & x, y \in [0, 1] \\ 0, & other wise \end{matrix}$

Then, $f$ is triangular α-admissible.

Definition 2.4.14

A function $ψ$ $: [0, \infty) \to [0, \infty)$ is referred to as an altering distance function if the given below principles are apply;

(i) $ψ$ is continuous and non decreasing;
(ii) $ψ$ $(t) = 0$ if and only if $t = 0$ .

where, $Ψ$ is the set of all altering distance functions.

Example 2.4.15

Let $ψ$ : $[0, \infty$ ) $\to [0, \infty$ ) given by;

$ψ (t) = {\begin{cases} \frac{t}{7}, & t < 3 \\ \frac{t^{2} + 3}{t^{2} + 4 t + 7}, & t \geq 3 \end{cases},$ then $ψ$ is an altering distance function.

Definition 2.5.16

Assuming that a b-metric space ( $X, d$ ) with s equal to or greater than one and $f : X \to X$ are given, then sequence ${x_{n}}$ with $x_{n} = f^{n} x_{0} = f x_{n - 1}$ for every $n \in ℕ$ and $x_{0} \in X$ is called a picard sequence.

Proposition 2.1.16

Suppose that a b-metric space $(X, d)$ with s equal to or greater than one and $f : X \to X$ are given. If a picard sequence ${x_{n}}$ of initial point $x_{0} \in X$ fulfills;

d (x_{n}, x_{n + 1}) \leq \frac{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + \frac{n}{s}}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})

Whenever $m, n \in ℝ^{+}$ and $n < m .$ Then ${x_{n}}$ is a Cauchy sequence.

Theorem 2.1.16

Assuming that a complete b-metric space $(X, d)$ with s equal to or greater than one and $f$ : X $\to$ X are given

sd (fx, fy) \leq \frac{d (x, fy) + d (x, f^{2} y) + d (y, fx) + d (y, f^{2} x) + n}{d (x, fx) + d (x, f^{2} x) + d (y, fy) + d (y, f^{2} y) + m} d (x, y)

for each $x, y \in X, if m, n \in ℝ^{+}$ and $n < m .$

Then, a fixed point of $f$ appears.

3. Main Results

Definition 3.1.8

Assuming that ( $X, d$ ) is a b-metric space with s equal to or greater than one and that $α : X \times X \to [0, \infty)$ are given, then $f : X \to X$ is referred to as “ $α$ - $ψ$ - $ϕ$ contractive mapping” if some $ψ,$ $ϕ$ $\in Ψ$ for all $x, y \in X$ with $α (x, y) \geq 1$ implies that

(3.1)

ψ (sd (fx, fy)) \leq ψ (M (x, y)) - ϕ (M (x, y))

where,

M (x, y) = \frac{d (x, fy) + d (x, f^{2} y) + d (y, fx) + d (y, f^{2} x) + n}{d (x, fx) + d (x, f^{2} x) + d (y, fy) + d (y, f^{2} y) + m} d (x, y)

for all

x, y \in X,

m, n \in ℝ^{+}

such that

n < m

and

ψ (t)

is greater than

ϕ (t)

for every t is greater than zero.

Theorem 3.1.

Assume that $(X, d)$ is a complete b-metric space with s greater than or equal to one, and mappings $f : X \to X$ and $α : X \times X \to [0, \infty$ ) are given;

Let’s the given below principles are apply;

(i) $f$ is $α$ - $ψ$ - $ϕ$ Contractive mapping;
(ii) $f$ is triangular $α$ -admissible mapping;
(iii) let $x_{0} \in X$ involves $α (x_{0}, f x_{0}$ ) $\geq 1$ ; and
(iv) $f$ is continuous mapping.

Then, a fixed point of $f$ appears

Proof:

Let $x_{0} \in X$ be given as (iii), i.e. $α (x_{0}, f x_{0}$ ) $\geq 1 .$

From the definition (2.6) $x_{n + 1} = {fx}_{n}$ and $x_{n} = {fx}_{n - 1}$ for every $n \in ℕ .$ If $x_{n_{o}} =$ $x_{n_{o + 1}}$ for $n_{o} \in ℕ,$ and ${fx}_{n_{o}} =$ $x_{n_{o}}$ ,then $x_{n_{o}}$ is a fixed point of $f .$

Next, $x_{n}$ Let is not equal with $x_{n + 1}$ , then $d (x_{n}, x_{n + 1})$ is also not equal to zero for every n belongs to $ℕ$

Due to the triangular $α$ -admissible mapping in (ii);

$α (x_{0,} f x_{0}) = α (x_{0,} x_{1}) \geq 1 ⟹ α (f x_{0}, f x_{1}) \geq 1$ ; for each $x_{0,} x_{1} \in X$ .

$α (x_{1}, f x_{1}) = α (x_{1}, x_{2}) \geq 1 ⟹$ $α (f x_{1}, f x_{2}) \geq 1$ ; for each $x_{1}, x_{2} \in X .$

by continuing in this process we get;

$α (x_{n}, f x_{n}) = α (x_{n}, x_{n + 1}) \geq 1 ⟹ α (f x_{n}, f x_{n + 1}) \geq 1$ for each $n \in ℕ$ .

From definition (3.1), obtain;

(3.2)

\begin{array}{l} ψ (sd (x_{n}, x_{n + 1})) = ψ (sd (f x_{n - 1}, f x_{n})) \\ \leq ψ (M (x_{n - 1}, x_{n})) - ϕ (M (x_{n - 1}, x_{n})) \forall n \in ℕ \end{array}

where,

\begin{matrix} M (x_{n - 1}, x_{n}) = \frac{d (x_{n - 1}, f x_{n}) + d (x_{n - 1}, f^{2} x_{n}) + d (x_{n}, f x_{n - 1}) + d (x_{n}, f^{2} x_{n - 1}) + n}{d (x_{n - 1}, f x_{n - 1}) + d (x_{n - 1}, f^{2} x_{n - 1}) + d (x_{n}, f x_{n}) + d (x_{n}, f^{2} x_{n}) + m} d (x_{n - 1}, x_{n}) \\ = \frac{d (x_{n - 1}, x_{n + 1}) + d (x_{n - 1}, x_{n + 2}) + d (x_{n}, x_{n}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n}) \\ = \frac{d (x_{n - 1}, x_{n + 2}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n}) \end{matrix}

Thus, from (3.2) we have;

\begin{matrix} ψ (sd (x_{n}, x_{n + 1})) \leq ψ (\frac{d (x_{n - 1}, x_{n + 2}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})) \\ - ϕ (\frac{d (x_{n - 1}, x_{n + 2}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})) \end{matrix}

But, $d (x_{n - 1}, x_{n + 2})$ ≤ $s [d (x_{n - 1}, x_{n}) + d (x_{n}, x_{n + 2})]$

\begin{array}{l} ψ (sd (x_{n}, x_{n + 1})) \leq ψ (\frac{s [d (x_{n - 1}, x_{n}) + d (x_{n}, x_{n + 2})] + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})) \\ - ϕ (\frac{s [d (x_{n - 1}, x_{n}) + d (x_{n}, x_{n + 2})] + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})) \\ ψ (sd (x_{n}, x_{n + 1})) \leq ψ (\frac{s [d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2})] + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})) \end{array}

Since, $ψ$ is continuous and increasing, we obtain;

(3.3)

sd (x_{n}, x_{n + 1}) \leq \frac{s [d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2})] + n}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})

(3.4)

d (x_{n}, x_{n + 1}) \leq \frac{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + \frac{n}{s}}{d (x_{n - 1}, x_{n}) + d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + m} d (x_{n - 1}, x_{n})

Since, the sequence ${d (x_{n}, x_{n + 1})}$ is limited from the lower. Therefore $\exists$ r ≥ 0 implies $lim_{n \to \infty} d (x_{n}, x_{n + 1}) = r .$

We want to show r = 0.

