F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172256.2 Research Article Articles New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone

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

 Bardan Mustafa K. Funding Acquisition Project Administration Software Validation Writing – Original Draft Preparation Writing – Review & Editing 1 Al-Jumaili Alaa M.F. Conceptualization Data Curation Formal Analysis Investigation Methodology Supervision https://orcid.org/0000-0002-9363-2562 a 1 Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Anbar, 31001, Iraq eps.alaamahmood.farhan@uoanbar.edu.iq

No competing interests were disclosed.

 14 2 2026 2025 14 1362 11 2 2026 Copyright: © 2026 Bardan MK and Al-Jumaili AMF 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

After witnessing the implementations of Banach fixed point theory which is stated that a mapping T: X→X has always a unique fixed point in X in giving the existence and uniqueness solutions for many integral and differential equations, various extensions of Banach fixed point theory were established. Consequently, the theory has evolved to encompass diverse extensions and is fruitful in many fields. One of the most significant advances in pure and applied mathematics is the discovery of solutions for linear and nonlinear systems as well fractal graphics, optimization theory, approximation theory, discrete dynamics, and numerous other areas. Our main outcomes in this manuscript represent one of the most important of these extensions.

 Methods and Results

Vital concepts are needed in the sequels which are playing a major role in verifying our major outcomes have been presented. Throughout this manuscript, ( X , ≼ ) indicates to a partially ordered set with the partially ordered ≼ . In this study, our main objective is to investigate and verify various new enhanced results of coupled fixed point theorems for continuous maps having the property of mixed monotone under the influence of extended contraction circumstances in the context of partially ordered D ∗ -complete metric spaces. Numerous characterizations of these types of coupled fixed point theorems have been verified. Additionally, an appropriate example that supports the major outcomes was prepared.

 Conclusions

Our main results in this manuscript have been explored novel various outcomes related to the uniqueness of various coupled fixed point theorems for continuous maps having the property of mixed monotone under the influence of extended contraction circumstances in the context of partially ordered D ∗ -complete metric spaces. We predict that the discoveries in this study will aid scientists in enhancing the research on popularized partially ordered metric spaces to elevate a universal framework for their practical implementations.

 D^*-metric spaces; partially ordered D^*-metric spaces; D^*-complete metric; coupled fixed point; property of mixed monotone; partially ordered set; D^*-continuous maps. The author(s) declared that no grants were involved in supporting this work. Amendments from Version 1

In this text, we will explain the most important modifications and different differences between the new version of our article and the previously published version, as shown below: 1- Some typographical errors may have occurred during reprinting and were not fully noticed. These errors are corrected in the final version. 2- The abstract (background) was rewritten according to the instructions of the honored evaluator. 3- We would like to assure that the manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 4- All the references which suggested by the honored evaluators have been added due to their importance in raising the scientific level for out manuscript and situate out work within the state-of-the-art. 5- We would like to assure that we thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. 6- The condition for the uniqueness of a continuous mapping to have a unique double fixed point that satisfies the mixed monotone property is explained in the final submitted manuscript 7- The relationships between the theorems presented in the research paper are evident from the results obtained through their proofs. 8- The conclusion of the research paper was summarized according to the honored reviewer's instructions. 9- All the errors pointed out according to honored evaluators instructions have been corrected by implementing all their valuable suggestions. 10- The manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts in the final version. 11-The uncommon wording in the research paper, as noted by the honored reviewer in his comments, has been corrected. Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions.

 1. Introduction

The idea of coupled fixed point theory is an extremely active branch of mathematics and is at the essence of nonlinear analysis because it provides an influential tool to confirm the existence and uniqueness of solutions for numerous nonlinear issues emerging with pure and applied mathematics and other branches of sciences, such as computer science, engineering, physics, differential equations, optimization, control theory, approximation theory, and discrete dynamics. One of the most important results of mathematical analysis is the Banach contraction mapping. It is a popular tool for solving existing issues in different fields of pure and applied mathematics. Numerous extensions of Banach contraction mapping have been proposed via some authors in the literature, such as. A, Ran, and M. Reurings 1 extended Banach contraction mapping in a partially ordered set with various implementations to linear and nonlinear matrix equations. Nieto and R. López 2 generalized the results of the authors in 1 and utilized their major outcomes to obtain a unique solution concerning satisfactory first-order differential periodic boundary circumstances. Subsequently, the notions of coupled fixed point and mixed monotone mappings were presented by T. Bhaskar and V. Lakshmikantham, 3 and several coupled fixed points resulted in partially ordered metric space, in addition to utilizing their outcomes on a first-order ordinary differential with periodic boundary circumstances. Presently, Shaban et al. 4 verified the idea D ∗ -metric spaces, which are extensions of ordinary metric spaces. However, after the publication of this work, numerous results related to coupled, common coupled, and coupled coincidence fixed points have been reported in the literature. 5– 9 Motivated by these facts, Shah et al. 10 investigated and proved various coupled fixed point theorems for integral-type contractive mappings in G-metric spaces. Furthermore, T. Oklah and A. Al-Jumaili 11 employed the notion of compatibility for hybrid pairs of mappings and verified several common coupled coincidence fixed point outcomes with satisfactory properties of mixed G -monotone in partially ordered D ∗ -metric spaces. Additionally, in the same year N. A. Majid, et al. 12 verified several novel outcomes of fixed points for monotone multi-valued maps in partially ordered D ∗ -complete metric spaces, and other various results of coupled fixed point of maps satisfying contractive conditions have been obtained, as well 13 they discussed and confirmed various outcomes of common and coincidence of fixed points theorems in S-complete metric spaces. Recently, Oklah and Al-Jumaili 14 offered and investigated some practical implementations for pairs of self maps satisfy extended contractive conditions of the integral kind in D ∗ -metric spaces and obtained some common and coupled fixed point results in such spaces. For future studies, we can expand and generalize our results to other spaces, such as. 15– 18 The inspiration for introducing this manuscript is to consider and verify novel extended categories of coupled fixed point outcomes for continuous maps satisfying the properties of mixed monotone under the influence of generalized contraction circumstances in context of partially D ∗ -complete metric spaces. In addition, introduce an appropriate example to support our main results. Our major outcomes, which are related to these categories of coupled fixed points, generalize and improve the various results existing in the literature.

 2. Materials and methods

This section is devoted to remembering various ideas and significant outcomes that play a vital role in this work and to confirming our major outcomes. Throughout this manuscript, ( X , ≼ ) indicates to a partially ordered set with the partially ordered ≼ . Via y ∗ ≺ ϰ ∗ holds, mean y ∗ ≼ ϰ ∗ holds, and via ϰ ∗ ≺ y ∗ holds, mean ϰ ∗ ≼ y ∗ holds, with ϰ ∗ ≠ y ∗ .

4 Suppose that D ∗ : X 3 → ℝ + , is a mapping described on X ≠ ∅ and satisfactory the next conditions ∀ ϰ ∗ , y ∗ , z ∗ , w ∗ ∈ X :

( D 1 ∗ ) D ∗ ( ϰ ∗ , y ∗ , z ∗ ) ≥ 0 , ∀ ϰ ∗ , y ∗ , z ∗ ∈ X ;

( D 2 ∗ ) D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = 0 iff ϰ ∗ = y ∗ = z ∗ ;

( D 3 ∗ ) D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = D ∗ ( P { ϰ ∗ , y ∗ , z ∗ } ) , (Symmetry) where P is permutation map,

( D 4 ∗ ) D ∗ ( ϰ ∗ , y ∗ , z ∗ ) ≤ D ∗ ( ϰ ∗ , y ∗ , w ∗ ) + D ∗ ( w ∗ , z ∗ , z ∗ ) .

Therefore, D ∗ is called D ∗ -metric and ( X , D ∗ ) called D ∗ -metric space.

4 Lineal examples of such a map are:

D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = max { | ϰ ∗ − y ∗ | , | y ∗ − z ∗ | , | ϰ ∗ − z ∗ | } ,

D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = | ϰ ∗ − y ∗ | + | y ∗ − z ∗ | + | ϰ ∗ − z ∗ | .

If X = ℝ . So describe: D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = { 0 if ϰ ∗ = y ∗ = z ∗ max { ϰ ∗ , y ∗ , z ∗ } otherwise

4 Presume that ( X , D ∗ ) is a D ∗ -metric space; therefore,

A sequence { ϰ s ∗ } is called converges to ϰ ∗ ∈ X iff D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ ∗ ) = D ∗ ( ϰ ∗ , ϰ ∗ , ϰ s ∗ ) → 0 as s → ∞ . i.e., equivalent with, ∀ ε > 0 , ∃ ℯ ∈ ℤ + (s. t), ∀ s , r ≥ ℯ ⟹ D ∗ ( ϰ ∗ , ϰ s ∗ , ϰ r ∗ ) < ε .

A sequence { ϰ s ∗ } is called D ∗ -Cauchy if ∀ ε > 0 , ∃ ℯ ∈ ℤ + ; ∀ s , r ≥ ℯ , D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ r ∗ ) < ε . That is, if D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ r ∗ ) → 0 as , s , r → ∞ .

( X , D ∗ ) is called D ∗ -complete metric space. If each D ∗ -Cauchy sequence is convergent in X .

In Ref. 4, it has been illustrated that D ∗ -metric space induces the Housdorff topology with convergence, as demonstrated in Definition 2.3, relative to this kind of topology. This topology is Housdorff, with { ϰ s ∗ } converge to only one point at most.

19 The next statements are equivalent in ( X , D ∗ ) :

{ ϰ s ∗ } is D ∗ -convergent to ϰ ∗ ;

D ∗ ( ϰ s ∗ , ϰ ∗ , ϰ ∗ ) → 0 , as ( s → ∞ ) ;

D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ ∗ ) → 0 , as ( s → ∞ ) ;

4 Next statements are equivalent in ( X , D ∗ ) :

The sequence { ϰ s ∗ } is D ∗ -Cauchy;

∀ ε > 0 , ∃ ℯ ∈ ℤ + (s. t), D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ r ∗ ) < ε ∀ s , r ≥ ℯ .

4 If ( X , D ∗ ) is D ∗ -metric space, thus D ∗ ( ϰ ∗ , ϰ ∗ , y ∗ ) ≤ 2 D ∗ ( y ∗ , y ∗ , ϰ ∗ ) ∀ ϰ ∗ , y ∗ ∈ X .

By combining Lemma 2.6 and Lemma 2.7 we obtain the next outcome:

If ( X , D ∗ ) is D ∗ -M-sp, hence { ϰ s ∗ } is D ∗ -Cauchy iff ∀ ε > 0 , ∃ ℯ ∈ ℤ + where, D ∗ ( ϰ s ∗ , ϰ s ∗ , ϰ r ∗ ) < ε ∀ s > r ≥ ℯ .

4 ( X , D ∗ ) called symmetric if D ∗ ( ϰ ∗ , ϰ ∗ , y ∗ ) = D ∗ ( ϰ ∗ , y ∗ , y ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X , As well ( X , D ∗ ) is said to be non-Symmetric if it’s not-Symmetric.

4 Each D ∗ -metric on X induces a metric d D ∗ on X via

d D ∗ ( ϰ ∗ , y ∗ ) = D ∗ ( ϰ ∗ , y ∗ , y ∗ ) + D ∗ ( y ∗ , ϰ ∗ , ϰ ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X , for D ∗ -Symmetric

d D ∗ ( ϰ ∗ , y ∗ ) = 2 D ∗ ( ϰ ∗ , y ∗ , y ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X .

Assume that X = ℤ + , describe: D ∗ ( ϰ ∗ , ϰ ∗ , y ∗ ) = { ϰ ∗ + y ∗ if ϰ ∗ < y ∗ ϰ ∗ + y ∗ + 1 2 if ϰ ∗ > y ∗ , D ∗ ( ϰ ∗ , y ∗ , z ∗ ) = { 0 if ϰ ∗ = y ∗ = z ∗ ϰ ∗ + y ∗ + z ∗ if ϰ ∗ ≠ y ∗ ≠ z ∗ with symmetry in each three variables

In this case, ( X , D ∗ ) is D ∗ -metric space and non-symmetric, because if ϰ ∗ < y ∗ , obtain D ∗ ( ϰ ∗ , ϰ ∗ , y ∗ ) = ϰ ∗ + y ∗ ≠ ϰ ∗ + y ∗ + 1 2 = D ∗ ( ϰ ∗ , y ∗ , y ∗ )

If ( X , D ∗ ) is non D ∗ -symmetric space, the D ∗ -metric properties demonstrate that 3 2 D ∗ ( ϰ ∗ , y ∗ , y ∗ ) ≤ d D ∗ ( ϰ ∗ , y ∗ ) = 3 D ∗ ( ϰ ∗ , y ∗ , y ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X .

20 Assume ( X , D ∗ ) and ( X ∗ , D ∗ ∗ ) are two D ∗ -metric space. Then, F : X ⟶ X ∗ is D ∗ -continuous at ϰ ∗ ∈ X iff it’s D ∗ -sequentially continuous at ϰ ∗ ,i.e., when { ϰ s ∗ } is D ∗ -convergent to ϰ ∗ , { F ( ϰ s ∗ ) } is D ∗ -convergent to F ( ϰ ∗ ) .