Suppose that r > 0.

Now, $n \to \infty$ of the two side of (3.4), gives:

\begin{array}{l} r \leq \frac{r + 2 sr + r + 2 sr + \frac{n}{s}}{r + 2 sr + r + 2 sr + m} r & = \frac{2 r + 4 sr + \frac{n}{s}}{2 r + 4 sr + m} r \\ \leq \frac{2 rs + 4 s^{2} r + n}{2 rs + 4 s^{2} r + m} r \end{array}

Since, $r$ > 0, $s$ ≥ 1 and $n$ < $m$ , $n, m \in ℝ^{+}$

This implies $\frac{2 rs + 4 s^{2} r + n}{2 rs + 4 s^{2} r + m} \in [0, 1)$ and $\frac{2 rs + 4 s^{2} r + n}{2 rs + 4 s^{2} r + m}$ < 1

$r \leq \frac{2 rs + 4 s^{2} r + n}{2 rs + 4 s^{2} r + m} r < r$ , a contradiction.

Hence, $lim_{n \to \infty} d (x_{n}, x_{n + 1}) = 0$ (3.5). So ${x_{n}}$ is a Cauchy sequence by (3.4) and Using proposition (2.1).

Let’s now demonstrate the fixed point $f$ ;-

$(\exists z)$ $(z \in$ X) implies $lim_{n \to \infty} d (x_{n}, z) = 0$ from a complete b-metric space

So, using definition (3.1), with $x = x_{n} and y = z$ , gives;

\begin{matrix} ψ (sd (x_{n + 1}, fz)) = ψ (sd (f x_{n}, fz)) \\ \leq ψ (\frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, f x_{n}) + d (z, f^{2} x_{n}) + n}{d (x_{n}, f x_{n}) + d (x_{n}, f^{2} x_{n}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z)) \\ - ϕ (\frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, f x_{n}) + (z, f^{2} x_{n}) + n}{d (x_{n}, f x_{n}) + d (x_{n}, f^{2} x_{n}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z)) \end{matrix}

ψ (sd (x_{n + 1}, fz)) \leq ψ (\frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, x_{n + 1}) + d (z, x_{n + 2}) + n}{d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z))

(3.6)

sd (x_{n + 1}, fz) \leq \frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, x_{n + 1}) + d (z, x_{n + 2}) + n}{d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z)

Taking the limit as $n \to \infty$ of the two side of (3.6), gives:

lim_{n \to \infty} (sd (x_{n + 1}, fz)) \leq lim_{n \to \infty} (\frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, x_{n + 1}) + d (z, x_{n + 2}) + n}{d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z))

0 \leq s lim_{n \to \infty} d (x_{n + 1}, fz) \leq lim_{n \to \infty} (\frac{d (x_{n}, fz) + d (x_{n}, f^{2} z) + d (z, x_{n + 1}) + d (z, x_{n + 2}) + n}{d (x_{n}, x_{n + 1}) + d (x_{n}, x_{n + 2}) + d (z, fz) + d (z, f^{2} z) + m} d (x_{n}, z)) = 0

0 \leq s lim_{n \to \infty} d (x_{n + 1}, fz) \leq 0

s lim_{n \to \infty} d (x_{n + 1}, fz) = 0

lim_{n \to \infty} d (x_{n + 1}, fz) = 0

lim_{n \to \infty} x_{n + 1} = fz

By the limit’s uniqueness and the Complete b-metric space it gives;

lim_{n \to \infty} x_{n + 1} = lim_{n \to \infty} x_{n}

fz = z

Hence, the fixed point of $f$ is $z$ .

Theorem 3.2.

Assume that $(X, d)$ is a complete b-metric space with s greater than or equal to one, and that $f : X \to X$ and $α : X \times X \to [0, \infty$ ) are given.

Let’s the given below principles are apply:-

(i) $f$ is $α$ - $ψ$ - $ϕ$ Contractive mapping;
(ii) $f$ is triangular $α$ -admissible mapping;
(iii) $x_{0} \in X$ implies $(x_{0}, f x_{0}$ ) $\geq 1$ ;
(iv) if ${x_{n}}$ in X implies $(x_{n}, x_{n + 1}$ ) is equal to or greater than one for every $n \in ℕ$ and $x_{n} \to x^{*} \in X$ as $n \to \infty$ , then there exist a subsequence ${x_{n_{k}}}$ of ${x_{n}}$ so that $α (x_{n_{k}}, x^{*}$ ) $\geq 1$ for every $k \in ℕ$ .

Then, a fixed point of $f$ appears.

Proof: Proceed analogously to the proof of first theorem and use (iv) $(x_{n_{k}}, x^{*}$ ) $\geq 1$ for a subsequence ${x_{n_{k}}}$ of ${x_{n}}$ and $k \in ℕ,$ we get;

(3.7)

\begin{array}{l} ψ (sd (x_{n_{k + 1}}, f x^{*})) = ψ (sd (f x_{n_{k}}, f x^{*})) \\ \leq ψ (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, f x_{n_{k}}) + d (x^{*}, f^{2} x_{n_{k}}) + n}{d (x_{n_{k}}, f x_{n_{k}}) + d (x_{n_{k}}, f^{2} x_{n_{k}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) \\ - ϕ (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, f x_{n_{k}}) + (x^{*}, f^{2} x_{n_{k}}) + n}{d (x_{n_{k}}, f x_{n_{k}}) + d (x_{n_{k}}, f^{2} x_{n_{k}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) \\ ψ (sd (x_{n_{k + 1}}, f x^{*})) = ψ (sd (f x_{n_{k}}, f x^{*})) \\ \leq ψ (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) \\ - ϕ (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) \\ ψ (sd (x_{n_{k + 1}}, f x^{*})) \leq ψ (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) \\ sd (x_{n_{k + 1}}, f x^{*}) \leq \frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*}) \end{array}

Taking the limit as $k \to \infty$ on two side of (3.7), gives:-

lim_{k \to \infty} (sd (x_{n_{k + 1}}, f x^{*})) \leq lim_{k \to \infty} (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*}))

0 \leq s lim_{k \to \infty} lim_{k \to \infty} (\frac{d (x_{n_{k}}, f x^{*}) + d (x_{n_{k}}, f^{2} x^{*}) + d (x^{*}, x_{n_{k + 1}}) + d (x^{*}, x_{n_{k + 2}}) + n}{d (x_{n_{k}}, x_{n_{k + 1}}) + d (x_{n_{k}}, x_{n_{k + 2}}) + d (x^{*}, f x^{*}) + d (x^{*}, f^{2} x^{*}) + m} d (x_{n_{k}}, x^{*})) = 0

0 \leq s lim_{k \to \infty} d (x_{n_{k + 1}}, f x^{*}) \leq 0

s lim_{k \to \infty} d (x_{n_{k + 1}}, f x^{*}) = 0

lim_{k \to \infty} d (x_{n_{k + 1}}, f x^{*}) = 0

lim_{k \to \infty} x_{n_{k + 1}} = f x^{*}

By uniqueness of limit and Complete b-metric space we have;

lim_{k \to \infty} x_{n_{k + 1}} = lim_{k \to \infty} x_{n_{k}} = x^{*} = f x^{*}

x^{*} = f x^{*}

Hence, the fixed point of $f$ is $x^{*} .$

Corollary 3.1.