11 Let ( X , D ∗ ) be D ∗ -metric space. A map F : X × X ⟶ X is called continuous if for arbitrary two D ∗ -convergent sequences { ϰ s ∗ } and { y s ∗ } converging to ϰ ∗ and y ∗ correspondingly, { F ( ϰ s ∗ , y s ∗ ) } is D ∗ -convergent to F ( ϰ ∗ , y ∗ ) .

3 Presume that ( X , ≼ ) partially ordered set (Concisely, P.O.S). A map F : X × X → X is said to have mixed monotone property (Concisely, M.M.P) if F ( ϰ ∗ , y ∗ ) is monotone nondecreasing in ϰ ∗ and is monotone nonincreasing of y ∗ ; that is ∀ ϰ ∗ , y ∗ ∈ X , ϰ 1 ∗ , ϰ 2 ∗ ∈ X , ϰ 1 ∗ ≼ ϰ 2 ∗ Implies F ( ϰ 1 ∗ , y ∗ ) ≼ F ( ϰ 2 ∗ , y ∗ ) , y 1 ∗ , y 2 ∗ ∈ X , y 1 ∗ ≼ y 2 ∗ Implies F ( ϰ ∗ , y 2 ∗ ) ≼ F ( ϰ ∗ , y 1 ∗ ) .

3 An ( ϰ ∗ , y ∗ ) ∈ X × X , when X ≠ ∅ , called coupled fixed point (Concisely, C.F.P) of a map F : X × X → X , if F ( ϰ ∗ , y ∗ ) = ϰ ∗ and F ( y ∗ , ϰ ∗ ) = y ∗ .

3. Some main results of new generalized categories of coupled fixed point theorems

This section is devoted to investigating and verifying various novel extended categories of coupled fixed point outcomes for continuous maps of the satisfactory properties of mixed monotones under various extended contraction conditions in partially ordered complete D ∗ -metric-spaces.

Let ( X , D ∗ ) be a D ∗ -complete metric space defined on (P.O.S) ( X , ≼ ) . Presume that F : ( X , D ∗ ) × ( X , D ∗ ) → ( X , D ∗ ) a continuous mapping containing the (M.M.P). Suppose that ∃ k ∈ [ 0 , 1 ) (s. t) for ϰ ∗ , y ∗ , z ∗ , p , q , w ∗ ∈ X , the following inequality holds: D ∗ ( F ( ϰ ∗ , y ∗ ) , F ( p , q ) , F ( w ∗ , z ∗ ) ) ≤ k 2 [ D ∗ ( ϰ ∗ , p , w ∗ ) + D ∗ ( y ∗ , q , z ∗ ) ]

∀ ϰ ∗ ≽ p ≽ w ∗ and y ∗ ≼ q ≼ z ∗ wherever either q ≠ z ∗ or p ≠ w ∗ . If ∃ ϰ 0 ∗ , y 0 ∗ ∈ X (s. t) ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) and y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) , so F has (C. F. P) in X .

Through the condition of above theorem ∃ ϰ 0 ∗ , y 0 ∗ ∈ X where ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) and y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) . Describe, ϰ 1 ∗ , y 1 ∗ ∈ X as ϰ 1 ∗ = F ( ϰ 0 ∗ , y 0 ∗ ) ≽ ϰ 0 ∗ and y 1 ∗ = F ( y 0 ∗ , ϰ 0 ∗ ) ≼ y 0 ∗ .

Presume that, ϰ 2 ∗ = F ( ϰ 1 ∗ , y 1 ∗ ) and y 2 ∗ = F ( y 1 ∗ , ϰ 1 ∗ ) , we write Ϝ 2 ( ϰ 0 ∗ , y 0 ∗ ) = Ϝ ( F ( ϰ 0 ∗ , y 0 ∗ ) , F ( y 0 ∗ , ϰ 0 ∗ ) ) = Ϝ ( ϰ 1 ∗ , y 1 ∗ ) = ϰ 2 ∗

And F 2 ( y 0 ∗ , ϰ 0 ∗ ) = F ( F ( y 0 ∗ , ϰ 0 ∗ ) , F ( ϰ 0 ∗ , y 0 ∗ ) = F ( y 1 ∗ , ϰ 1 ∗ ) = y 2 ∗ .

Utilizing, the (M. M. P) of F we obtain, ϰ 2 ∗ = F 2 ( ϰ 0 ∗ , y 0 ∗ ) = F ( ϰ 1 ∗ , y 1 ∗ ) ≽ F ( ϰ 0 ∗ , y 0 ∗ ) = ϰ 1 ∗ ≽ ϰ 0 ∗

And y 2 ∗ = F 2 ( y 0 ∗ , ϰ 0 ∗ ) = F ( y 1 ∗ , ϰ 1 ∗ ) ≼ F ( y 0 ∗ , ϰ 0 ∗ ) = y 1 ∗ ≼ y 0 ∗ .

Ongoing the above proceedings we get repeatedly, ∀ s ≥ 0 , ϰ s + 1 ∗ = F s + 1 ( ϰ 0 ∗ , y 0 ∗ ) = Ϝ ( F s ( ϰ 0 ∗ , y 0 ∗ ) , F s ( y 0 ∗ , ϰ 0 ∗ ) )

And y s + 1 ∗ = F s + 1 ( y 0 ∗ , ϰ 0 ∗ ) = Ϝ ( F s ( y 0 ∗ , ϰ 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) ) .

In that case, ∀ s ≥ 0 , ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) = ϰ 1 ∗ ≼ F 2 ( ϰ 0 ∗ , y 0 ∗ ) = ϰ 2 ∗ ≼ ∙ ∙ ∙ ≼ F s + 1 ( ϰ 0 ∗ , y 0 ∗ ) = ϰ s + 1 ∗ ≼ ∙ ∙ ∙

With y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) = y 1 ∗ ≽ F 2 ( y 0 ∗ , ϰ 0 ∗ ) = y 2 ∗ ≽ ∙ ∙ ∙ ≽ F s + 1 ( y 0 ∗ , ϰ 0 ∗ ) = y s + 1 ∗ ≽ ∙ ∙ ∙

If ( ϰ s + 1 ∗ , y s + 1 ∗ ) = ( ϰ s ∗ , y s ∗ ) , in that case F has (C. F. P), as a result we presume

( ϰ s + 1 ∗ , y s + 1 ∗ ) ≠ ( ϰ s ∗ , y s ∗ ) for each s ≥ 0 , that is, we presume that either ϰ s + 1 ∗ = F ( ϰ s ∗ , y s ∗ ) ≠ ϰ s ∗ or y s + 1 ∗ = F ( y s ∗ , ϰ s ∗ ) ≠ y s ∗ .

Next, we verify that, ∀ s ≥ 0 , { D ∗ ( F s + 1 ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) ) ≤ k s 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) }

And { D ∗ ( F s + 1 ( y 0 ∗ , ϰ 0 ∗ ) , F s ( y 0 ∗ , ϰ 0 ∗ ) , F s ( y 0 ∗ , ϰ 0 ∗ ) ) ≤ k s 2 [ D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) + D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) }

For, s = 1 , we obtain { D ∗ ( F 2 ( ϰ 0 ∗ , y 0 ∗ ) , F ( ϰ 0 ∗ , y 0 ∗ ) , F ( ϰ 0 ∗ , y 0 ∗ ) ) = D ∗ ( F ( F ( ϰ 0 ∗ , y 0 ∗ ) , F ( y 0 ∗ , ϰ 0 ∗ ) , F ( ϰ 0 ∗ , y 0 ∗ ) , F ( ϰ 0 ∗ , y 0 ∗ ) ) ) ≤ k 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] }

Utilizing, inequality (3.1), because ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) and y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) and since either ϰ s + 1 ∗ = F ( ϰ s ∗ , y s ∗ ) ≠ ϰ s ∗ or y s + 1 ∗ = F ( y s ∗ , ϰ s ∗ ) ≠ y s ∗ ∀ s that is D ∗ ( ϰ 2 ∗ , ϰ 1 ∗ , ϰ 1 ∗ ) ≤ k 2 [ D ∗ ( ϰ 1 ∗ , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( y 1 ∗ , y 0 ∗ , y 0 ∗ ) ] .

Likewise, establish D ∗ ( y 2 ∗ , y 1 ∗ , y 1 ∗ ) ≤ k 2 [ D ∗ ( y 1 ∗ , y 0 ∗ , y 0 ∗ ) + D ∗ ( ϰ 1 ∗ , ϰ 0 ∗ , ϰ 0 ∗ ) ] .

Consequently inequalities (3.4) and (3.5) hold for s = 1 .

Next, presume that inequalities- (3.4) and (3.5) hold, for s = r .

Utilizing the truths F r + 1 ( ϰ 0 ∗ , y 0 ∗ ) ≽ F r ( ϰ 0 ∗ , y 0 ∗ ) and F r + 1 ( y 0 ∗ , ϰ 0 ∗ ) ≼ F r ( y 0 ∗ , ϰ 0 ∗ ) , we obtain { D ∗ ( F r + 2 ( ϰ 0 ∗ , y 0 ∗ ) , F r + 1 ( ϰ 0 ∗ , y 0 ∗ ) , F r + 1 ( ϰ 0 ∗ , y 0 ∗ ) ) = D ∗ [ F ( F r + 1 ( ϰ 0 ∗ , y 0 ∗ ) , F r + 1 ( y 0 ∗ , ϰ 0 ∗ ) ) , F ( F r ( ϰ 0 ∗ , y 0 ∗ ) , F r ( y 0 ∗ , ϰ 0 ∗ ) ) , F ( F r ( ϰ 0 ∗ , y 0 ∗ ) , F r ( y 0 ∗ , ϰ 0 ∗ ) ) ] ≤ k 2 [ D ∗ ( F r + 1 ( ϰ 0 ∗ , y 0 ∗ ) , F r ( ϰ 0 ∗ , y 0 ∗ ) , F r ( ϰ 0 ∗ , y 0 ∗ ) ) + D ∗ ( F r + 1 ( y 0 ∗ , ϰ 0 ∗ ) , F r ( y 0 ∗ , ϰ 0 ∗ ) , F r ( y 0 ∗ , ϰ 0 ∗ ) ) ] }

Utilizing, inequality (3.1), because

F r ( ϰ 0 ∗ , y 0 ∗ ) = F ( ϰ r ∗ , y r ∗ ) ≽ ϰ r ∗ and F r ( y 0 ∗ , ϰ 0 ∗ ) = F ( y r ∗ , ϰ r ∗ ) ≼ y r ∗ , and because either ϰ r + 1 ∗ = F ( ϰ r ∗ , y r ∗ ) ≠ ϰ r ∗ or y r + 1 ∗ = F ( y r ∗ , ϰ r ∗ ) ≠ y r ∗ ≤ k 2 { k r 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] + k r 2 [ D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) + D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) ] }

Because inequality (3.4) and (3.5) are presumed to hold for s = r , so obtain ≤ k r + 1 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] .

Likewise, we can establish that { D ∗ ( F r + 2 ( y 0 ∗ , ϰ 0 ∗ ) , F r + 1 ( y 0 ∗ , ϰ 0 ∗ ) , F r + 1 ( y 0 ∗ , ϰ 0 ∗ ) ) ≤ k r + 1 2 [ D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) + D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) ] }

In that case, via induction, inequalities (3.4) and (3.5) are verified ∀ s ≥ 0 .

In addition, for each positive integer s , r ; s < r , we have through the rectangle inequality( ( D 4 ∗ ) of Definition 2.1) that { D ∗ ( F r ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) ) ≤ D ∗ ( F r ( ϰ 0 ∗ , y 0 ∗ ) , F r − 1 ( ϰ 0 ∗ , y 0 ∗ ) , F r − 1 ( ϰ 0 ∗ , y 0 ∗ ) ) + D ∗ ( F r − 1 ( ϰ 0 ∗ , y 0 ∗ ) , F r − 2 ( ϰ 0 ∗ , y 0 ∗ ) , F r − 2 ( ϰ 0 ∗ , y 0 ∗ ) ) + ∙ ∙ ∙ + D ∗ ( F s + 1 ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) , F s ( ϰ 0 ∗ , y 0 ∗ ) ) ≤ k r − 1 + ∙ ∙ ∙ + k s 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] }

Utilizing inequality (3.4) = { k s ( 1 + k + ∙ ∙ ∙ + k r − s − 1 ) 2 [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] < k s 2 ( 1 − k ) [ D ∗ ( F ( ϰ 0 ∗ , y 0 ∗ ) , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( F ( y 0 ∗ , ϰ 0 ∗ ) , y 0 ∗ , y 0 ∗ ) ] }

This mean, D ∗ ( ϰ r ∗ , ϰ s ∗ , ϰ s ∗ ) < k s 2 ( 1 − k ) [ D ∗ ( ϰ 1 ∗ , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( y 1 ∗ , y 0 ∗ , y 0 ∗ ) ] .

Therefore, lim r , s → ∞ D ∗ ( ϰ r ∗ , ϰ s ∗ , ϰ s ∗ ) = 0 .

Consequently, utilizing Lemma-2.8, { ϰ s ∗ } that is, { F s ( ϰ 0 ∗ , y 0 ∗ ) } is Cauchy sequence and thus is convergent in D ∗ -complete-metric space X . Assume that ϰ s ∗ → ϰ ∗ , as s → ∞ .