Let $f : X \to X$ and $α : X \times X \to [0, \infty$ ) are the predetermined mappings, while $(X, d)$ is a complete metric space;

Let’s the given below principles are apply:-

(i) $f$ is $α$ - $ψ$ - $ϕ$ Contractive mapping;
(ii) $f$ is triangular $α$ -admissible mapping;
(iii) let $x_{0} \in X$ implies $α (x_{0}, f x_{0}$ ) $\geq 1$ ; and
(iv) if ${x_{n}}$ in X implies $(x_{n}, x_{n + 1}$ ) $\geq 1$ for every $n \in ℕ$ and $x_{n} \to x^{*} \in X$ as $n \to \infty,$

For some subsequence ${x_{n_{k}}}$ of ${x_{n}}$ so that $α (x_{n_{k}}, x^{*}$ ) $\geq 1$ for every $k \in ℕ$ .

Then, a fixed point of $f$ appears

Proof:

If s is one, From Theorems 3.1 and 3.2 One finds this consequence.

Example 3.1.

If X = {0,3,4}, $d : X \times X$ $\to [0, \infty)$ is given by $d (x, y) = {| x - y |}^{2}, (X, d)$ is a complete b-metric space with s = 2, and also $α : X \times X$ $\to [0, \infty)$ and $f : X \to X$ are given as follows; ${\begin{cases} α (x, y) = 1, & if x, y \in {0, 3, 4} \\ α (x, y) = 0, & other wise \end{cases}$

f 0 = 0, f 3 = 3, f 4 = 4

again, If

ψ (t) = t

and

ϕ (t) = \frac{t}{3}

, whenever

ψ, ϕ : [0, \infty) \to [0, \infty) \forall t \in [0, \infty)

, then

f

α

ψ

ϕ

Contractive Mapping.

First, show that $f$ is α-admissible mapping.

If $x$ = 0, $y = 3$ , $α (0, 3) \geq 1 \Rightarrow α (f 0, f 3) = α (0, 3) = 1 \geq 1$

$\Rightarrow f$ is α-admissible mapping.

Secondly, If $x$ = 0, $y = 3 and z = 4$ , the given equation holds.

$\Rightarrow$ $f$ is triangular α-admissible.

Lastly, check that $f$ is $α$ - $ψ$ - $ϕ$ Contractive Mapping;

(*)

ψ (sd (fx, fy)) = ψ (2 d (f 0, f 3)) = ψ (2 d (0, 3)) = ψ (18) = 18

(**)

\begin{array}{l} ψ (M (0, 3)) - ϕ (M (0, 3)) = ψ (\frac{324 + 9 n}{m}) - ϕ (\frac{324 + 9 n}{m}) \\ = \frac{324 + 9 n}{m} - \frac{324 + 9 n}{3 m}, choose n = \frac{1}{2} and m = 1 . \\ = \frac{324 + \frac{9}{2}}{1} - \frac{324 + \frac{9}{2}}{3} \\ = \frac{648 + 9}{2} - \frac{648 + 9}{6} = \frac{657}{2} - \frac{657}{6} \\ = \frac{657 (3) - 657}{6} = \frac{1,971 - 657}{6} \\ = \frac{1,314}{6} = 219 \end{array}

From, (*) and (**), $ψ (2 d (f 0, f 3)) \leq ψ (M (0, 3)) - ϕ (M (0, 3))$

18 < 219

$\Rightarrow$ $f$ is $α$ - $ψ$ - $ϕ$ Contractive Mapping.

Conclusion

In conclusion, this research has introduced the concept of α-ψ-ϕ 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 α-ψ-ϕ contractive mappings opens new avenues for research with broad potential implications.

Authors contribution

All the authors equally contributed towards this work.

Availability of data and materials

The data is provided on request to the authors.

References

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Version 2

VERSION 2 PUBLISHED 03 Jun 2024

Author details Author details

Tamene Raji
Roles: Conceptualization

Nasir Ali
Roles: Data Curation, Formal Analysis

Gudeta Hanchalu
Roles: Investigation, Methodology, Writing – Original Draft Preparation

Fikadu Tesgera Tolasa
Roles: Supervision, Validation

Berhanu Seboka
Roles: Resources

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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https://doi.org/10.12688/f1000research.150979.1

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Raji T, Ali N, Hanchalu G et al. Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces [version 1; peer review: 2 approved with reservations]. F1000Research 2024, 13:566 (https://doi.org/10.12688/f1000research.150979.1)

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Reviewer Report 09 Oct 2024

Konsalraj Julietraja, Presidency University, Bengaluru, India

Agilan P, Mathematics, St. Joseph's College of Engineering, Chennai, Tamil Nadu, India

Approved with Reservations

https://doi.org/10.5256/f1000research.165597.r320358

The article "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces" delves into advanced fixed point theory, focusing on the development of novel contractive conditions involving α, ψ, and ϕ mappings in the framework of complete b-metric spaces. The authors aim to generalize and extend classical fixed-point theorems, such as the Banach Contraction Principle, by introducing new types of contractive conditions, thus broadening the scope of potential applications in mathematics and its interdisciplinary fields. The article requires the following major revisions in order to make it suitable for indexing:

The authors should detail the contribution of the article in the abstract.
The authors should improve the Introduction section with the necessary references.
The authors should provide a clear definition of contractive mapping and α−ψ−ϕ contractive mapping:
The authors should justify the role of ϕ.
Theorem 3.1 - The statement is not clear. The authors should provide an unambiguous statement.
The authors should include the comparison between Theorem 3.1 and 3.2
The authors should compare α−ψ Contractive Mapping and α−ψ−ϕ Contractive Mapping and explore the advantages of α−ψ−ϕ Contractive Mapping.
The authors should cite the article entitled “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” 2013, 2013:151.
The authors should justify the relevance of the references 1, 6, 9,17 in their article.

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

We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above.

CITE

Report a concern

Author Response 16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

16 Dec 2024

Author Response
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which ... Continue reading
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which will help enhance the clarity and impact of our work. Below, we address each of your major revision points in detail:

Contribution in the Abstract: We have revised the abstract to explicitly outline the key contributions of our article. It now clearly states how our novel contractive conditions extend classical fixed-point theorems and emphasizes the broader implications for mathematics and interdisciplinary applications.

Improvement of the Introduction: The introduction has been enhanced with additional references that provide context and support for our research. We have incorporated relevant literature to establish a stronger foundation for our work.

Definition of Contractive Mappings: We have added a clear and precise definition of contractive mappings, including a detailed explanation of α−ψ−ϕ contractive mappings. This section has been formulated to ensure that readers can easily understand the distinctions and relevance of these mappings within our framework.

Justification of the Role of ϕ: A new subsection has been included to justify the role of the function ϕ in our contractive mappings. We explain how its inclusion adds complexity and depth to the mapping conditions, enabling more generalized applications and insights into fixed-point behaviors.