Likewise, { y s ∗ } , mean { F s ( y 0 ∗ , ϰ 0 ∗ ) } is as well a Cauchy sequence and consequently is convergent in D ∗ -complete-metric space X . Assume that y s ∗ → y ∗ , as s → ∞ .

Next we illustrate that F has (C. F. P) in X .

Utilizing inequalities- (3.4) and (3.6) we obtain D ∗ ( F ( ϰ s ∗ , y s ∗ ) , ϰ s ∗ , ϰ s ∗ ) ≤ k s 2 [ D ∗ ( ϰ 1 ∗ , ϰ 0 ∗ , ϰ 0 ∗ ) + D ∗ ( y 1 ∗ , y 0 ∗ , y 0 ∗ ) ] .

Selecting the limit as s → ∞ and utilizing the fact that map F is continuous, we obtain: D ∗ ( F ( ϰ ∗ , y ∗ ) , ϰ ∗ , ϰ ∗ ) ≤ 0 , ⟹ F ( ϰ ∗ , y ∗ ) = ϰ ∗ .

Similarly, obtain F ( y ∗ , ϰ ∗ ) = y ∗ . Therefore, we verified ( ϰ ∗ , y ∗ ) is (C. F. P) of F .

Let ( X , D ∗ ) be a D ∗ -complete metric space defined on (P.O.S) ( X , ≼ ) . Presume that F : ( X , D ∗ ) × ( X , D ∗ ) → ( X , D ∗ ) satisfying all the conditions in Theorem 3.1, as well as the next conditions: ( i ) If a nondecreasing { ϰ s ∗ } ⟶ ϰ ∗ . So , ϰ s ∗ ≼ ϰ ∗ , ∀ s , ( ii ) If a nonincreasing { y s ∗ } ⟶ y ∗ . So , y s ∗ ≽ y ∗ , ∀ s .

In that case F has (C. F. P).

According to the procedures followed in Theorem 3.1, exactly we can reach inequalities (3.6) and (3.7) exactly. So by inequalities- (3.8) and (3.9), we obtain, ϰ s ∗ ≼ ϰ ∗ and y s ∗ ≽ y ∗ , ∀ s ≥ 0 .

If ϰ s ∗ = ϰ ∗ and y s ∗ = y ∗ for some s, hence, via structures, ϰ s + 1 ∗ = ϰ ∗ and y s + 1 ∗ = y ∗ and ( ϰ ∗ , y ∗ ) is a (C. F. P). As a result we presume either ϰ s ∗ ≠ ϰ ∗ or y s ∗ ≠ y ∗ .

In that case we obtain { D ∗ ( F ( ϰ ∗ , y ∗ ) , F ( ϰ ∗ , y ∗ ) , ϰ ∗ ) ≤ D ∗ ( F ( ϰ ∗ , y ∗ ) , F ( ϰ ∗ , y ∗ ) , F ( ϰ s ∗ , y s ∗ ) ) + D ∗ ( F ( ϰ ∗ , y ∗ ) , F ( ϰ ∗ , y ∗ ) , ϰ ∗ ) = D ∗ ( F ( ϰ s ∗ , y s ∗ ) , F ( ϰ ∗ , y ∗ ) , F ( ϰ ∗ , y ∗ ) ) + D ∗ ( F ( ϰ s ∗ , y s ∗ ) , F ( ϰ s ∗ , y s ∗ ) , ϰ ∗ ) ≤ k 2 [ D ∗ ( ϰ s ∗ , ϰ ∗ , ϰ ∗ ) + D ∗ ( y s ∗ , y ∗ , y ∗ ) ] + D ∗ ( ϰ s + 1 ∗ , ϰ s + 1 ∗ , ϰ ∗ ) by inequality − ( 3.1 ) }

Selecting, s → ∞ in the above inequality, we get D ∗ ( F ( ϰ ∗ , y ∗ ) , F ( ϰ ∗ , y ∗ ) , ϰ ∗ ) = 0 , this mean F ( ϰ ∗ , y ∗ ) = ϰ ∗ . Likewise, we obtain F ( y ∗ , ϰ ∗ ) = y ∗ .

Let ( X , D ∗ ) be D ∗ -complete metric space defined on (P.O.S) ( X , ≼ ) . Presume that F : ( X , D ∗ ) × ( X , D ∗ ) → ( X , D ∗ ) a continuous map containing the (M.M.P) on X , (s. t) F ( ϰ ∗ , y ∗ ) ≼ F ( y ∗ , ϰ ∗ ) when ϰ ∗ ≼ y ∗ . Suppose that ∃ k ∈ [ 0 , 1 ) (s. t), ∀ ϰ ∗ , y ∗ , z ∗ , p , q , w ∗ ∈ X , (3.1) holds, when ϰ ∗ ≽ p ≽ w ∗ , y ∗ ≼ q ≼ z ∗ and ϰ ∗ ≺ y ∗ wherever p ≠ w ∗ or q ≠ z ∗ .

If ∃ ϰ 0 ∗ , y 0 ∗ ∈ X , (s. t) ϰ 0 ∗ ≼ y 0 ∗ , ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) and y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) , so F has (C. F. P) in X .

Through the circumstances of above theorem ∃ ϰ 0 ∗ , y 0 ∗ ∈ X (s. t) ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) and y 0 ∗ ≽ F ( y 0 ∗ , ϰ 0 ∗ ) .

We describe ϰ 1 ∗ , y 1 ∗ ∈ X as ϰ 1 ∗ = F ( ϰ 0 ∗ , y 0 ∗ ) ≽ ϰ 0 ∗ and y 1 ∗ = F ( y 0 ∗ , ϰ 0 ∗ ) ≼ y 0 ∗ .

Because, ϰ 0 ∗ ≼ y 0 ∗ , we obtain, via circumstances of the theorem, F ( ϰ 0 ∗ , y 0 ∗ ) ≼ F ( y 0 ∗ , ϰ 0 ∗ ) .

For this reason, ϰ 0 ∗ ≼ ϰ 1 ∗ = F ( ϰ 0 ∗ , y 0 ∗ ) ≼ F ( y 0 ∗ , ϰ 0 ∗ ) = y 1 ∗ ≼ y 0 ∗ .

Ongoing the above processes we get two sequences { ϰ s ∗ } and { y s ∗ } recursively as follows. ∀ s ≥ 1 , ϰ s ∗ = F ( ϰ s − 1 ∗ , y s − 1 ∗ ) and y s ∗ = F ( y s − 1 ∗ , ϰ s − 1 ∗ ) ,

Such that { ϰ 0 ∗ ≼ F ( ϰ 0 ∗ , y 0 ∗ ) = ϰ 1 ∗ ≼ ∙ ∙ ∙ ≼ F ( ϰ s − 1 ∗ , y s − 1 ∗ ) = ϰ s ∗ ≼ ∙ ∙ ∙ ≼ y s ∗ = F ( y s − 1 ∗ , ϰ s − 1 ∗ ) ≼ ∙ ∙ ∙ ≼ y 1 ∗ = F ( y 0 ∗ , ϰ 0 ∗ ) ≼ y 0 ∗ }

In special, we have ∀ s ≥ 0 , ϰ s ∗ ≼ F ( ϰ s ∗ , y s ∗ ) = ϰ s + 1 ∗ ≼ y s + 1 ∗ = F ( y s ∗ , ϰ s ∗ ) ≼ y s ∗ .

When, ϰ s ∗ = y s ∗ = c for some s , in that case c ≼ F ( c , c ) ≼ F ( c , c ) ≼ c . It's illustrates, c = F ( c , c ) . Therefore ( c , c ) is (C. F. P). Therefore, we assume that ϰ s ∗ ≼ y s ∗ , ∀ s ≥ 0 .

Additional, via identical cause as declared in Theorem 3.1, presume ( ϰ s ∗ , y s ∗ ) ≠ ( ϰ s + 1 ∗ , y s + 1 ∗ ) .

In that case, in sight of inequality (3.12), ∀ s ≥ 0 , inequality (3.1) hold and ϰ ∗ = ϰ s + 2 ∗ , p = ϰ s + 1 ∗ , w ∗ = ϰ s ∗ , y ∗ = y s ∗ , q = y s + 1 ∗ and z ∗ = y s + 2 ∗ .

Rest of evidence is achieved via reiterating similar procedures as in Theorem 3.1.

Let ( X , D ∗ ) be D ∗ -complete metric space defined on (P.O.S) ( X , ≼ ) . Presume that F : ( X , D ∗ ) × ( X , D ∗ ) ⟶ ( X , D ∗ ) satisfing all the conditions in Theorem 3.3, as well the next conditions:

If nondecreasing { ϰ s ∗ } ⟶ ϰ ∗ . So, ϰ s ∗ ≼ ϰ ∗ , ∀ s ,

If nonincreasing { y s ∗ } ⟶ y ∗ . So, y s ∗ ≽ y ∗ , ∀ s .

In that case F has (C. F. P).

This evidence is straightforward and analogous to that of Theorem 3.2.

Next, discuss the following example, which extends to that of 21 :

Suppose that X = ℕ 0 and D ∗ : X 3 → [ 0 , ∞ ) is described as follows: { 0 if ϰ ∗ = y ∗ = z ∗ z ∗ + 1 if ϰ ∗ = 0 , y ∗ = 0 , z ∗ ≠ 0 y ∗ + 2 if ϰ ∗ = 0 , y ∗ = z ∗ ≠ 0 y ∗ + z ∗ + 1 if ϰ ∗ = 0 , y ∗ ≠ z ∗ , ( s . t ) y ∗ , z ∗ diverse of 0 ϰ ∗ + z ∗ if ϰ ∗ = y ∗ ≠ z ∗ , each diverse of 0 , ϰ ∗ + y ∗ + z ∗ if ϰ ∗ , y ∗ , z ∗ , are each diverse with diverse of 0

Because, ( X , D ∗ ) satisfies (iii) in Definition 2.3, in that case ( X , D ∗ ) is D ∗ -complete-metric space. Assume that a (P. O) ≼ described on X as follows: ∀ ϰ ∗ , y ∗ ∈ X , such that ϰ ∗ ≼ y ∗ holds, if ϰ ∗ > y ∗ with 0 ≼ 1 and 3 ≼ 1 hold.

Assume a map F : X × X → X is described as: F ( ϰ ∗ , y ∗ ) = { 1 if ϰ ∗ ≺ y ∗ 0 if otherwise

Presume that, w ∗ ≼ p ≼ ϰ ∗ ≺ y ∗ ≼ q ≼ z ∗ hold, so via equivalent form, obtain w ∗ ≥ p ≥ ϰ ∗ > y ∗ ≥ q ≥ z ∗ . In that case F ( ϰ ∗ , y ∗ ) = F ( p , q ) = F ( w ∗ , z ∗ ) = 1 . Consequently , the left-hand side of inequality (3.1) is D ∗ ( 1 , 1 , 1 ) = 0 , thus (3.1) is satisfied.

After that with ϰ 0 ∗ = 81 and y 0 ∗ = 0 . Theorem 3.4 is appropriate for this Example 3.5. It may be viewed in this example that the (C. F. P) is not unique. Therefore, (0, 0) and (1,0) are two (C. F. P) of map F .

The next Remarks are analogy of the Remarks in [21] in D ∗ -metric space:

We observed through Lemma 2.10 that D ∗ -metric induces a metric d D ∗ on X via

d D ∗ ( ϰ ∗ , y ∗ ) = D ∗ ( ϰ ∗ , y ∗ , y ∗ ) + D ∗ ( y ∗ , ϰ ∗ , ϰ ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X , for D ∗ -Symmetric d D ∗ ( ϰ ∗ , y ∗ ) = 2 D ∗ ( ϰ ∗ , y ∗ , y ∗ ) , ∀ ϰ ∗ , y ∗ ∈ X . Because of the circumstance p ≠ w ∗ or q ≠ z ∗ , inequality- (3.1) doesn’t minimize to any metric inequality with metric d D ∗ . Therefore, our theorems do not minimize fixed point issues analogous to ( X , d D ∗ ) .

4. Conclusions

The theorems of coupled fixed points in generalized partially ordered metric spaces represent a significant part of confirming the existence and uniqueness of solutions for various integral type equations in pure mathematics and applied sciences such as mathematical models, optimization, control theory, approximation theory, discrete dynamics, and economic theories. Therefore, novel extended categories of coupled fixed point theorems for continuous maps satisfying the property of mixed monotone in the context of extended partially ordered D ∗ -complete-M-sp have been investigated and proven. In addition, to reinforce our major outcomes, a suitable example is provided. Additionally, our major results in Theorems 3.1 and 3.2 are not appropriate for Example 3.5. This is apparent from the fact that inequality (3.1) is not satisfied when w ∗ = p = ϰ ∗ = y ∗ = 3 , z ∗ = 1 and q = 0 . Finally, our main results, which are related to these types of extended coupled fixed point theorems, extend and improve the various outcomes in the literature, also for other research works we refer the authors to see [ 22– 24]. We predict that the discoveries in this study will aid scientists in enhancing the research on popularized partially ordered metric spaces to elevate a universal framework for their practical implementation.

 Ethical approval

We would like to inform you that our study does not require any ethical approval.

 Data availability

No datasets were generated or analyzed during the current study (Our manuscript type does not require data).