Clarification of Theorem 3.1: We have rephrased Theorem 3.1 to provide a clearer and more unambiguous statement. The revised formulation emphasizes the key principles involved and their implications for the existence of fixed points.

Comparison Between Theorem 3.1 and Theorem 3.2: A comparison between Theorems 3.1 and 3.2 has been added to the manuscript. This section discusses the similarities and differences in the conditions and implications of each theorem, highlighting their respective contributions to fixed-point theory.

Comparison of α−ψ and α−ψ−ϕ Contractive Mappings: We have included a discussion comparing α−ψ and α−ψ−ϕ contractive mappings. This analysis explores the advantages of the α−ψ−ϕ framework in terms of flexibility and potential for broader applications in various mathematical contexts.

Citation of Relevant Articles: We have cited the article “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” (2013:151) as recommended. We believe that this reference complements our discussion and strengthens our literature review.

Relevance of References 1, 6, 9, and 17: Justification of References:

Felhi, A., Aydi, H., & Zhang, D. (2016): This reference is critical as it discusses fixed points for α-admissible contractive mappings, which directly relates to the theoretical foundation of our work. The methodologies and results presented in this paper provide a comparative backdrop for our exploration of α−ψ−ϕ contractive mappings, emphasizing the evolution of fixed-point theory through varying mapping conditions.

Güngör, G. D., & Altun, I. (2024): This article presents new fixed-point theorems applicable to quasi-metric spaces, showcasing the versatility and adaptability of fixed-point theory in different mathematical frameworks. Our inclusion of this reference serves to highlight the ongoing advancements in fixed-point theorems and their relevance to the broader mathematical community, particularly in how they relate to our findings in complete b-metric spaces.

Ji, D., Yu, Y., & Li, C. (2024): The authors of this paper address fixed-point and endpoint theorems in convex b-metric spaces, which aligns closely with our study's context. This reference is particularly significant as it extends the applicability of fixed-point theory to multivalued mappings, reinforcing the importance of diverse mapping approaches in achieving generalizable results. It also helps position our work within a contemporary research landscape that seeks to unify various contractive conditions.

Huang, M., Wang, X., Wang, Z., & Xu, Z. (2023): This research explores locally bi-Hölder continuous maps and their embeddings between Besov spaces, contributing to our understanding of continuity and its implications for fixed-point theorems. The techniques discussed in this paper may offer potential methodologies for further investigating the properties of α−ψ−ϕ contractive mappings. Including this reference enriches our discussion on the continuity conditions necessary for the application of fixed-point theorems in various mathematical settings.

We hope that these revisions address your concerns and enhance the clarity and quality of our article. Thank you once again for your valuable feedback, and we look forward to your further comments.
Sincerely,
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which will help enhance the clarity and impact of our work. Below, we address each of your major revision points in detail:

Contribution in the Abstract: We have revised the abstract to explicitly outline the key contributions of our article. It now clearly states how our novel contractive conditions extend classical fixed-point theorems and emphasizes the broader implications for mathematics and interdisciplinary applications.

Improvement of the Introduction: The introduction has been enhanced with additional references that provide context and support for our research. We have incorporated relevant literature to establish a stronger foundation for our work.

Definition of Contractive Mappings: We have added a clear and precise definition of contractive mappings, including a detailed explanation of α−ψ−ϕ contractive mappings. This section has been formulated to ensure that readers can easily understand the distinctions and relevance of these mappings within our framework.

Justification of the Role of ϕ: A new subsection has been included to justify the role of the function ϕ in our contractive mappings. We explain how its inclusion adds complexity and depth to the mapping conditions, enabling more generalized applications and insights into fixed-point behaviors.

Clarification of Theorem 3.1: We have rephrased Theorem 3.1 to provide a clearer and more unambiguous statement. The revised formulation emphasizes the key principles involved and their implications for the existence of fixed points.

Comparison Between Theorem 3.1 and Theorem 3.2: A comparison between Theorems 3.1 and 3.2 has been added to the manuscript. This section discusses the similarities and differences in the conditions and implications of each theorem, highlighting their respective contributions to fixed-point theory.

Comparison of α−ψ and α−ψ−ϕ Contractive Mappings: We have included a discussion comparing α−ψ and α−ψ−ϕ contractive mappings. This analysis explores the advantages of the α−ψ−ϕ framework in terms of flexibility and potential for broader applications in various mathematical contexts.

Citation of Relevant Articles: We have cited the article “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” (2013:151) as recommended. We believe that this reference complements our discussion and strengthens our literature review.

Relevance of References 1, 6, 9, and 17: Justification of References:

Felhi, A., Aydi, H., & Zhang, D. (2016): This reference is critical as it discusses fixed points for α-admissible contractive mappings, which directly relates to the theoretical foundation of our work. The methodologies and results presented in this paper provide a comparative backdrop for our exploration of α−ψ−ϕ contractive mappings, emphasizing the evolution of fixed-point theory through varying mapping conditions.

Güngör, G. D., & Altun, I. (2024): This article presents new fixed-point theorems applicable to quasi-metric spaces, showcasing the versatility and adaptability of fixed-point theory in different mathematical frameworks. Our inclusion of this reference serves to highlight the ongoing advancements in fixed-point theorems and their relevance to the broader mathematical community, particularly in how they relate to our findings in complete b-metric spaces.

Ji, D., Yu, Y., & Li, C. (2024): The authors of this paper address fixed-point and endpoint theorems in convex b-metric spaces, which aligns closely with our study's context. This reference is particularly significant as it extends the applicability of fixed-point theory to multivalued mappings, reinforcing the importance of diverse mapping approaches in achieving generalizable results. It also helps position our work within a contemporary research landscape that seeks to unify various contractive conditions.

Huang, M., Wang, X., Wang, Z., & Xu, Z. (2023): This research explores locally bi-Hölder continuous maps and their embeddings between Besov spaces, contributing to our understanding of continuity and its implications for fixed-point theorems. The techniques discussed in this paper may offer potential methodologies for further investigating the properties of α−ψ−ϕ contractive mappings. Including this reference enriches our discussion on the continuity conditions necessary for the application of fixed-point theorems in various mathematical settings.

We hope that these revisions address your concerns and enhance the clarity and quality of our article. Thank you once again for your valuable feedback, and we look forward to your further comments.
Sincerely,
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

16 Dec 2024

Author Response
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which ... Continue reading
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which will help enhance the clarity and impact of our work. Below, we address each of your major revision points in detail:

Contribution in the Abstract: We have revised the abstract to explicitly outline the key contributions of our article. It now clearly states how our novel contractive conditions extend classical fixed-point theorems and emphasizes the broader implications for mathematics and interdisciplinary applications.

Improvement of the Introduction: The introduction has been enhanced with additional references that provide context and support for our research. We have incorporated relevant literature to establish a stronger foundation for our work.

Definition of Contractive Mappings: We have added a clear and precise definition of contractive mappings, including a detailed explanation of α−ψ−ϕ contractive mappings. This section has been formulated to ensure that readers can easily understand the distinctions and relevance of these mappings within our framework.

Justification of the Role of ϕ: A new subsection has been included to justify the role of the function ϕ in our contractive mappings. We explain how its inclusion adds complexity and depth to the mapping conditions, enabling more generalized applications and insights into fixed-point behaviors.

Clarification of Theorem 3.1: We have rephrased Theorem 3.1 to provide a clearer and more unambiguous statement. The revised formulation emphasizes the key principles involved and their implications for the existence of fixed points.

Comparison Between Theorem 3.1 and Theorem 3.2: A comparison between Theorems 3.1 and 3.2 has been added to the manuscript. This section discusses the similarities and differences in the conditions and implications of each theorem, highlighting their respective contributions to fixed-point theory.

Comparison of α−ψ and α−ψ−ϕ Contractive Mappings: We have included a discussion comparing α−ψ and α−ψ−ϕ contractive mappings. This analysis explores the advantages of the α−ψ−ϕ framework in terms of flexibility and potential for broader applications in various mathematical contexts.

Citation of Relevant Articles: We have cited the article “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” (2013:151) as recommended. We believe that this reference complements our discussion and strengthens our literature review.

Relevance of References 1, 6, 9, and 17: Justification of References:

Felhi, A., Aydi, H., & Zhang, D. (2016): This reference is critical as it discusses fixed points for α-admissible contractive mappings, which directly relates to the theoretical foundation of our work. The methodologies and results presented in this paper provide a comparative backdrop for our exploration of α−ψ−ϕ contractive mappings, emphasizing the evolution of fixed-point theory through varying mapping conditions.

Güngör, G. D., & Altun, I. (2024): This article presents new fixed-point theorems applicable to quasi-metric spaces, showcasing the versatility and adaptability of fixed-point theory in different mathematical frameworks. Our inclusion of this reference serves to highlight the ongoing advancements in fixed-point theorems and their relevance to the broader mathematical community, particularly in how they relate to our findings in complete b-metric spaces.

Ji, D., Yu, Y., & Li, C. (2024): The authors of this paper address fixed-point and endpoint theorems in convex b-metric spaces, which aligns closely with our study's context. This reference is particularly significant as it extends the applicability of fixed-point theory to multivalued mappings, reinforcing the importance of diverse mapping approaches in achieving generalizable results. It also helps position our work within a contemporary research landscape that seeks to unify various contractive conditions.

Huang, M., Wang, X., Wang, Z., & Xu, Z. (2023): This research explores locally bi-Hölder continuous maps and their embeddings between Besov spaces, contributing to our understanding of continuity and its implications for fixed-point theorems. The techniques discussed in this paper may offer potential methodologies for further investigating the properties of α−ψ−ϕ contractive mappings. Including this reference enriches our discussion on the continuity conditions necessary for the application of fixed-point theorems in various mathematical settings.

We hope that these revisions address your concerns and enhance the clarity and quality of our article. Thank you once again for your valuable feedback, and we look forward to your further comments.
Sincerely,
Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which will help enhance the clarity and impact of our work. Below, we address each of your major revision points in detail:

Contribution in the Abstract: We have revised the abstract to explicitly outline the key contributions of our article. It now clearly states how our novel contractive conditions extend classical fixed-point theorems and emphasizes the broader implications for mathematics and interdisciplinary applications.

Improvement of the Introduction: The introduction has been enhanced with additional references that provide context and support for our research. We have incorporated relevant literature to establish a stronger foundation for our work.

Definition of Contractive Mappings: We have added a clear and precise definition of contractive mappings, including a detailed explanation of α−ψ−ϕ contractive mappings. This section has been formulated to ensure that readers can easily understand the distinctions and relevance of these mappings within our framework.

Justification of the Role of ϕ: A new subsection has been included to justify the role of the function ϕ in our contractive mappings. We explain how its inclusion adds complexity and depth to the mapping conditions, enabling more generalized applications and insights into fixed-point behaviors.

Clarification of Theorem 3.1: We have rephrased Theorem 3.1 to provide a clearer and more unambiguous statement. The revised formulation emphasizes the key principles involved and their implications for the existence of fixed points.

Comparison Between Theorem 3.1 and Theorem 3.2: A comparison between Theorems 3.1 and 3.2 has been added to the manuscript. This section discusses the similarities and differences in the conditions and implications of each theorem, highlighting their respective contributions to fixed-point theory.

Comparison of α−ψ and α−ψ−ϕ Contractive Mappings: We have included a discussion comparing α−ψ and α−ψ−ϕ contractive mappings. This analysis explores the advantages of the α−ψ−ϕ framework in terms of flexibility and potential for broader applications in various mathematical contexts.

Citation of Relevant Articles: We have cited the article “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” (2013:151) as recommended. We believe that this reference complements our discussion and strengthens our literature review.

Relevance of References 1, 6, 9, and 17: Justification of References:

Felhi, A., Aydi, H., & Zhang, D. (2016): This reference is critical as it discusses fixed points for α-admissible contractive mappings, which directly relates to the theoretical foundation of our work. The methodologies and results presented in this paper provide a comparative backdrop for our exploration of α−ψ−ϕ contractive mappings, emphasizing the evolution of fixed-point theory through varying mapping conditions.

Güngör, G. D., & Altun, I. (2024): This article presents new fixed-point theorems applicable to quasi-metric spaces, showcasing the versatility and adaptability of fixed-point theory in different mathematical frameworks. Our inclusion of this reference serves to highlight the ongoing advancements in fixed-point theorems and their relevance to the broader mathematical community, particularly in how they relate to our findings in complete b-metric spaces.

Ji, D., Yu, Y., & Li, C. (2024): The authors of this paper address fixed-point and endpoint theorems in convex b-metric spaces, which aligns closely with our study's context. This reference is particularly significant as it extends the applicability of fixed-point theory to multivalued mappings, reinforcing the importance of diverse mapping approaches in achieving generalizable results. It also helps position our work within a contemporary research landscape that seeks to unify various contractive conditions.

Huang, M., Wang, X., Wang, Z., & Xu, Z. (2023): This research explores locally bi-Hölder continuous maps and their embeddings between Besov spaces, contributing to our understanding of continuity and its implications for fixed-point theorems. The techniques discussed in this paper may offer potential methodologies for further investigating the properties of α−ψ−ϕ contractive mappings. Including this reference enriches our discussion on the continuity conditions necessary for the application of fixed-point theorems in various mathematical settings.

We hope that these revisions address your concerns and enhance the clarity and quality of our article. Thank you once again for your valuable feedback, and we look forward to your further comments.
Sincerely,
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 27 Sep 2024

Tanakit Thianwan, University of Phayao, Mueang Phayao District, Phayao, Thailand

Approved with Reservations

https://doi.org/10.5256/f1000research.165597.r320355

REPORT FOR THE PAPER
I reviewed the document and identified the following mistakes:

In the keywords section, "maping" should be "mapping."
In the definitions, some equations lack proper spacing between variables and symbols,

REPORT FOR THE PAPER
I reviewed the document and identified the following mistakes:

In the keywords section, "maping" should be "mapping."
In the definitions, some equations lack proper spacing between variables and symbols, making them harder to read.
Page 4 of 11, line-2, Inconsistent use of semicolons, such as in "Let’s the given below principles are apply;" which should be rephrased for clarity.
Page 4 of 11, line-2, "Let’s the given below principles are apply" should be "Let the following principles apply."
The language used in the introduction is formal and appropriate for an academic paper. However, there are minor grammar issues, such as in the sentence "Through a meticulous synthesis of internet sources, scholarly publications, and pertinent literature, we establish the theoretical framework for this novel concept," where "internet sources" may not be the best fit for a research paper. Replacing it with "previous studies" or "existing research" might be more suitable.
Ensure that all symbols and notation are consistent across the entire document. For example, double-check that the use of ψ and ϕ functions is consistent in both definitions and examples.
Furthermore, there are many typos. The authors should be revised carefully.

Content Accuracy: The introduction provides a solid background but would benefit from a clearer and more concise explanation of how α-ψ-ϕ contractive mappings expand upon traditional contraction theory. It would be helpful to explain why the generalization to b-metric spaces is significant for real-world applications.

Suggestion: Include a brief mention of practical applications of these mappings in areas like data science or machine learning to ground the mathematical theory in broader fields.

In addition, the following references should be added.

Baewnoi K, et al., 2024 (Ref 1).

Chairatsiripong C, et al., 2023 (Ref 2)

Yambangwai, D., et al., 2024 (Ref 3)

Paimsang P, et al., 2023 (Ref 4)

By citing these works, helps establish the background and significance of introducing an alternative method, as the properties and applications of nonexpansive mappings in Banach spaces have strong implications for contractive mappings in b-metric spaces.

In essence, these citations establish a comprehensive framework for fixed point theory and iterative methods, which is crucial for understanding and situating the novel contributions made by exploring α−ψ−ϕ contractive mappings in complete b-metric spaces. They showcase related iterative techniques, their convergence behaviors, and their applications across different spaces, thereby providing both a theoretical foundation and practical motivations for the new theorems being developed.

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 Text
2. Chairatsiripong C, Kittiratanawasin L, Yambangwai D, Thianwan T: Convergence theorem and convergence rate of a new faster iteration method for continuous functions on an arbitrary interval. Miskolc Mathematical Notes. 2023; 24 (2). Publisher Full Text
3. D, Yambangwai T, Thianwan: A parallel inertial SP-iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings and its application in linear system, differential, and signal recovery problems. https://carpathian.cunbm.utcluj.ro/wp-content/uploads/carpathian_2024_40_2_535_557.pdf. 2024. Publisher Full Text
4. Paimsang P, Thianwan T: Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces. Demonstratio Mathematica. 2023; 56 (1). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Fixed point theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

CITE

Report a concern

Author Response 16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

16 Dec 2024

Author Response
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below ... Continue reading
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below is a summary of the revisions made in response to your suggestions:

Keyword Correction: We corrected the typo in the keywords section by changing "maping" to "mapping."

Spacing in Equations: We reviewed all equations and ensured that proper spacing between variables and symbols is applied for better readability.

Clarity and Grammar:

The sentence "Let’s the given below principles are apply" has been rephrased to "Let the following principles apply" for clarity.

We reviewed the manuscript for consistent use of semicolons and corrected them where needed to ensure uniformity and clarity in punctuation.

Introduction Refinement: As per your suggestion, we replaced "internet sources" with "previous studies" to maintain the formal tone appropriate for an academic paper. Additionally, we provided a clearer and more concise explanation of how α−ψ−ϕ contractive mappings expand upon traditional contraction theory, and why their generalization to b-metric spaces is significant for real-world applications.

Consistency in Symbols and Notation: We carefully reviewed the use of all symbols and notations throughout the document, including the ψ\psiψ and ϕ\phiϕ functions, and ensured consistency in both definitions and examples.

Grammar and Typos: We have carefully proofread the manuscript to eliminate any remaining typos and improve overall grammar.

Content Accuracy and Practical Applications: We revised the introduction to more clearly explain the importance of generalizing contraction theory to b-metric spaces and included a brief discussion on the practical applications of these mappings in fields such as data science and machine learning, grounding the mathematical theory in broader, real-world contexts.

Addition of References: We have added the suggested references:

Baewnoi K, et al., 2024 (Ref 1).

Chairatsiripong C, et al., 2023 (Ref 2).

Yambangwai, D., et al., 2024 (Ref 3).

Paimsang P, et al., 2023 (Ref 4).

These references strengthen the theoretical framework and provide important background for understanding the significance of introducing α−ψ−ϕ\alpha-\psi-\phiα−ψ−ϕ contractive mappings in complete b-metric spaces. They also establish connections to related iterative techniques and their convergence behaviors.

We have uploaded the revised version of the manuscript, incorporating all these changes. We believe these revisions have improved the clarity, consistency, and overall quality of the paper, and we hope that it now meets the expectations of the review process.
Thank you again for your insightful comments and suggestions.
Sincerely,
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below is a summary of the revisions made in response to your suggestions:

Keyword Correction: We corrected the typo in the keywords section by changing "maping" to "mapping."

Spacing in Equations: We reviewed all equations and ensured that proper spacing between variables and symbols is applied for better readability.

Clarity and Grammar:

The sentence "Let’s the given below principles are apply" has been rephrased to "Let the following principles apply" for clarity.

We reviewed the manuscript for consistent use of semicolons and corrected them where needed to ensure uniformity and clarity in punctuation.

Introduction Refinement: As per your suggestion, we replaced "internet sources" with "previous studies" to maintain the formal tone appropriate for an academic paper. Additionally, we provided a clearer and more concise explanation of how α−ψ−ϕ contractive mappings expand upon traditional contraction theory, and why their generalization to b-metric spaces is significant for real-world applications.

Consistency in Symbols and Notation: We carefully reviewed the use of all symbols and notations throughout the document, including the ψ\psiψ and ϕ\phiϕ functions, and ensured consistency in both definitions and examples.

Grammar and Typos: We have carefully proofread the manuscript to eliminate any remaining typos and improve overall grammar.

Content Accuracy and Practical Applications: We revised the introduction to more clearly explain the importance of generalizing contraction theory to b-metric spaces and included a brief discussion on the practical applications of these mappings in fields such as data science and machine learning, grounding the mathematical theory in broader, real-world contexts.

Addition of References: We have added the suggested references:

Baewnoi K, et al., 2024 (Ref 1).

Chairatsiripong C, et al., 2023 (Ref 2).

Yambangwai, D., et al., 2024 (Ref 3).

Paimsang P, et al., 2023 (Ref 4).

These references strengthen the theoretical framework and provide important background for understanding the significance of introducing α−ψ−ϕ\alpha-\psi-\phiα−ψ−ϕ contractive mappings in complete b-metric spaces. They also establish connections to related iterative techniques and their convergence behaviors.

We have uploaded the revised version of the manuscript, incorporating all these changes. We believe these revisions have improved the clarity, consistency, and overall quality of the paper, and we hope that it now meets the expectations of the review process.
Thank you again for your insightful comments and suggestions.
Sincerely,
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

16 Dec 2024

Author Response
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below ... Continue reading
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below is a summary of the revisions made in response to your suggestions:

Keyword Correction: We corrected the typo in the keywords section by changing "maping" to "mapping."

Spacing in Equations: We reviewed all equations and ensured that proper spacing between variables and symbols is applied for better readability.

Clarity and Grammar:

The sentence "Let’s the given below principles are apply" has been rephrased to "Let the following principles apply" for clarity.

We reviewed the manuscript for consistent use of semicolons and corrected them where needed to ensure uniformity and clarity in punctuation.

Introduction Refinement: As per your suggestion, we replaced "internet sources" with "previous studies" to maintain the formal tone appropriate for an academic paper. Additionally, we provided a clearer and more concise explanation of how α−ψ−ϕ contractive mappings expand upon traditional contraction theory, and why their generalization to b-metric spaces is significant for real-world applications.

Consistency in Symbols and Notation: We carefully reviewed the use of all symbols and notations throughout the document, including the ψ\psiψ and ϕ\phiϕ functions, and ensured consistency in both definitions and examples.

Grammar and Typos: We have carefully proofread the manuscript to eliminate any remaining typos and improve overall grammar.

Content Accuracy and Practical Applications: We revised the introduction to more clearly explain the importance of generalizing contraction theory to b-metric spaces and included a brief discussion on the practical applications of these mappings in fields such as data science and machine learning, grounding the mathematical theory in broader, real-world contexts.

Addition of References: We have added the suggested references:

Baewnoi K, et al., 2024 (Ref 1).

Chairatsiripong C, et al., 2023 (Ref 2).

Yambangwai, D., et al., 2024 (Ref 3).

Paimsang P, et al., 2023 (Ref 4).

These references strengthen the theoretical framework and provide important background for understanding the significance of introducing α−ψ−ϕ\alpha-\psi-\phiα−ψ−ϕ contractive mappings in complete b-metric spaces. They also establish connections to related iterative techniques and their convergence behaviors.

We have uploaded the revised version of the manuscript, incorporating all these changes. We believe these revisions have improved the clarity, consistency, and overall quality of the paper, and we hope that it now meets the expectations of the review process.
Thank you again for your insightful comments and suggestions.
Sincerely,
Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below is a summary of the revisions made in response to your suggestions:

Keyword Correction: We corrected the typo in the keywords section by changing "maping" to "mapping."

Spacing in Equations: We reviewed all equations and ensured that proper spacing between variables and symbols is applied for better readability.

Clarity and Grammar:

The sentence "Let’s the given below principles are apply" has been rephrased to "Let the following principles apply" for clarity.

We reviewed the manuscript for consistent use of semicolons and corrected them where needed to ensure uniformity and clarity in punctuation.

Introduction Refinement: As per your suggestion, we replaced "internet sources" with "previous studies" to maintain the formal tone appropriate for an academic paper. Additionally, we provided a clearer and more concise explanation of how α−ψ−ϕ contractive mappings expand upon traditional contraction theory, and why their generalization to b-metric spaces is significant for real-world applications.

Consistency in Symbols and Notation: We carefully reviewed the use of all symbols and notations throughout the document, including the ψ\psiψ and ϕ\phiϕ functions, and ensured consistency in both definitions and examples.

Grammar and Typos: We have carefully proofread the manuscript to eliminate any remaining typos and improve overall grammar.

Content Accuracy and Practical Applications: We revised the introduction to more clearly explain the importance of generalizing contraction theory to b-metric spaces and included a brief discussion on the practical applications of these mappings in fields such as data science and machine learning, grounding the mathematical theory in broader, real-world contexts.

Addition of References: We have added the suggested references:

Baewnoi K, et al., 2024 (Ref 1).

Chairatsiripong C, et al., 2023 (Ref 2).

Yambangwai, D., et al., 2024 (Ref 3).

Paimsang P, et al., 2023 (Ref 4).

These references strengthen the theoretical framework and provide important background for understanding the significance of introducing α−ψ−ϕ\alpha-\psi-\phiα−ψ−ϕ contractive mappings in complete b-metric spaces. They also establish connections to related iterative techniques and their convergence behaviors.

We have uploaded the revised version of the manuscript, incorporating all these changes. We believe these revisions have improved the clarity, consistency, and overall quality of the paper, and we hope that it now meets the expectations of the review process.
Thank you again for your insightful comments and suggestions.
Sincerely,
Competing Interests: No competing interests were disclosed. Close
Report a concern

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Version 2

VERSION 2 PUBLISHED 03 Jun 2024

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Version 2 (revision) 16 Dec 24	read		read
Version 1 03 Jun 24	read	read

Tanakit Thianwan, University of Phayao, Mueang Phayao District, Thailand
Konsalraj Julietraja, Presidency University, Bengaluru, India

Agilan P, St. Joseph's College of Engineering, Chennai, India
Ashutosh Mishra, Amity University, Lucknow, India

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Reviewer Report

14 Views

24 Jan 2025 | for Version 2

Ashutosh Mishra, Amity University, Lucknow, India

14 Views Cite this report Responses(0)

Approved

The work is purely theoretical, no practical application is mentioned. If some real-world problems were detailed then it would be beneficial to the scientific society. Although, the paper adequately describes and proves, as well, the abstracted notion. In Definition 3.1, condition 3.1 has some printing error. As a whole, I approve the work.

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.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report

4 Views

18 Dec 2024 | for Version 2

Tanakit Thianwan, University of Phayao, Mueang Phayao District, Phayao, Thailand

4 Views Cite this report Responses(0)

Approved

No further comments.

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fixed point theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report

13 Views

09 Oct 2024 | for Version 1

Konsalraj Julietraja, Presidency University, Bengaluru, India

Agilan P, Mathematics, St. Joseph's College of Engineering, Chennai, Tamil Nadu, India

13 Views Cite this report Responses(1)

Approved With Reservations

The authors should detail the contribution of the article in the abstract.
The authors should improve the Introduction section with the necessary references.
The authors should provide a clear definition of contractive mapping and α−ψ−ϕ contractive mapping:
The authors should justify the role of ϕ.
Theorem 3.1 - The statement is not clear. The authors should provide an unambiguous statement.
The authors should include the comparison between Theorem 3.1 and 3.2
The authors should compare α−ψ Contractive Mapping and α−ψ−ϕ Contractive Mapping and explore the advantages of α−ψ−ϕ Contractive Mapping.
The authors should cite the article entitled “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” 2013, 2013:151.
The authors should justify the relevance of the references 1, 6, 9,17 in their article.

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

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Author Response

16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

Dear Reviewer,
Thank you for your thoughtful and constructive feedback on our article, "Exploring α−ψ−ϕ Contractive Mapping: Novel Fixed Point Theorems in Complete b-Metric Spaces." We appreciate your insights, which will help enhance the clarity and impact of our work. Below, we address each of your major revision points in detail:

Contribution in the Abstract: We have revised the abstract to explicitly outline the key contributions of our article. It now clearly states how our novel contractive conditions extend classical fixed-point theorems and emphasizes the broader implications for mathematics and interdisciplinary applications.
Improvement of the Introduction: The introduction has been enhanced with additional references that provide context and support for our research. We have incorporated relevant literature to establish a stronger foundation for our work.
Definition of Contractive Mappings: We have added a clear and precise definition of contractive mappings, including a detailed explanation of α−ψ−ϕ contractive mappings. This section has been formulated to ensure that readers can easily understand the distinctions and relevance of these mappings within our framework.
Justification of the Role of ϕ: A new subsection has been included to justify the role of the function ϕ in our contractive mappings. We explain how its inclusion adds complexity and depth to the mapping conditions, enabling more generalized applications and insights into fixed-point behaviors.
Clarification of Theorem 3.1: We have rephrased Theorem 3.1 to provide a clearer and more unambiguous statement. The revised formulation emphasizes the key principles involved and their implications for the existence of fixed points.
Comparison Between Theorem 3.1 and Theorem 3.2: A comparison between Theorems 3.1 and 3.2 has been added to the manuscript. This section discusses the similarities and differences in the conditions and implications of each theorem, highlighting their respective contributions to fixed-point theory.
Comparison of α−ψ and α−ψ−ϕ Contractive Mappings: We have included a discussion comparing α−ψ and α−ψ−ϕ contractive mappings. This analysis explores the advantages of the α−ψ−ϕ framework in terms of flexibility and potential for broader applications in various mathematical contexts.
Citation of Relevant Articles: We have cited the article “Modified α-ψ-contractive mappings with applications Fixed Point Theory and Applications” (2013:151) as recommended. We believe that this reference complements our discussion and strengthens our literature review.
Relevance of References 1, 6, 9, and 17: Justification of References:

Felhi, A., Aydi, H., & Zhang, D. (2016): This reference is critical as it discusses fixed points for α-admissible contractive mappings, which directly relates to the theoretical foundation of our work. The methodologies and results presented in this paper provide a comparative backdrop for our exploration of α−ψ−ϕ contractive mappings, emphasizing the evolution of fixed-point theory through varying mapping conditions.

Güngör, G. D., & Altun, I. (2024): This article presents new fixed-point theorems applicable to quasi-metric spaces, showcasing the versatility and adaptability of fixed-point theory in different mathematical frameworks. Our inclusion of this reference serves to highlight the ongoing advancements in fixed-point theorems and their relevance to the broader mathematical community, particularly in how they relate to our findings in complete b-metric spaces.

Ji, D., Yu, Y., & Li, C. (2024): The authors of this paper address fixed-point and endpoint theorems in convex b-metric spaces, which aligns closely with our study's context. This reference is particularly significant as it extends the applicability of fixed-point theory to multivalued mappings, reinforcing the importance of diverse mapping approaches in achieving generalizable results. It also helps position our work within a contemporary research landscape that seeks to unify various contractive conditions.

Huang, M., Wang, X., Wang, Z., & Xu, Z. (2023): This research explores locally bi-Hölder continuous maps and their embeddings between Besov spaces, contributing to our understanding of continuity and its implications for fixed-point theorems. The techniques discussed in this paper may offer potential methodologies for further investigating the properties of α−ψ−ϕ contractive mappings. Including this reference enriches our discussion on the continuity conditions necessary for the application of fixed-point theorems in various mathematical settings.

We hope that these revisions address your concerns and enhance the clarity and quality of our article. Thank you once again for your valuable feedback, and we look forward to your further comments.
Sincerely,

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

12 Views

27 Sep 2024 | for Version 1

Tanakit Thianwan, University of Phayao, Mueang Phayao District, Phayao, Thailand

12 Views Cite this report Responses(1)

Approved With Reservations

REPORT FOR THE PAPER
I reviewed the document and identified the following mistakes:

In the keywords section, "maping" should be "mapping."
In the definitions, some equations lack proper spacing between variables and symbols, making them harder to read.
Page 4 of 11, line-2, Inconsistent use of semicolons, such as in "Let’s the given below principles are apply;" which should be rephrased for clarity.
Page 4 of 11, line-2, "Let’s the given below principles are apply" should be "Let the following principles apply."
The language used in the introduction is formal and appropriate for an academic paper. However, there are minor grammar issues, such as in the sentence "Through a meticulous synthesis of internet sources, scholarly publications, and pertinent literature, we establish the theoretical framework for this novel concept," where "internet sources" may not be the best fit for a research paper. Replacing it with "previous studies" or "existing research" might be more suitable.
Ensure that all symbols and notation are consistent across the entire document. For example, double-check that the use of ψ and ϕ functions is consistent in both definitions and examples.
Furthermore, there are many typos. The authors should be revised carefully.

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

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fixed point theory

Respond to this report

Responses (1)

Author Response

16 Dec 2024

Nasir Ali, Mathematics, COMSATS University Islamabad, Vehari Campus, 61100, Pakistan

Dear Reviewer,
Thank you for your thorough review and valuable feedback on our manuscript. We appreciate your constructive comments and have carefully addressed each one in our revised submission. Below is a summary of the revisions made in response to your suggestions:

Keyword Correction: We corrected the typo in the keywords section by changing "maping" to "mapping."
Spacing in Equations: We reviewed all equations and ensured that proper spacing between variables and symbols is applied for better readability.
Clarity and Grammar:
- The sentence "Let’s the given below principles are apply" has been rephrased to "Let the following principles apply" for clarity.
- We reviewed the manuscript for consistent use of semicolons and corrected them where needed to ensure uniformity and clarity in punctuation.
Introduction Refinement: As per your suggestion, we replaced "internet sources" with "previous studies" to maintain the formal tone appropriate for an academic paper. Additionally, we provided a clearer and more concise explanation of how α−ψ−ϕ contractive mappings expand upon traditional contraction theory, and why their generalization to b-metric spaces is significant for real-world applications.
Consistency in Symbols and Notation: We carefully reviewed the use of all symbols and notations throughout the document, including the ψ\psiψ and ϕ\phiϕ functions, and ensured consistency in both definitions and examples.
Grammar and Typos: We have carefully proofread the manuscript to eliminate any remaining typos and improve overall grammar.
Content Accuracy and Practical Applications: We revised the introduction to more clearly explain the importance of generalizing contraction theory to b-metric spaces and included a brief discussion on the practical applications of these mappings in fields such as data science and machine learning, grounding the mathematical theory in broader, real-world contexts.
Addition of References: We have added the suggested references:
- Baewnoi K, et al., 2024 (Ref 1).
- Chairatsiripong C, et al., 2023 (Ref 2).
- Yambangwai, D., et al., 2024 (Ref 3).
- Paimsang P, et al., 2023 (Ref 4).
These references strengthen the theoretical framework and provide important background for understanding the significance of introducing α−ψ−ϕ\alpha-\psi-\phiα−ψ−ϕ contractive mappings in complete b-metric spaces. They also establish connections to related iterative techniques and their convergence behaviors.

We have uploaded the revised version of the manuscript, incorporating all these changes. We believe these revisions have improved the clarity, consistency, and overall quality of the paper, and we hope that it now meets the expectations of the review process.
Thank you again for your insightful comments and suggestions.
Sincerely,

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

[1] 1. Ali N, Siddiqui HMA, Riaz MB, et al.: A graph-theoretic approach to ring analysis: dominant metric dimensions in zero-divisor graphs. Heliyon. 2024; 10: e30989. Publisher Full Text

[2] 2. Güngör GD, Altun I: Some Fixed Point Results for α-Admissible Mappings on Quasi Metric Space Via θ-Contractions. Mathematical Sciences and Applications E-Notes. 2024; 12(1): 12–19.

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Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces

Abstract

Background

Methods

Results

Conclusions

Keywords

1. Introduction

2. Preliminaries

Definition 2.1.2,3

Definition 2.2.11

Example 2.1.7

Definition 2.3.12

Example 2.2.13

Definition 2.4.14

Example 2.4.15

Definition 2.5.16

Proposition 2.1.16

Theorem 2.1.16

3. Main Results

Definition 3.1.8

(3.1)

Theorem 3.1.

Proof:

(3.2)

(3.3)

(3.4)

(3.6)

Theorem 3.2.

(3.7)

Corollary 3.1.

Proof:

Example 3.1.

(*)

(**)

Conclusion

Authors contribution

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