 References Ran ACM Reurings MCB : A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004;132(5):1435–1443. 10.1090/S0002-9939-03-07220-4 Nieto JJ López RR : Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 2005;22:223–239. 10.1007/s11083-005-9018-5 Bhaskar TG Lakshmikantham V : fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006;65:1379–1393. 10.1016/j.na.2005.10.017 Shaban S Nabi S Haiyun Z : A common Fixed Point Theorem in D *-Metric Spaces. Hindawi Publishing Corporation. Fixed Point Theory Appl. 2007;13. Article ID 27906. Abbas M Khan MA Radenović S : Common coupled fixed point theorem in cone metric space for w-compatible mappings. Appl. Math. Comput. 2010;217:195–202. 10.1016/j.amc.2010.05.042 Saadati R Vaezpour SM Vetro P : Fixed point theorems in generalized partially ordered G-metric spaces. Math. Comput. Model. 2010;52:797–801. 10.1016/j.mcm.2010.05.009 Aydi H Damjanovic B Samet B : Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model. 2011;54(9-10):2443–2450. 10.1016/j.mcm.2011.05.059 Shah R Zada A Li T : New common coupled fixed point results of integral type contraction in generalized metric spaces. J. Anal. Num. Theor. 2016;4(2):145–152. 10.18576/jant/040210 Zada A Shah R Li T : Integral type contraction and coupled coincidence fixed point theorems for two pairs in G-metric spaces. Hacet. J. Math. Stat. 2016;5(5):1475–1484. 10.15672/HJMS.20164514280 Shah R Zada A : Coupled fixed point theorems involving contractive condition of integral type in generalized metric spaces. J. Linear. Topological. Algebra. 2017;6(1):45–53. Oklah TN Al-Jumaili AMF : On Compatible Mappings and Common Coupled Coincidence Fixed Point Results in Partially Ordered D *-Metric Spaces. 6th Iraqi International Conference on Engineering Technology and its Applications, IICETA 2023. 2023; pp.669–674. Majid NA AL-Jumaili AMF Ng ZC : Some applications of fixed point results for monotone multivalued and integral type contractive mappings. Fixed Point Theory Algorithms Sci. Eng. 2023;2023(1):11. 10.1186/s13663-023-00748-9 Majid NA AL-Jumaili AMF Ng ZC : Enhanced Results in Common and Coincidence Fixed Point Theory with Applications to Simulation Mappings. Eur. J. Pure Appl. Math. 2024;17(3):1877–1893. 10.29020/nybg.ejpam.v17i3.5268 Oklah TN Al-Jumaili AMF : Fixed Point Results of Integral Type Contractive Mappings in D∗-Metric Spaces. AIP Conference Proceedings, 2nd International Conference on Scientific Research and Innovation 2023, 2ICSRI 2023. 2025;3169(1). Mohsen SD Thiyab YH : Some Characteristics of Completeness Property in Fuzzy Soft b-Metric Space. J. Appl. Eng. Sci. 2023;27(3):2227–2232. Abed AH Al-Jumaili AMF : Occasionally Weakly Compatible Maps and common Fixed Point Theorems in Certain New Generalized Symmetric Space. Iraqi J. Sci. 2024;65(8):4419–4427. 10.24996/ijs.2024.65.8.24 Majid NA Al-Jumaili AMF Ng ZC : New Results of Fixed-Point Theorems and Their Applications in Complete Complex D c ∗-Metric Spaces. J. Funct. Spaces. 2024;2024:5532624. 10.1155/2024/5532624 Hijab AA Shaakir LK Aljohani S : Results on Common Fixed Points in Strong-Composed-Cone Metric Spaces. Int. J. Anal. Appl. 2025;23(75):17. 10.28924/2291-8639-23-2025-75 Aage CT Salunke JN : some fixed points theorems in generalized D *-metric spaces. Appl. Sci. 2010;2010:1–13. 10.1155/2010/621469 Abed AH Al-Jumaili AMF : On Expansive Mappings and Common Fixed Point Results in D*-Metric Spaces, 3rd Information Technology to Enhance e-Learning and Other Application. IT-ELA. 2022;2022:238–241. Choudhury BS Maity P : Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011;54(1–2):73–79. 10.1016/j.mcm.2011.01.036 Li S Zada A Shah R : Fixed point theorems in dislocated quasi-metric spaces. J. Nonlinear Sci. Appl. 2017;10(09):4695–4703. 10.22436/jnsa.010.09.12 Shah R Zada A Khan I : Some Fixed Point Theorems of Integral Type Contraction in Cone b-metric Spaces. Turkish Journal of Analysis and Number Theory. 2016;3(6):165–169. 10.12691/tjant-3-6-5 George R Aranđelović I Mišić V : Some Fixed Points Results in-Metric and Quasi-Metric Spaces. J. Funct. Spaces. 2022;2022:1–6. 10.1155/2022/1803348 10.5256/f1000research.196441.r459041 Reviewer response for version 2 Padaliya Sanjay Kumar 1 Referee S.G.R.R. (P.G.) College, Dehradun, Uttarakhand, India

Competing interests: No competing interests were disclosed.

 26 2 2026 Copyright: © 2026 Padaliya SK 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

Comments for Manuscript (New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone)

 

The abstract mentions result under "extended contraction circumstances". These need to be explicitly defined or referred to by specific theorem numbers early on to avoid uncertainty.

The manuscript inconsistently uses D^*-metric spaces, partially ordered D^*-metric spaces, and D^ *-complete metric in keywords. Authors must choose correct notation and apply it uniformly throughout the entire manuscript.

 Lemma 2.8 requires a more detailed proof to be fully acceptable for technical soundness.

Authors must explicitly explain the significance of the inequality in remark 2.12 within a non-D*-symmetric space to satisfy the technical correctness.

The current conclusion is vague, stating that "main results... have been explored novel various outcomes". It needs to be rewritten to highlight the specific, concrete mathematical objective achieved.

While authors predict the study will aid scientists in "practical implementations", Authors must specify which practical areas are most impacted.

It is currently required to determine and explain the relationships between the four theorems presented, rather than just listing them as separate results.

Authors need to address whether the major results remain valid if a weaker type of continuity is used, as this is a key point of interest for improving the paper's scope.

 Correct the use of "uncommon forms" and English throughout the manuscript (e.g., phrases like "After witnessing the implementations..."). Also check for spelling mistakes throughout the manuscript (for example it is “Hausdorff” not “Housdorff” in remark 2.4.)

Remove repetitive words, such as the multiple uses of "various" in the same sentence in the abstract and conclusion.

If authors address above points the manuscript can be indexed.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

No

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Fixed Point Theory, Fuzzy Analysis and Fractal Geometry.

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.

 10.5256/f1000research.196441.r458567 Reviewer response for version 2 Shah Rahim 1 Referee https://orcid.org/0009-0001-9044-5470 Kohsar University Murree, Murree, Pakistan

Competing interests: No competing interests were disclosed.

 24 2 2026 Copyright: © 2026 Shah R 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The suggested revisions have been completed, and the paper is now suitable for acceptance for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

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

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Analysis, Fixed Point Theory, Applied Mathematics

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.

AL-Jumaili Alaa Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Alanbar, Iraq

Competing interests: The authors declare no competing interests.

 25 2 2026

We extend our sincere thanks to the esteemed evaluator Prof... Rahim Shah for his valuable corrections and important suggestions due to their importance in raising the scientific level and rigor of the manuscript.  As well  for what he said regarding the paper being good and its results being original.

 10.5256/f1000research.189971.r440891 Reviewer response for version 1 Mahmoud Jamil 1 Referee https://orcid.org/0000-0003-4539-2370 University of Diyala, Diyala, Iraq

Competing interests: No competing interests were disclosed.

 12 1 2026 Copyright: © 2026 Mahmoud J 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

Comments for Manuscript  (New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone)

 

Uniqueness: which conditions must be met for a continuous mapping to have a unique double fixed point that satisfies the mixed monotone property.

 

Relationships: the authors offered three details proofs for each of the four theorems. it is essential to determine their relationships and give examples support their arguments.

  

The major results that authors presented in this paper can be valid when we deploy a weaker type of continuity.

  

Conclusions: the conclusion that given by authors can be summarized in the following: first, the applications of coupled fixed point in generalized partially ordered metric space. Second, viewing an extended category of couple fixed point theorems for a continuous map satisfying the mixed monotone property. Third, the authors claim that these results will help future researchers for practical implementation. The article's main objective should be highlighted in the conclusion, along with a brief explanation of how it would benefit them. The conclusion section needs to be more specific.

  

Linguistic issues: Although the article was written in good English overall, the authors committed a few mistakes, such as utilizing an uncommon form.

  

Definition 2.1 ( D 3 ) (Symmetry) where P is permutation map. Since both have the same meaning, there is no need for both. Use P as permutation map.

  

Lemma 2.8: giving a detailed proof.

  

Lemma 2.10 should be put in this way ( D * -symmetric, d D * x * , y * =2 D * x * , y * , y *  ).

  

Remark 2.12: it stated that X, D *  is non D * -symmetric space, then D * -metric properties demonstrate that 3 2 D * x * , y * , y * ≤ 3D * x * , y * , y *  ( is this a required result? Explain).

 I suggest that this work be accepted since it introduces and establishes new generalized results of linked fixed-point theorems for continuous functions with the property of mixed monotone in the context of partially ordered D * -complete metric spaces. Additionally, I advise authors to evaluate and address the submitted comments.

Best wishes

Your sincerely

Assist. Prof. Dr. Jamil Mahmoud Jamil

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Main fields of research: topology : fuzzy topology, soft topology, generalized topology, fuzzy Algebra , advanced functional 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.

AL-Jumaili Alaa Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Alanbar, Iraq

Competing interests: Competing interests: The authors declare no competing interests.

 27 1 2026

Dear Prof...

Best greetings.....

I hope this message finds you well.

Subject \ comments responses about the manuscript-f1000res172256

 (New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172256)

First: We extend our sincere thanks to the esteemed evaluator for what he said regarding the paper being good and its results being original, as well as the originality of the examples and applications presented in it.      

                                                          

The following is the response to the comments kindly provided by the esteemed resident, and they have been carefully considered due to their importance.

1- The condition for the uniqueness of a continuous mapping to have a unique double fixed point that satisfies the mixed monotone property is explained in the submitted manuscript.

2- The relationships between the theorems presented in the research paper are evident from the results obtained through their proofs.

3- Yes, surely, thank you for your feedback.

4- The conclusion of the research paper was summarized according to the honored reviewer's instructions.

5- The uncommon wording in the research paper, as noted by the honored reviewer in his comments, has been corrected.

6- The definition [2.1] is as stated in the main reference.

7- Researchers can access the complete proof of the Lemma 2.8 by referring to its reference [4].

8- Ok, Thank you for all the feedback.

9- Yes, it is required result.

Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions  

 10.5256/f1000research.189971.r440893 Reviewer response for version 1 Shah Rahim 1 Referee https://orcid.org/0009-0001-9044-5470 Kohsar University Murree, Murree, Pakistan

Competing interests: No competing interests were disclosed.

 31 12 2025 Copyright: © 2025 Shah R 2025 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

The paper is well written and technically sound. To further improve its quality, the following minor revisions are recommended. After addressing these comments, the paper may be considered for acceptance for indexing.

Rewrite the Abstract (Background) in a standard and structured form.

Carefully review the manuscript to ensure grammatical correctness throughout.

To strengthen the Introduction section, the authors should include additional relevant references, such as:

Li, et al., 2017 - Ref 1

Shah, et al., 2016 - Ref 2

George, et al., 2022 - Ref 3

Is the work clearly and accurately presented and does it cite the current literature?

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

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Analysis, Fixed Point Theory, Applied Mathematics

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.

References : Fixed point theorems in dislocated quasi-metric spaces. The Journal of Nonlinear Sciences and Applications .2017;10(09) : 10.22436/jnsa.010.09.12 4695-4703 10.22436/jnsa.010.09.12 : Some Fixed Point Theorems of Integral Type Contraction in Cone b-metric Spaces. Turkish Journal of Analysis and Number Theory .2016;3(6) : 10.12691/tjant-3-6-5 165-169 10.12691/tjant-3-6-5 : Some Fixed Points Results in-Metric and Quasi-Metric Spaces. Journal of Function Spaces .2022;2022: 10.1155/2022/1803348 1-6 10.1155/2022/1803348 AL-Jumaili Alaa Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Alanbar, Iraq

Competing interests: Competing interests: The authors declare no competing interests.

 12 1 2026

Dear Editor...

Best greetings.....

I hope this message finds you well.

Subject \ comments responses about the manuscript-f1000res172256

  (New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172256)

First: We extend our sincere thanks to the esteemed evaluator for what he said regarding the paper being good and its results being original, as well as the originality of the examples and applications presented in it.                                                                

The following is the response to the comments kindly provided by the esteemed resident, and they have been carefully considered due to their importance.

1-Answer : We extend our sincere thanks to the esteemed evaluator for what he said regarding the paper being well written and technically sounds.

2- Answer : The abstract (background) was rewritten in a standardized and organized manner according to the instructions of the esteemed evaluator.

3-Answer : The manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts.

4- Answer : All the references (Ref-1, Ref-2 and Ref-3) which suggested by the honored reviewer have been added due to their importance in raising the scientific level and rigor of the manuscript.

We assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions.