Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces

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Abbas" }, { "@type": "Person", "name": "Ali A. Shihab" }, { "@type": "Person", "name": "Alaa M.F. Al-Jumaili" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": " 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 such as D d ∗ -Symmetric spaces and weakly compatible maps are reviewed to establish the framework for our main results. The major objective of the present study is to investigate and verify the uniqueness of some common fixed point theorems for three pairs of self-maps under the influence of other enhanced categories of extended contractive conditions in the context of D d ∗ -Symmetric spaces. Our first main outcomes were established by applying the concepts of weak compatibility and common limit in the range property, whereas we obtained our second major results by utilizing the notion of occasionally weakly compatible mapping. Additionally, various common fixed point outcomes for the two pairs of self-maps were determined. Conclusion This manuscript explores novel outcomes regarding the uniqueness of various common fixed point theorems for three pairs of self-maps under the influence of other enhanced types of extended contractive conditions in the context of D d ∗ -Symmetric spaces. We anticipate that the discoveries in this manuscript will aid scientists in enhancing the authors on popularized extended symmetric-spaces to elevate a universal framework for their practical implementations in each advanced branches of science. 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F1000Research 2026, 14 :1363 ( https://doi.org/10.12688/f1000research.172242.2 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Revised Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] Abduljabar K. Abbas https://orcid.org/0000-0001-9248-2411 1 , Ali A. Shihab 2 , Alaa M.F. Al-Jumaili https://orcid.org/0000-0002-9363-2562 3 Abduljabar K. Abbas https://orcid.org/0000-0001-9248-2411 1 , Ali A. Shihab 2 , Alaa M.F. Al-Jumaili https://orcid.org/0000-0002-9363-2562 3 PUBLISHED 07 Feb 2026 Author details Author details 1 Department of mathematics, College of Computer science and Mathematics, University of Tikrit, Tikrit, Salah Al-Deen, 34001, Iraq 2 Department of mathematics, College of Education for pure science, University of Tikrit, Tikrit, Salah Al-Deen, 34001, Iraq 3 Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Anbar, 31001, Iraq Abduljabar K. Abbas Roles: Methodology, Writing – Original Draft Preparation, Writing – Review & Editing Ali A. Shihab Roles: Investigation, Software, Supervision, Validation Alaa M.F. Al-Jumaili Roles: Conceptualization, Data Curation, Project Administration OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract 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 such as D d ∗ -Symmetric spaces and weakly compatible maps are reviewed to establish the framework for our main results. The major objective of the present study is to investigate and verify the uniqueness of some common fixed point theorems for three pairs of self-maps under the influence of other enhanced categories of extended contractive conditions in the context of D d ∗ -Symmetric spaces. Our first main outcomes were established by applying the concepts of weak compatibility and common limit in the range property, whereas we obtained our second major results by utilizing the notion of occasionally weakly compatible mapping. Additionally, various common fixed point outcomes for the two pairs of self-maps were determined. Conclusion This manuscript explores novel outcomes regarding the uniqueness of various common fixed point theorems for three pairs of self-maps under the influence of other enhanced types of extended contractive conditions in the context of D d ∗ -Symmetric spaces. We anticipate that the discoveries in this manuscript will aid scientists in enhancing the authors on popularized extended symmetric-spaces to elevate a universal framework for their practical implementations in each advanced branches of science. READ ALL READ LESS Keywords D*-metric spaces; D_d*-symmetric spaces; common fixed points; (CLR) property; weakly compatibility; (O.W.C) maps;(〖CLR〗_((FF,GJ))) property; uniqueness of common fixed points. Corresponding Author(s) Alaa M.F. Al-Jumaili ( [email protected] ) Close Corresponding author: Alaa M.F. Al-Jumaili Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 K. Abbas A 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: K. Abbas A, A. Shihab A and Al-Jumaili AMF. Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.12688/f1000research.172242.2 ) First published: 05 Dec 2025, 14 :1363 ( https://doi.org/10.12688/f1000research.172242.1 ) Latest published: 07 Feb 2026, 14 :1363 ( https://doi.org/10.12688/f1000research.172242.2 ) Revised 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- Some lengthy definitions have been simplified to improve their mathematical precision in the final version. 3- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 4- We would like to assure that all of labeling of inequalities (1), (2) and (3) are referenced correctly throughout the text in the final version. 5- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the in the final version. 6- All the errors pointed out according to honored evaluators instructions have been corrected by implementing all their valuable suggestions. 7- The manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts in the final version. 8-The term "gratify" and the symbol (&) will be replaced in all chapters of the paper in the final version. 9- All names of journals in the source list have been checked and verified as directed by the respected reviewers. 10- 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. 11- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions. 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- Some lengthy definitions have been simplified to improve their mathematical precision in the final version. 3- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 4- We would like to assure that all of labeling of inequalities (1), (2) and (3) are referenced correctly throughout the text in the final version. 5- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the in the final version. 6- All the errors pointed out according to honored evaluators instructions have been corrected by implementing all their valuable suggestions. 7- The manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts in the final version. 8-The term "gratify" and the symbol (&) will be replaced in all chapters of the paper in the final version. 9- All names of journals in the source list have been checked and verified as directed by the respected reviewers. 10- 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. 11- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions. See the authors' detailed response to the review by Khairul Habib Alam See the authors' detailed response to the review by Yousif Yaqoub Yousif See the authors' detailed response to the review by Choonkil Park READ REVIEWER RESPONSES 1. Introduction The common fixed point theory under influence various classes of extended contractive conditions has been developed over the decades to include various extensions and productive implementations in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems, making it a cornerstone of mathematical analysis and topological spaces. In 1976, Jungck 1 extended the celebrated Banach contraction principle by exploiting the idea of commuting maps and established a common fixed point theory. Subsequently, in 1982, Sessa 2 started the tradition of modifying commutativity in fixed point theorems by offering the idea of weakly commuting maps. As an extension of commuting maps, the idea of compatible maps was presented by Jungck in 1986, 3 which has been frequently applied to verify the existence of common fixed point theorems. In 2002, Aamri and El Moutawakil 4 introduced (E-A) property for pairs of self-maps, which is a true extension of non-compatible maps under contractive concisions. Consequently, Liu et al. 5 introduced the concept of the common (E-A) property, which contains (E-A) property presented. 4 In addition, Jungck and Rhoades 6 defined the idea of occasionally weakly compatible maps, which is more general among commutativity ideas, and obtained various common fixed point theorems. Numerous researchers have presented different extensions of the concept of metric spaces. In particular, Shaban et al. in 2007 7 defined the context of D ∗ -metric spaces. Cho et al. 8 verified a number of common fixed point theorems for weakly compatible maps and presented various counter examples. Chandra and Bhatt 9 confirmed the fixed point theory for extended contraction under restrictive conditions. On the other hand, W. Sintunavarat, 10 presented the idea of common limit in the range ( C L R ξ ) property for pair of self-maps in metric spacers. Later, in 2013, Karapinar et al. 11 expanded the idea ( C L R ξ ) for two couples of self-maps in symmetric spaces. In addition, Eke 12 proved various common fixed point theorems for contraction maps in uniform space. The fixed point theory has expanded rapidly in extended metric spaces talented with partial ordering. Jumaili 13 applied D ∗ -metric space and offered various coincidence fixed point theorems for maps satisfying contractive conditions in partially ordered complete D ∗ -metric spaces. Recently (2019), Al-Jumaili et al. 14 extended D ∗ -metric-sp by modifying ℝ via an ordered Banach space. Additionally, in 2020, Latif and Abed 15 studied fixed point of set-valued contractions in ordered G-metric spaces. In addition, Nagaraju 16 defined the idea of weakly contractive maps and proved several common fixed point theorems for three pairs of self-maps in G-metric spaces. As promised studies, we could generalize our outcomes to other spaces, such as. 17 – 23 Motivated by above facts, N. A. Majid et al. in 2023 24 verified various original fixed point outcomes for monotone multi-valued maps in partially ordered D ∗ -metric spaces, and investigated various existence and uniqueness of coupled fixed point outcomes of maps satisfying contractive conditions, additionally 25 they investigated and proved various outcomes of common and coincidence of fixed points in S -metric spaces. Subsequently, in 2024, Abed and Al-Jumaili 26 defined a novel type of extended metric space, namely D d ∗ -Symmetric space, and established some common fixed point results for maps satisfying extended contractive conditions in D d ∗ -Symmetric spaces. Recent developments in 2025 have further expanded fixed point theory into complex generalized spaces with robust applications. For instance, new results in m v b -metric spaces have addressed multivalued integral contractions “ https://doi.org/10.15672/hujms.1471688 ” and Fredholm integral inclusions “ https://doi.org/10.3934/math.2025926 ”. Furthermore, significant advances have been made in the application of fuzzy metric spaces to nonlinear differential inclusions “ https://doi.org/10.1177/18758967251366331 ” and proximal contractions to optimization problems “ https://doi.org/10.1186/s13663-025-00799-0 ”. These works highlight the continued relevance of extending contractive conditions in generalized spaces. The major goal of the present manuscript is to discuss and verify various common fixed-point theorems for several categories of self-maps under influence of other extended weakly contractive conditions in the context of D d ∗ -Symmetric spaces. Additionally, employing the notions of weak compatibility and common limit in the range property, our first major outcomes have been verified, while other major results utilizing the idea of occasionally weakly compatible maps have been obtained. Lastly, our major outcomes, which are related to these categories of common fixed-point theorems for numerous kinds of single-valued maps, improve and extend various recognized analogous outcomes in the literature. 2. Materials and methods This section presents the various definitions and motivations that are needed in the sequel, which will help us in the outcomes that follow and play a major role in verifying our major outcomes. Definition 2.1: 26 A; D d ∗ -Symmetric on X ≠ ∅ is a map D d ∗ : X × X × X → [ 0 , ∞ ) (s. t) ∀ x ∗ , y ∗ , z ∗ ∈ X , the next axioms are satisfied: ( D d 1 ∗ ) D d ∗ ( x ∗ , y ∗ , z ∗ ) ≥ 0 , ∀ x ∗ , y ∗ , z ∗ ∈ X ; ( D d 2 ∗ ) D d ∗ ( x ∗ , y ∗ , z ∗ ) = 0 iff x ∗ = y ∗ = z ∗ ; ( D d 3 ∗ ) D d ∗ ( x ∗ , y ∗ , z ∗ ) = D d ∗ ( P { x ∗ , y ∗ , z ∗ } ) , ( symmetry ) ( s . t ) P is a permutation map . In this case, D d ∗ is called D d ∗ -Symmetric and ( X , D d ∗ ) is called symmetric space. Example 2.2: 26 Presume that X = [ 0 , 1 ] equipped with D ∗ -Symmetric map described through D d ∗ ( x ∗ , y ∗ , z ∗ ) = ( x ∗ − y ∗ ) 2 + ( y ∗ − z ∗ ) 2 + ( z ∗ − x ∗ ) 2 , ∀ x ∗ , y ∗ , z ∗ ∈ X . Here, ( D d ∗ , X ) are D d ∗ -Symmetric space. Definition 2.3: 26 Assume that ( X , D d ∗ ) is a D d ∗ -Symmetric space, therefore a sequence { x s ∗ } is called D d ∗ -converges to x ∗ ∈ X iff D d ∗ ( x s ∗ , x s ∗ , x ∗ ) = D d ∗ ( x ∗ , x ∗ , x s ∗ ) ⟶ 0 as s ⟶ ∞ . Remark 2.4: Equivalent to Wilson axiom’s 27 in D d ∗ -Symmetric space as ( W 1 ) Given { x s ∗ } , x ∗ , y ∗ ∈ X , D d ∗ ( x s ∗ , x ∗ , x ∗ ) ⟶ 0 with D d ∗ ( x s ∗ , y ∗ , y ∗ ) ⟶ 0 ⟹ x ∗ = y ∗ . ( W 2 ) Given { x s ∗ } & { y s ∗ } , such that x ∗ , y ∗ ∈ X , D d ∗ ( x s ∗ , x ∗ , x ∗ ) ⟶ 0 with D d ∗ ( x s ∗ , y s ∗ , y s ∗ ) ⟶ 0 ⟹ D d ∗ ( y s ∗ , x ∗ , x ∗ ) ⟶ 0 . ( W 3 ) Assume that ( X , D d ∗ ) are complete, D d ∗ -Symmetric space. For an arbitrary sequence { x s ∗ } in X , we have lim s , r → ∞ D d ∗ ( x s ∗ , x r ∗ , x r ∗ ) = 0 , iff lim s → ∞ D d ∗ ( x s ∗ , x s + 1 ∗ , x s + 1 ∗ ) = 0 . The next Lemma is analogue of Lemma 27 in ( X , D d ∗ ) : Lemma 2.5: Each D d ∗ -Symmetric ( X , D d ∗ ) , describes a symmetric d D d ∗ on X via: d D d ∗ ( x ∗ , y ∗ ) = D d ∗ ( x ∗ , y ∗ , y ∗ ) + D d ∗ ( y ∗ , x ∗ , x ∗ ) , ∀ x ∗ , y ∗ ∈ X , that is: d D d ∗ ( x ∗ , y ∗ ) = 2 D d ∗ ( x ∗ , y ∗ , y ∗ ) , ∀ x ∗ , y ∗ ∈ X . Definition 2.6: 28 Assume that F , G : ( X , D d ∗ ) ⟶ ( X , D d ∗ ) are single-valued self-maps If F ( x ∗ ) = G ( x ∗ ) = w ∗ for x ∗ ∈ X , thus x ∗ is said to be a coincidence point of F & G , and w ∗ is the point of coincidence of F and G . Definition 2.7: 29 Assume that F , G : ( X , D d ∗ ) → ( X , D d ∗ ) are single-valued self-maps. If F ( x ∗ ) = G ( x ∗ ) = x ∗ for x ∗ ∈ X , consequently x ∗ is called the common fixed point (C.F.P) of F and G . Definition 2.8: 26 A pair ( F , G ) of self-maps of D d ∗ -Symmetric ( X , D d ∗ ) are called weakly compatible maps (Concisely, W.C.M) if they are commute at their coincidence points. Definition 2.9: 6 Self-maps F and G of ( X , D d ∗ ) are called occasionally weakly compatible iff ∃ x ∈ X which is coincidence point of F and G which F and G are commute. Remark 2.10: It is clear that each pair of weakly compatible maps is (O.W.C) map, other than the opposite, and not generally correct. Definition 2.11: 30 μ : [ 0 , ∞ ) → [ 0 , ∞ ) is said to be an altering distance if μ is continuous and nondecreasing with μ ( t ) = 0 iff t = 0 . Definition 2.12: 10 Two self-maps η and ξ of ( X , d ) satisfy the property of the common limit in the range of ξ , indicated via ( C L R ξ ) , if ∃ { x s ∗ } in X (s. t) lim s → ∞ η x s ∗ = lim s → ∞ ξ x s ∗ = ξ p for p ∈ X . Definition 2.13: 11 The pairs ( ξ , η ) and ( F , G ) of self-maps in Symmetric space ( X , d ) are called gratify property of common limit range related to the maps η and G , indicated via ( C L R ( η , G ) ) , if ∃ { x s ∗ } and { y s ∗ } (s. t) lim s → ∞ ξ x s ∗ = lim s → ∞ η x s ∗ = lim s → ∞ F y s ∗ = lim s → ∞ G y s ∗ = t with t = η p = Gq for some t , p , q ∈ X . Remark 2.14: 10 If ξ = F and η = G , in that case Definition-2.12 reduces to ( C L R η ) property. 3. Main results of new extended categories of common fixed point theorems In this section, we establish the uniqueness of various common fixed-point theorems for three pairs of self-maps satisfying ( C L R ( F F , GJ ) ) properties under influence of other enhanced categories of extended contractive maps in D d ∗ -Symmetric space. Additionally, various common fixed-point outcomes for two pairs of self-maps were identified. Remark 3.1: Throughout this manuscript, assume that the next properties are hold: (i) Let ℂ = { ψ / ψ : [ 0 , ∞ ) → [ 0 , ∞ ) is lower semi continuous and nondecreasing map where ψ ( t ) = 0 ⟺ t = 0 } (ii) Assume that; S , ℚ , F , F , G and J are three pairs of self-maps of a D d ∗ -Symmetric ( X , D d ∗ ) , where (1) μ ( D d ∗ ( S x ∗ , ℚ y ∗ , ℚ z ∗ ) ) ≤ μ ( L ( x ∗ , y ∗ , z ∗ ) ) − ψ ( L ( x ∗ , y ∗ , z ∗ ) ) , ∀ x ∗ , y ∗ , z ∗ ∈ X Wherever, μ is altering distance map, ψ ∈ ℂ with L ( x ∗ , y ∗ , z ∗ ) = max { D d ∗ ( F F x ∗ , GJ y ∗ , GJ z ∗ ) , D d ∗ ( F F x ∗ , F F x ∗ , S x ∗ ) , D d ∗ ( GJ y ∗ , ℚ y ∗ , ℚ z ∗ ) , 1 3 { D d ∗ ( F F x ∗ , ℚ y ∗ , ℚ z ∗ ) + D d ∗ ( GJ y ∗ , GJ y ∗ , S x ∗ ) } } Theorem 3.2: Let S , ℚ , F , F , G and J are three pairs of self-maps of a D d ∗ -Symmetric space ( X , D d ∗ ) satisfy inequality (1) and the following conditions: (i) If ( S , F F ) and ( ℚ , GJ ) gratify C L R ( F F , GJ ) property. (ii) If ( S , F F ) and ( ℚ , GJ ) are weakly compatible maps. In this case, the maps S , ℚ , F F and GJ have a unique (C. F. P) in X . Additionally, S , ℚ , F , F , G and J contain a unique(C. F. P) provided that the pairs of maps ( F , F ) , ( S , F ) , ( S , F ) , ( G , J ) , ( ℚ , G ) and ( ℚ , J ) are commuting. Proof: Because ( S , F F ) and ( ℚ , GJ ) satisfy ( C L R ( F F , GJ ) ) property, we can discover two sequences { x s ∗ } and { y s ∗ } in X (s. t), lim s → ∞ S x s ∗ = lim s → ∞ F F x s ∗ = lim s → ∞ ℚ y s ∗ = lim s → ∞ GJ y s ∗ = t with t = F Fp = GJq for some t , p , q ∈ X . Firstly verify F Fp = S p . From inequality (1) and x ∗ = p , y ∗ = z ∗ = y s ∗ , we get (2) μ ( D d ∗ ( S p , S p , ℚ y ∗ ) ) ≤ μ ( L ( p , p , y s ∗ ) ) − ψ ( L ( p , p , y s ∗ ) ) Such that L ( p , p , y s ∗ ) = max { D d ∗ ( F Fp , F Fp , GJ y s ∗ ) , D d ∗ ( F Fp , F Fp , S p ) , D d ∗ ( GJ y s ∗ , GJ y s ∗ , ℚ y s ∗ ) , 1 3 { D d ∗ ( F Fp , F Fp , ℚ y s ∗ ) + D d ∗ ( GJ y s ∗ , GJ y s ∗ , S p ) } } Consequently lim s → ∞ L ( p , p , y s ∗ ) = max { D d ∗ ( t , t , t ) , D ∗ ( t , t , S p ) , D d ∗ ( t , t , t ) , 1 3 { D d ∗ ( t , t , t ) + D d ∗ ( t , t , S p ) } } = max { 0 , D d ∗ ( t , t , S p ) , 0 , 1 3 { D d ∗ ( t , t , S p ) } } = D d ∗ ( S p , S p , t ) , by using ( D d 3 ∗ ) of Definition 2.1 . Let s → ∞ in inequality (2) , μ ( D d ∗ ( S p , S p , t ) ) ≤ μ ( D d ∗ ( S p , S p , t ) ) − ψ ( D d ∗ ( S p , S p , t ) ) which implies ψ ( D d ∗ ( S p , S p , t ) ) = 0 , as a result D d ∗ ( S p , S p , t ) = 0 , that is, S p = t = F Fp , illustrating p is the coincidence point of S and F F . Because ( S , F F ) is weakly compatible, we have S ( F F ) p = ( F F ) S p , therefore S t = F Ft . Secondly verify: ℚ q = GJq . From inequality (1) (s. t), x ∗ = x s ∗ , y ∗ = z ∗ = q , get that (3) μ ( D d ∗ ( S x s ∗ , S x s ∗ , ℚ q ) ) ≤ μ ( L ( x s ∗ , x s ∗ , q ) ) − ψ ( L ( x s ∗ , x s ∗ , q ) ) Where L ( x s ∗ , x s ∗ , q ) = max { D d ∗ ( F F x s ∗ , F F x s ∗ , GJq ) , D d ∗ ( F F x s ∗ , F F x s ∗ , S x s ∗ ) , D d ∗ ( GJq , GJq , ℚ q ) , 1 3 { D d ∗ ( F F x s ∗ , F F x s ∗ , ℚ q ) + D d ∗ ( GJq , GJq , S x s ∗ ) } } Therefore, lim s → ∞ L ( x s ∗ , x s ∗ , q ) = max { D d ∗ ( t , t , t ) , D ∗ ( t , t , t ) , D d ∗ ( t , t , ℚ q ) , 1 3 { D d ∗ ( t , t , ℚ q ) + D d ∗ ( t , t , t ) } } = max { 0 , 0 , D ∗ ( t , t , ℚ q ) , 1 3 { D d ∗ ( t , t , ℚ q ) + 0 } } = D d ∗ ( t , t , ℚ q ) . Selecting limit as s → ∞ in inequality (3) , μ ( D d ∗ ( t , t , ℚ q ) ) ≤ μ ( D d ∗ ( t , t , ℚ q ) ) − ψ ( D d ∗ ( t , t , ℚ q ) ) This implies that ( D d ∗ ( t , t , ℚ q ) ) = 0 , as a result D d ∗ ( t , t , ℚ q ) = 0 ,that is, ℚ q = t = GJq , explaining q is the coincidence point of ℚ and GJ . As ( ℚ , GJ ) is (W.C.M), we obtain ℚ ( GJ ) q = ( GJ ) ℚ q , and consequently ℚ t = GJt . Third verify: S t = F Ft = t . From inequality (1) (s. t) x ∗ = t , y ∗ = z ∗ = q , obtain μ ( D d ∗ ( S t , S t , ℚ q ) ) ≤ μ ( L ( t , t , q ) ) − ψ ( L ( t , t , q ) ) Or μ ( D d ∗ ( S t , S t , t ) ) ≤ μ ( L ( t , t , q ) ) − ψ ( L ( t , t , q ) ) Such that L ( t , t , q ) = max { D d ∗ ( F Ft , F Ft , GJq ) , D d ∗ ( F Ft , F Ft , S t ) , D d ∗ ( GJq , GJq , ℚ q ) , 1 3 { D d ∗ ( F Ft , F Ft , ℚ q ) + D d ∗ ( GJq , GJq , S t ) } } Consequently max { D d ∗ ( S t , S t , t ) , D d ∗ ( S t , S t , S t ) , D d ∗ ( t , t , t ) , 1 3 { D d ∗ ( S t , S t , t ) + D d ∗ ( t , t , S t ) } } = max { D d ∗ ( S t , S t , t ) , 0 , 0 , 2 3 { D d ∗ ( S t , S t , t ) } } , by using ( D d 3 ∗ ) of Definition 2.1 . = D d ∗ ( S t , S t , t ) . Thus, μ ( D d ∗ ( S t , S t , t ) ) ≤ μ ( D d ∗ ( S t , S t , t ) ) − ψ ( D d ∗ ( S t , S t , t ) ) ⟹ ψ ( D d ∗ ( S t , S t , t ) ) = 0 , as a result D ∗ ( S t , S t , t ) = 0 , that is, S t = t . Consequently, S t = F Ft = t . Finally verify: ℚ t = GJt = t . From inequality- (1) (s. t) x ∗ = p , y ∗ = z ∗ = t , we obtain μ ( D d ∗ ( S p , S p , ℚ t ) ) ≤ μ ( L ( p , p , t ) ) − ψ ( L ( p , p , t ) ) Orμ ( D d ∗ ( t , t , ℚ t ) ) ≤ μ ( L ( p , p , t ) ) − ψ ( L ( p , p , t ) ) Such that L ( p , p , t ) = max { D d ∗ ( F Fp , F Fp , GJt ) , D d ∗ ( F Fp , F Fp , S p ) , D d ∗ ( GJt , GJt , ℚ t ) , 1 3 { D d ∗ ( F Fp , F Fp , ℚ t ) + D d ∗ ( GJt , GJt , S p ) } } = max { D d ∗ ( t , t , ℚ t ) , D d ∗ ( t , t , t ) , D ∗ ( ℚ t , ℚ t , ℚ t ) , 1 3 { D d ∗ ( t , t , ℚ t ) + D d ∗ ( ℚ t , ℚ t , t ) } } = max { D d ∗ ( t , t , ℚ t ) , 0 , 0 , 1 3 { D d ∗ ( t , t , ℚ t ) + D d ∗ ( ℚ t , ℚ t , t ) } } = max { D d ∗ ( t , t , ℚ t ) , 2 3 { D d ∗ ( t , t , ℚ t ) } } , by using ( D d 3 ∗ ) of Definition 2.1 . = D d ∗ ( t , t , ℚ t ) . Consequently, μ ( D d ∗ ( t , t , ℚ t ) ) ≤ μ ( D d ∗ ( t , t , ℚ t ) ) − ψ ( D d ∗ ( t , t , ℚ t ) ) ⟹ ( D d ∗ ( t , t , ℚ t ) ) = 0 . Therefore D d ∗ ( t , t , ℚ t ) = 0 , that is, ℚ t = t . Consequently, ℚ t = GJt = t , thus S t = F Ft = ℚ t = GJt = t , demonstrating t is (C. F. P) of S , ℚ , F F and JG . Uniqueness: Assume that ( n ≠ t ) is different (C. F. P) for S , ℚ , F F and JG . Consequently, we have S n = F Fn = ℚ n = GJn = n . Now, from inequality (1) with x ∗ = t , y ∗ = z ∗ = n , obtain μ ( D d ∗ ( S t , S t , ℚ n ) ) ≤ μ ( L ( t , t , n ) ) − ψ ( L ( t , t , n ) ) Or μ ( D d ∗ ( t , t , n ) ) ≤ μ ( L ( t , t , n ) ) − ψ ( L ( t , t , n ) ) , Such that L ( t , t , n ) = max { D d ∗ ( F Ft , F Ft , GJn ) , D d ∗ ( F Ft , F Ft , S t ) , D d ∗ ( GJn , GJn , ℚ n ) , 1 3 { D d ∗ ( F Ft , F Ft , ℚ n ) + D d ∗ ( GJn , GJn , S t ) } } = max { D d ∗ ( t , t , n ) , D d ∗ ( t , t , t ) , D d ∗ ( n , n , n ) , 1 3 { D d ∗ ( t , t , n ) + D d ∗ ( n , n , t ) } } = max { D d ∗ ( t , t , n ) , 0 , 0 , 1 3 { D d ∗ ( t , t , n ) + D d ∗ ( n , n , t ) } } = max { D d ∗ ( t , t , n ) , 2 3 { D d ∗ ( t , t , n ) } } = D d ∗ ( t , t , n ) , by using ( D d 3 ∗ ) of Definition 2.1 . Therefore, μ ( D d ∗ ( t , t , n ) ) ≤ μ ( D d ∗ ( t , t , n ) ) − ψ ( D d ∗ ( t , t , n ) ) ⟹ ψ ( D d ∗ ( t , t , n ) ) = 0 . Consequently, D d ∗ ( t , t , n ) = 0 , that is, t = n . Thus, S , F F , ℚ , JG have unique common fixed point in X . Next, we shall verify S , ℚ , F , F , G and J have unique common fixed point. Because, ( F , F ) , ( S , F ) , ( S , F ) are commuting, so F t = F ( F Ft ) = ( F F ) F t with F t = F ( S t ) = S ( F t ) . In addition, Ft = F ( F Ft ) = ( F F ) Ft with Ft = F ( S t ) = S ( Ft ) . This illustrates F t and Ft are (C. F. Ps) of F F and S . Therefore, via the uniqueness of common fixed point, we obtain F t = Ft = t . Likewise, because ( G , J ) , ( ℚ , G ) and ( ℚ , J ) are commuting, obtain Gt = G ( ℚ t ) = ( ℚ G ) t with Gt = G ( GJt ) = GJ ( Gt ) . Moreover, Jt = J ( ℚ t ) = ℚ ( Jt ) and Jt = J ( GJt ) = GJ ( Jt ) . This explains Gt and Jt are common fixed point of GJ and ℚ . Consequently, via the uniqueness of common fixed point, obtain Gt = Jt = t . As a result, S t = ℚ t = F t = Ft = Gt = Jt = t , verifying t is unique common fixed point to S , ℚ , F , F , G and J . Corollary 3.3: If S , ℚ , F and J are two pairs of self-maps of D d ∗ -Symmetric ( X , D d ∗ ) satisfying the following conditions: (i) If μ ( D d ∗ ( S x ∗ , ℚ y ∗ , ℚ z ∗ ) ) ≤ μ ( L ( x ∗ , y ∗ , z ∗ ) ) − ψ ( L ( x ∗ , y ∗ , z ∗ ) ) ∀ x ∗ , y ∗ , z ∗ ∈ X . (s. t), μ alters the distance map with ψ : ℝ + → ℝ + , is a lower semi-continuous and nondecreasing map (s. t) ψ ( t ) = 0 iff t = 0 with L ( x ∗ , y ∗ , z ∗ ) = max { D d ∗ ( F x ∗ , J y ∗ , J z ∗ ) , D d ∗ ( F x ∗ , F x ∗ , S x ∗ ) , D d ∗ ( J y ∗ , ℚ y ∗ , ℚ z ∗ ) , 1 3 { D d ∗ ( F x ∗ , ℚ y ∗ , ℚ z ∗ ) + D d ∗ ( J y ∗ , J y ∗ , S x ∗ ) } } (ii) If ( S , F ) and ( ℚ , J ) satisfy ( C L R ( F , J ) ) property. (iii) If ( S , F ) and ( ℚ , J ) are (W. C. M). In this case, S , ℚ , F and J contained unique (C. F. P) in X . Proof: Its follows immediately from Theorem-3.2 , via putting G = F = I X (Identity map). Lemma 3.4: Assume that; X ≠ ∅ , with ξ and η are (O. W. C) maps. If ξ and η include a unique point of coincidence w ∗ = ξ t = η t for t ∈ X , then w ∗ is unique (C. F. P) of ξ and η . Theorem 3.5: If S , ℚ , F , F , G and J are three pairs of self-maps in D d ∗ -symmetric ( X , D d ∗ ) satisfies the following conditions: (i) If μ ( D d ∗ ( S x ∗ , ℚ y ∗ , ℚ z ∗ ) ) ≤ μ ( M ( x ∗ , y ∗ , z ∗ ) ) − ψ ( M ( x ∗ , y ∗ , z ∗ ) ) , ∀ x ∗ , y ∗ , z ∗ ∈ X , (s. t), μ μ alters the distance map with ψ : ℝ + → ℝ + , is a lower semi-continuous and nondecreasing map (s. t) ψ ( t ) = 0 iff t = 0 with M ( x ∗ , y ∗ , z ∗ ) = max { D d ∗ ( F F x ∗ , GJ y ∗ , GJ z ∗ ) , D d ∗ ( F F x ∗ , F F x ∗ , S x ∗ ) , D d ∗ ( GJ y ∗ , ℚ y ∗ , ℚ z ∗ ) , D d ∗ ( GJ y ∗ , GJ y ∗ , S x ∗ ) } (ii) If ( S , F F ) and ( ℚ , GJ ) are (O. W. C) maps. In this case, the maps S , ℚ , F F and GJ have a unique (C. F. P) in X . Additional S , ℚ , F , F , G and J include unique (C.F.P), provided the pairs of maps ( F , F ) , ( S , F ) , ( S , F ) , ( G , J ) , ( ℚ , G ) and ( ℚ , J ) are commuting. Proof: Assuming that ( S , F F ) and ( ℚ , GJ ) are (O. W. C), we can discover p and q in X (s. t) S p = F Fp = p 1 and S ( F F ) p = ( F F ) S p with ℚ q = GJq = q 1 and ℚ ( GJ ) q = ( GJ ) ℚ q . Firstly, verify: S p = ℚ q . From condition (i), (s. t) x ∗ = p , y ∗ = z ∗ = q , we get (4) μ ( D d ∗ ( S p , S p , ℚ q ) ) ≤ μ ( M ( p , p , q ) ) − ψ ( M ( p , p , q ) ) Such that M ( p , p , q ) = max { D d ∗ ( F Fp , F Fp , GJq ) , D d ∗ ( F Fp , F Fp , S p ) , D d ∗ ( GJq , GJq , ℚ q ) , D d ∗ ( GJq , GJq , S p ) } = max { D d ∗ ( S p , S p , ℚ q ) , D d ∗ ( S p , S p , S p ) , D d ∗ ( ℚ q , ℚ q , ℚ q ) , D d ∗ ( ℚ q , ℚ q , S p ) } = max { D d ∗ ( S p , S p , ℚ q ) , 0 , 0 , D d ∗ ( ℚ q , ℚ q , S p ) } = D d ∗ ( S p , S p , ℚ q ) , by using ( D d 3 ∗ ) of Definition 2.1 . Consequently, μ ( D d ∗ ( S p , S p , ℚ q ) ) ≤ μ ( D d ∗ ( S p , S p , ℚ q ) ) − ψ ( D d ∗ ( S p , S p , ℚ q ) ) which implies that ψ ( D d ∗ ( S p , S p , ℚ q ) ) = 0 , as a result D d ∗ ( S p , S p , ℚ q ) = 0 , that is, S p = ℚ q . Therefore, S p = F Fp = ℚ q = GJq . Suppose that p 1 is another point where S p 1 = F F p 1 . In this case, obtain from condition (i), that S p 1 = F F p 1 = ℚ q = GJq . Thus , S p = S p 1 , that is, p = p 1 , demonstrating S and F F include a unique point of coincidence. Consequently, via Lemma-3.4 , get S and F F include unique (C.F.P), namely, t . Likewise, it can be verified ℚ and GJ include unique (C.F.P), say t 1 . Secondly verify, t = t 1 . From condition (i), (s. t) x ∗ = t , y ∗ = z ∗ = t 1 , get (5) μ ( D d ∗ ( S t , S t , ℚ t 1 ) ) ≤ μ ( M ( t , t , t 1 ) ) − ψ ( M ( t , t , t 1 ) ) Or μ ( D d ∗ ( t , t , t 1 ) ) ≤ μ ( M ( t , t , t 1 ) ) − ψ ( M ( t , t , t 1 ) ) Where M ( t , t , t 1 ) = max { D d ∗ ( F Ft , F Ft , GJ t 1 ) , D d ∗ ( F Ft , F Ft , S t ) , D d ∗ ( GJ t 1 , GJ t 1 , ℚ t 1 ) , D d ∗ ( GJ t 1 , GJ t 1 , S t ) } = max { D d ∗ ( t , t , t 1 ) , D d ∗ ( t , t , t ) , D d ∗ ( t 1 , t 1 , t 1 ) , D d ∗ ( t 1 , t 1 , t ) } = max { D d ∗ ( t , t 1 , t 1 ) , 0 , 0 , D d ∗ ( t 1 , t , t ) } = D d ∗ ( t , t , t 1 ) , by using ( D d 3 ∗ ) of Definition 2.1 . Therefore, μ ( D d ∗ ( t , t , t 1 ) ) ≤ μ ( D d ∗ ( t , t , t 1 ) ) − ψ ( D d ∗ ( t , t , t 1 ) ) ⟹ ψ ( D d ∗ ( t , t , t 1 ) ) = 0 , as a result D d ∗ ( t , t , t 1 ) = 0 , that is, t = t 1 . Consequently, S , ℚ , F F and GJ are unique (C. F. P). The rest of the evidence is similar to of Theorem-3.2 , for this reason, obtain S t = ℚ t = F t = Ft = Gt = Jt = t , verifying t is unique(C.F.P) to S , ℚ , F , F , G and J . Corollary 3.6: If S , ℚ , F and J are two pairs of self-maps of a D d ∗ -symmetric ( X , D d ∗ ) satisfying the next conditions: (i) If μ ( D d ∗ ( S x ∗ , ℚ y ∗ , ℚ z ∗ ) ) ≤ μ ( M ( x ∗ , y ∗ , z ∗ ) ) − ψ ( M ( x ∗ , y ∗ , z ∗ ) ) , ∀ x ∗ , y ∗ , z ∗ ∈ X , (s. t), μ μ altering distance map with ψ : ℝ + ⟶ ℝ + , is lower semi-continuous and nondecreasing map (s. t) ψ ( t ) = 0 , iff t = 0 , with M ( x ∗ , y ∗ , z ∗ ) = max { D d ∗ ( F x ∗ , J y ∗ , J z ∗ ) , D d ∗ ( F x ∗ , F x ∗ , S x ∗ ) , D d ∗ ( J y ∗ , ℚ y ∗ , ℚ z ∗ ) , D d ∗ ( J y ∗ , J y ∗ , S x ∗ ) } (ii) If ( S , F ) and ( ℚ , J ) are (O. W. C) maps. In this case, maps S , ℚ , F and J have a unique common fixed point in X . Proof: This follows directly from Theorem-3.5 , by putting F = G = I X (Identity map). 4. Conclusion The study of common fixed point theory for various categories of single-valued mappings under the influence of generalized contractive conditions in extended symmetric-spaces has witnessed a spectacular development of interest in the past few decades. Apparently, it is crucial in numerous disciplines in pure and applied mathematics, and it has various originative implementations in other branches of science. One of the most significant advances in pure and applied mathematics is the discovery of solutions for linear and nonlinear systems as well as fractal graphics, optimization theory, approximation theory, discrete dynamics, and numerous other areas. Therefore, some novel common fixed point theorems for three pairs of self-maps under influence of new extended contractive conditions in the context of D d ∗ -Symmetric spaces were investigated and verified. In addition, utilizing the concepts of weak compatibility and a common limit in the range property, our first outcomes are established, whereas (O.W.C) maps are applied to obtain our second outcomes. Our results generalize and improve various recent outcomes of (C.F.P) theorems under extended weakly contractive maps in the literature. We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Authors’ declaration All research studies on humans (individuals, samples) No human (individuals and samples) studies are present in the manuscript. 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 1. Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976; 83 : 261–263. Publisher Full Text 2. Sessa S: on a weak commutativity condition of mappings in fixed point considerations. Publ. Inst. Math. 1982; 32 : 149–153. 3. Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986; 9 : 771–779. Publisher Full Text 4. Aamri M, El Moutawakil D: some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002; 270 : 181–188. Publisher Full Text 5. Liu Y, Wu J, Li Z: common fixed points of single-valued and multi-valued maps. Int. J. Math. Math. Sci. 2005; 2005 : 3045–3055. Publisher Full Text 6. Jungck G, Rhoades BE: fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory. 2006; 7 (2): 287–296. 7. Shaban S, Nabi S, Haiyun Z: A common Fixed Point Theorem in D * -Metric Spaces. Hindawi Publishing Corporation, Fixed Point Theory and Applications. 2007; p. 13. Article ID 27906. 8. Cho SH, Lee GY, Bae JS: on coincidence and fixed point theorems in symmetric space. Fixed Point Theory Appl. 2008; 2008 ; 9. Art.ID 562130. Publisher Full Text 9. Chandra H, Bhatt A: Some fixed point theorems for set valued maps in symmetric spaces. Int. J. Math. Anal. 2009; 3 : 839–846. 10. Sintunavarat W, Kumam P: common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011; 2011 . Article ID 637958. Publisher Full Text 11. Karapinar E, Patel DK, Imdad M, et al. : Some non unique fixed point theorems in symmetric spaces through CLR(S,T) property. Int. J. Math. Math. Sci. 2013; 2013 ; 1–8. Article ID 753965. Publisher Full Text 12. Eke KS: common fixed point theorems for generalized contraction mappings on uniform spaces. Far East J. Appl. Math. 2016; 99 (11): 1753–1760. Publisher Full Text 13. Al Jumaili AMF: Some Coincidence and Fixed Point Results in Partially Ordered Complete Generalized D * - Metric Spaces. Eur. J. Pure Appl. Math. 2017; 10 (5): 1024–1035. 14. Al-Jumaili AM, Abed MM, Al-sharqi FG: on fixed point theorems and ∇-distance in complete partially ordered G-cone metric spaces. J. Anal. Appl. 2019; 17 (1): 1–20. 15. Latif SQ, Abed SS: Types of Fixed Points of Set-Valued Contraction Mappings for Comparable Elements. Iraqi J. Sci. 2020; 190–195. Special Issue. Publisher Full Text 16. Nagaraju V: Common Fixed Point Theorems for Six Self-Maps in G-Metric Spaces. Ann. Pure Appl. Math. 2020; 22 (1): 57–64. Publisher Full Text 17. Mohsen SD: some generalizations of fuzzy soft (k ∗ –Â)-quasi normal operators in fuzzy soft Hilbert spaces. J. Interdiscip. Math. 2023; 26 (6): 1133–1143. Publisher Full Text 18. Mohsen SD, Thiyab YH: some Characteristics of Completeness Property in Fuzzy Soft b-Metric Space. J. Appl. Sci. Eng. 2023; 27 (3): 2227–2232. 19. Hijab AA, Shaakir LK, Aljohani S, et al. : Fredholm integral equation in composed-cone metric spaces. Bound. Value Probl. 2024; 2024 (64): 1–12. Publisher Full Text 20. Hijab AA, Shaakir LK: New Generalization of Strong-Composed Metric Type Spaces with Special (ψ,∅)-Contraction. Adv. Fixed Point Theory. 2025; 15 (5): 1–20. 21. Majid NA, Al-Jumaili AMF, Ng ZC, et al. : New Results of Fixed-Point Theorems and Their Applications in Complete Complex D ∗ c -Metric Spaces. J. Funct. Spaces 2024; 2024 : 5532624. Publisher Full Text 22. 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). 23. Abed AH, Al-Jumaili AMF: On Weakly Compatible Maps and Some Common Fixed Point Theorems in D*-Metric Spaces. AIP Conference Proceedings, 2nd International Conference on Scientific Research and Innovation 2023, 2ICSRI 2023. 2025; 3169 (1). 24. Majid NA, Alaa MF, Al-Jumaili ZC, et al. : Lee, some applications of fixed point results for monotone multivalued and integral type contractive mappings. Fixed Point Theory Algorithms Sci. Eng. 2023; 2023 (1): 1–11. Publisher Full Text 25. Majid NA, Al-Jumaili A, Ng ZC, et al. : Enhanced Results in Common and Coincidence Fixed Point Theory with Applications to Simulation Mappings. Eur. J. Pure Appl. Math. 2024; 17 (3): 1877–1893. Publisher Full Text 26. 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. Publisher Full Text 27. Wilson WA: On Semi-Metric Spaces. Am. J. Math. 1931; 53 (2): 361–373. Publisher Full Text 28. Abbas M, Jungck G: common fixed point results for non-commuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 2008; 341 (1): 416–420. Publisher Full Text 29. Jungck G: common fixed points for non-continuous non-self maps on non-metric spaces. Far East J. Math. Sci. 1996; 4 (2): 199–215. 30. Khan MS, Swaleh M, Sessa S: fixed point theorems by altering distance between the points. Bull. Aust. Math. Soc. 1984; 30 : 1–9. Publisher Full Text Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 05 Dec 2025 ADD YOUR COMMENT Comment Author details Author details 1 Department of mathematics, College of Computer science and Mathematics, University of Tikrit, Tikrit, Salah Al-Deen, 34001, Iraq 2 Department of mathematics, College of Education for pure science, University of Tikrit, Tikrit, Salah Al-Deen, 34001, Iraq 3 Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, Anbar, 31001, Iraq Abduljabar K. Abbas Roles: Methodology, Writing – Original Draft Preparation, Writing – Review & Editing Ali A. Shihab Roles: Investigation, Software, Supervision, Validation Alaa M.F. Al-Jumaili Roles: Conceptualization, Data Curation, Project Administration Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (2) version 2 Revised Published: 07 Feb 2026, 14:1363 https://doi.org/10.12688/f1000research.172242.2 version 1 Published: 05 Dec 2025, 14:1363 https://doi.org/10.12688/f1000research.172242.1 Copyright © 2026 K. Abbas A 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. Download Export To Sciwheel Bibtex EndNote ProCite Ref. 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Key to Reviewer Statuses VIEW HIDE 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 Version 2 VERSION 2 PUBLISHED 07 Feb 2026 Revised Views 0 Cite How to cite this report: Yadav K. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456794 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456794 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 14 Mar 2026 Kapil Yadav , Lovely Professional University, Phagwara, Punjab, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.195954.r456794 Article Title: Some Extended Results of Common Fixed-Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces Article summary: The manuscript establishes common fixed-point results for pairs of contractive mappings satisfying weak compatibility and ... Continue reading READ ALL Article Title: Some Extended Results of Common Fixed-Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces Article summary: The manuscript establishes common fixed-point results for pairs of contractive mappings satisfying weak compatibility and CLR property using altering distance functions in the setting of Dd*symmetric spaces. In addition, the authors derive common fixed-point results for occasionally weakly compatible mappings in this framework. Several corollaries are also presented as consequences of the main theorem. The results generalize and improve various recent outcomes of (C.F.P) theorems under extended weakly contractive maps in the literature. The proofs of the theorems are well illustrated and are presented in a clear and understandable manner. In order to enhance the readability of the manuscript, the following suggestions are provided. Suggestions: The topic is relevant and falls within the scope of fixed-point theory, and the results may be of interest to researchers working in generalized metric structures and nonlinear analysis. Grammatical/Typo: Some suggestions related to grammar and typographical corrections are listed below. The authors are encouraged to carefully review the entire manuscript and implement similar corrections wherever necessary to improve the overall presentation. 1 In Definition 2.9 add the abbreviation for "occasionally weakly compatible" (O.W.C.) 2 In Definition 2.7 correct the font of "x" 3 In Definition 2.13 replace "are called gratify property of common limit range related" with "have common limit range property wrt" 4 In Remark 3.1 is the order of the pairs of three mappings have some significance? if not then just use "are self-mappings". Addition of Illustrative Examples: The authors are encouraged to include some illustrative examples to verify the fixed-point results established in the main section of the manuscript. Addition of a Remark or Section: The authors are encouraged to add a remark or a separate Section (or in the Conclusion section) highlighting the results that are extended by the main theorem of the manuscript. If possible, the authors should clearly indicate the earlier results that are generalized by their work, along with appropriate citations to the related literature. For example, a statement such as “Our result extends Theorem XX of [\cite].” may be included where applicable. 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? I cannot comment. A qualified statistician is required. 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: Fixed Point Theory of Contraction Mappings, Generalized Metric Structure, Stability of Functional Equations 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Yadav K. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456794 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456794 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Yousif YY. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456310 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456310 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 16 Feb 2026 Yousif Yaqoub Yousif , University of Baghdad, Baghdad, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.195954.r456310 The paper can now ... Continue reading READ ALL The paper can now be accepted for indexing. Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Yousif YY. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456310 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456310 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Alam KH. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456308 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456308 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 10 Feb 2026 Khairul Habib Alam , Mathematics, Indian Institute of Science Education and Research Berhampur (Ringgold ID: 486382), Brahmapur, Odisha, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.195954.r456308 I am satisfied with this version. Kindly cite the following papers for improving the literature. (refer to 1,2,3,4) References 1. Alam K, Yumnam R, Tomar A: Fixed point and its geometry and application for multivalued integral type contractions in $m_v^b$-metric spaces. ... Continue reading READ ALL I am satisfied with this version. Kindly cite the following papers for improving the literature. (refer to 1,2,3,4) References 1. Alam K, Yumnam R, Tomar A: Fixed point and its geometry and application for multivalued integral type contractions in $m_v^b$-metric spaces. Hacettepe Journal of Mathematics and Statistics . 2025; 54 (5): 1708-1724 Publisher Full Text 2. Alam K, Rohen Y, Tomar A, Sajid M, et al.: Set-valued contractions with an application to Fredholm integral inclusions in $ {m_v^b}- $metric spaces. AIMS Mathematics . 2025; 10 (9): 20742-20758 Publisher Full Text 3. Alam K, Rohen Y, Tomar A, Ahmad J: A new extended metric space and solution of rocket’s ascending motion. Asian-European Journal of Mathematics . 2025; 18 (06). Publisher Full Text 4. Alam K, Rohen Y, Tomar A, Sajid M: On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models. Ain Shams Engineering Journal . 2025; 16 (1). Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fixed point theory, generalized metric structures, iterative algorithms, and applied optimization 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Alam KH. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456308 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456308 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Version 1 VERSION 1 PUBLISHED 05 Dec 2025 Views 0 Cite How to cite this report: Alam KH. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441497 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441497 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 10 Jan 2026 Khairul Habib Alam , Mathematics, Indian Institute of Science Education and Research Berhampur (Ringgold ID: 486382), Brahmapur, Odisha, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.189957.r441497 Peer Review Report Article Title: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D_d*-Symmetric Spaces The manuscript investigates standard fixed point theorems for three pairs of self-mappings within the framework of ... Continue reading READ ALL Peer Review Report Article Title: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D_d*-Symmetric Spaces The manuscript investigates standard fixed point theorems for three pairs of self-mappings within the framework of D_d*-Symmetric Spaces. The authors utilize the property of Common Limit in the Range (CLR) and the concept of Occasionally Weakly Compatible (OWC) mappings to establish unique common fixed point results. Language and Grammar: The manuscript suffers from significant grammatical errors, awkward phrasing, and sentence structure issues that impede readability. A thorough proofreading is required. Literature Review: While the historical context is provided, the bibliography lacks recent developments in related generalized metric spaces (specifically, 2025 advancements) and applications to differential/integral inclusions, which the authors claim their work supports. Clarity of Definitions: Some definitions are verbose and could be streamlined for better mathematical precision. Grammatical Errors and Sentence Clarity The English usage throughout the manuscript needs substantial polishing. Below are specific examples of errors and suggested corrections. Current Text: "The Banach fixed point theory stated that a mapping T: X to X always has a unique fixed point..." Correction: "Banach fixed point theory states that..." Current Text: "Various vital concepts such as (D_d*-Symmetric and weakly compatible maps) that are needed in the sequel, which will help us in the outcomes that follow and play a major role in verifying our major outcomes." Correction: "Vital concepts, such as D_d*-Symmetric Spaces and weakly compatible maps, are reviewed to establish the framework for our main results." Current Text: "Our main results in this manuscript have explored novel various outcomes..." Correction: "This manuscript explores novel outcomes regarding the uniqueness of..." Current Text: "...various extensions of Banach fixed point theory were carried out. Therefore, fixed point theory has been developed and diversified..." Correction: "...various extensions of Banach fixed point theory were established. Consequently, the theory has evolved to encompass diverse extensions..." Current Text: "...talented with partial ordering." Correction: "... endowed with a partial ordering." Current Text: "Our motivation for introducing this segment is to study and verify..." Correction: "In this section, we establish the uniqueness..." Current Text: "Let S, Q, F, F, G and J be three pairs of self-maps..." Correction: "Let S, Q, F, G, and J be three pairs of self-maps..." Current Text: "We anticipate that the discoveries in this manuscript will aid scientists in enhancing the authors' understanding of popularized extended symmetric spaces..." Correction: "We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences." Improvement of Literature and Citations The Introduction and Conclusion claim that fixed point theory is vital for finding solutions to "linear and nonlinear systems," "integral and differential equations," and "optimization theory." However, the current references do not sufficiently reflect the most recent advancements in these specific applications (particularly in 2025). To substantiate the paper's claims regarding applicability and to situate the work within the state-of-the-art, the following recent papers must be cited. These works deal with similar generalizations (M_v^b-metric spaces, fuzzy spaces) and explicitly demonstrate the applications (Fredholm integrals, differential inclusions) that the authors allude to. Suggested Additions: Context: When discussing applications to integral equations and set-valued mappings (e.g., Introduction, paragraph 1 or 5), cite: https://doi.org/10.15672/hujms.1471688 Context: When mentioning extensions to fuzzy structures or applications to nonlinear Cauchy differential inclusions (e.g., Introduction, near the discussion of metric space generalizations), cite: https://doi.org/10.1177/18758967251366331 Context: When discussing contraction principles and their application to Fredholm integral inclusions (specifically to support the claim in the "Background" section about integral equations), cite: https://doi.org/10.3934/math.2025926 Context: When referring to "optimization theory" or proximal contractions (e.g., Introduction or Conclusion), cite: https://doi.org/10.1186/s13663-025-00799-0 Integration Strategy: The authors should add a paragraph in the Introduction (around the discussion of "Promised studies" or "Motivated by above facts") that reads: "Recent developments in 2025 have further expanded fixed point theory into complex generalized spaces with robust applications. For instance, new results in M_v^b-metric spaces have addressed multivalued integral contractions [1] and Fredholm integral inclusions [3]. Furthermore, significant advances have been made in the application of fuzzy metric spaces to nonlinear differential inclusions [2] and proximal contractions to optimization problems [4]. These works highlight the continued relevance of extending contractive conditions in generalized spaces." Other Comments Symmetry of D_d*: In the proofs (e.g., Theorem 3.2), the authors frequently state "because D_d* is symmetric." It would be beneficial to explicitly reference the specific property number from Definition 2.1 (e.g., $D_3$) to assist the reader. Inequality Numbering: Ensure that the labeling of inequalities (1), (2), (3) is consistent and that they are referenced correctly throughout the text. Uniqueness Proof: The uniqueness proof in Theorem 3.2 is generally sound, but the notation for the arbitrary fixed points (neq t) should be defined clearly at the start of the uniqueness paragraph. Examples: Authors have to add numerical examples to validate the results. Graphical Representation: Authors can add a graphical representation in the numerical examples. Application: The Author can add an application to the real-world problem. Decision The paper presents interesting results in the domain of symmetric spaces. However, the presentation is currently not up to the standard of a high-impact journal. I recommend Major Revisions . 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? Yes If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fixed point theory, generalized metric structures, iterative algorithms, and applied optimization 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Alam KH. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441497 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441497 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 07 Feb 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 07 Feb 2026 Author Response Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point ... Continue reading Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd*- Symmetric Spaces)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 2- Some lengthy definitions have been simplified to improve their mathematical precision such as (Definition 2.3 and 2. 8). 3- All the errors pointed out by the esteemed evaluator have been corrected by implementing all his valuable suggestions. 4- All the references [27-30] which suggested by the esteemed evaluator have been added due to their importance in raising the scientific level for out paper and rigor of the manuscript y and to situate out work within the state-of-the-art. 5- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 6- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the paper 7- We would like to assure that all of labeling of inequalities (1), (2) and (3) are consistent and they are referenced correctly throughout the text. 8- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. 9- We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Acknowledgments: thankful to reviewers for their valuable corrections and important suggestions Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd*- Symmetric Spaces)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 2- Some lengthy definitions have been simplified to improve their mathematical precision such as (Definition 2.3 and 2. 8). 3- All the errors pointed out by the esteemed evaluator have been corrected by implementing all his valuable suggestions. 4- All the references [27-30] which suggested by the esteemed evaluator have been added due to their importance in raising the scientific level for out paper and rigor of the manuscript y and to situate out work within the state-of-the-art. 5- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 6- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the paper 7- We would like to assure that all of labeling of inequalities (1), (2) and (3) are consistent and they are referenced correctly throughout the text. 8- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. 9- We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Acknowledgments: thankful to reviewers for their valuable corrections and important suggestions Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 07 Feb 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 07 Feb 2026 Author Response Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point ... Continue reading Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd*- Symmetric Spaces)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 2- Some lengthy definitions have been simplified to improve their mathematical precision such as (Definition 2.3 and 2. 8). 3- All the errors pointed out by the esteemed evaluator have been corrected by implementing all his valuable suggestions. 4- All the references [27-30] which suggested by the esteemed evaluator have been added due to their importance in raising the scientific level for out paper and rigor of the manuscript y and to situate out work within the state-of-the-art. 5- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 6- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the paper 7- We would like to assure that all of labeling of inequalities (1), (2) and (3) are consistent and they are referenced correctly throughout the text. 8- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. 9- We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Acknowledgments: thankful to reviewers for their valuable corrections and important suggestions Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd*- Symmetric Spaces)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 2- Some lengthy definitions have been simplified to improve their mathematical precision such as (Definition 2.3 and 2. 8). 3- All the errors pointed out by the esteemed evaluator have been corrected by implementing all his valuable suggestions. 4- All the references [27-30] which suggested by the esteemed evaluator have been added due to their importance in raising the scientific level for out paper and rigor of the manuscript y and to situate out work within the state-of-the-art. 5- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 6- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the paper 7- We would like to assure that all of labeling of inequalities (1), (2) and (3) are consistent and they are referenced correctly throughout the text. 8- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. 9- We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Acknowledgments: thankful to reviewers for their valuable corrections and important suggestions Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Yousif YY. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441499 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441499 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 05 Jan 2026 Yousif Yaqoub Yousif , University of Baghdad, Baghdad, Iraq Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.189957.r441499 I would like to inform you that several comments have been registered about the following paper: ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) ... Continue reading READ ALL I would like to inform you that several comments have been registered about the following paper: ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) General Comments: In this manuscript, some new generalized common fixed point theorems for three pairs of self-maps under the influence of other enhanced categories of extended contractive conditions in the context of D d * -Symmetric spaces have been presented and verified. Also, by applying the concepts of weak compatibility and common limit in the range property some main results have been established. The presentation method of the abstract is a bit disorganized and I believe it needs some expansion and focus on the main objectives of the manuscript. Please, state the most important results obtained at the final of the introduction so as to display the latest results obtained compared to previous studies. I think that the first condition in definition (2. 1) is extra. It is evident. Remove it Could you please add the Lemma in the reference [27] in the last of Remark 2.4 to the Fundamentals chapter? Please, could you cite the reference for the remark (3. 1) observation into the main results chapter? Could you please extend the proof of the theorem (3. 2) to be easier for authors to follow? Why wasn't the proof for the Corollary (3.3) mentioned in detail? Please check all punctuation marks in the all sections in the manuscript. The authors should be replacing the word (gratify) by (satisfy) and the symbol (&) by (and) in all sections of their manuscript. I think contributions and research objectives/questions are not clearly presented. Are there any practical applications for this type of research? Please explain them. Does the underlying space exhibit the Hausdorff Topology? In the conclusion, the authors have discussed that the fixed point theory has various applications in science and engineering. This should be discussed in the introduction part Check the names of all the journals in the list of references including the abbreviation of ones With kind regards, Yours sincerely Prof. Dr. Yousif Yaqoub Yousif University of Baghdad College of Education For Pure science (Ibn Al-Haitham) Department of Mathematics – Baghdad – Iraq Mathematics – Topology and its Applications E-mail : [email protected] Scopes : https://www.scopus.com/authid/detail.uri?authorId=54790328600 ORCID : https://orcid.org/0000-0002-6834-3515 https://orcid.org/0000-0002-6834-3515 Google Scholar : https://scholar.google.com/citations?hl=en&user=Eax_5YIAAAAJ Research Gate : https://www.researchgate.net/profile/Yousif_Yousif3 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 Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Yousif YY. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441499 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441499 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 12 Jan 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 12 Jan 2026 Author Response Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via ... Continue reading Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 : Yes, This represents an abstract of the main objectives of our manuscript . 2- Answer : The main objectives of our manuscript are highlighted in the abstract and cannot be expanded further because the journal has specified the word count for the abstract, and it has also been formatted for optimal clarity. 3- Answer : The main findings of this paper are listed at the end of the introduction, as directed by the respected reviewer. 4- Answer : The first condition in definition (2. 1) is essential condition and it is found in the original paper [26]. 5- Answer : The Lemma which was mentioned at the end of Remark (2.4) in our manuscript is already present in reference [27] and any author can View it. 6- Answer : The remark (3. 1) is from our presenting and has been included because we needed it in our main results. 7- Answer : I think that the proof of Theorem No. (3. 2) is easy and clear and can be easily followed by researchers. 8- Answer : Because the Corollary (3.3) is a direct result of Theorem (3. 2) and its proof follows immediately from Theorem-3.2, via putting G = F = I X (Identity mapping). 9- Answer : All punctuation marks in the all sections of the manuscript have been checked and verified as directed by the respected reviewer. 10- Answer : The term "gratify" and the symbol ( & ) will be replaced in all chapters of the paper in the final version. 11- Answer : Our contributions and research objectives have been presented in the end of the introduction. 12- Answer : Yes, there are certainly many any practical applications for the topic of this paper in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems. 13- Answer : Yes, as it is known to all scientists and researchers, every metric space is T2 a topological space, as shown in the link attached below . https://math.uchicago.edu/~may/REU2013/REUPapers/Clarke.pdf 14 Answer : The applications of fixed point theory in the fields of mathematics and other branches of sciences have demonstrated in the introduction part. 15- Answer : All names of journals in the source list have been checked and verified as directed by the respected reviewer. Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 : Yes, This represents an abstract of the main objectives of our manuscript . 2- Answer : The main objectives of our manuscript are highlighted in the abstract and cannot be expanded further because the journal has specified the word count for the abstract, and it has also been formatted for optimal clarity. 3- Answer : The main findings of this paper are listed at the end of the introduction, as directed by the respected reviewer. 4- Answer : The first condition in definition (2. 1) is essential condition and it is found in the original paper [26]. 5- Answer : The Lemma which was mentioned at the end of Remark (2.4) in our manuscript is already present in reference [27] and any author can View it. 6- Answer : The remark (3. 1) is from our presenting and has been included because we needed it in our main results. 7- Answer : I think that the proof of Theorem No. (3. 2) is easy and clear and can be easily followed by researchers. 8- Answer : Because the Corollary (3.3) is a direct result of Theorem (3. 2) and its proof follows immediately from Theorem-3.2, via putting G = F = I X (Identity mapping). 9- Answer : All punctuation marks in the all sections of the manuscript have been checked and verified as directed by the respected reviewer. 10- Answer : The term "gratify" and the symbol ( & ) will be replaced in all chapters of the paper in the final version. 11- Answer : Our contributions and research objectives have been presented in the end of the introduction. 12- Answer : Yes, there are certainly many any practical applications for the topic of this paper in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems. 13- Answer : Yes, as it is known to all scientists and researchers, every metric space is T2 a topological space, as shown in the link attached below . https://math.uchicago.edu/~may/REU2013/REUPapers/Clarke.pdf 14 Answer : The applications of fixed point theory in the fields of mathematics and other branches of sciences have demonstrated in the introduction part. 15- Answer : All names of journals in the source list have been checked and verified as directed by the respected reviewer. Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 12 Jan 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 12 Jan 2026 Author Response Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via ... Continue reading Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 : Yes, This represents an abstract of the main objectives of our manuscript . 2- Answer : The main objectives of our manuscript are highlighted in the abstract and cannot be expanded further because the journal has specified the word count for the abstract, and it has also been formatted for optimal clarity. 3- Answer : The main findings of this paper are listed at the end of the introduction, as directed by the respected reviewer. 4- Answer : The first condition in definition (2. 1) is essential condition and it is found in the original paper [26]. 5- Answer : The Lemma which was mentioned at the end of Remark (2.4) in our manuscript is already present in reference [27] and any author can View it. 6- Answer : The remark (3. 1) is from our presenting and has been included because we needed it in our main results. 7- Answer : I think that the proof of Theorem No. (3. 2) is easy and clear and can be easily followed by researchers. 8- Answer : Because the Corollary (3.3) is a direct result of Theorem (3. 2) and its proof follows immediately from Theorem-3.2, via putting G = F = I X (Identity mapping). 9- Answer : All punctuation marks in the all sections of the manuscript have been checked and verified as directed by the respected reviewer. 10- Answer : The term "gratify" and the symbol ( & ) will be replaced in all chapters of the paper in the final version. 11- Answer : Our contributions and research objectives have been presented in the end of the introduction. 12- Answer : Yes, there are certainly many any practical applications for the topic of this paper in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems. 13- Answer : Yes, as it is known to all scientists and researchers, every metric space is T2 a topological space, as shown in the link attached below . https://math.uchicago.edu/~may/REU2013/REUPapers/Clarke.pdf 14 Answer : The applications of fixed point theory in the fields of mathematics and other branches of sciences have demonstrated in the introduction part. 15- Answer : All names of journals in the source list have been checked and verified as directed by the respected reviewer. Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 : Yes, This represents an abstract of the main objectives of our manuscript . 2- Answer : The main objectives of our manuscript are highlighted in the abstract and cannot be expanded further because the journal has specified the word count for the abstract, and it has also been formatted for optimal clarity. 3- Answer : The main findings of this paper are listed at the end of the introduction, as directed by the respected reviewer. 4- Answer : The first condition in definition (2. 1) is essential condition and it is found in the original paper [26]. 5- Answer : The Lemma which was mentioned at the end of Remark (2.4) in our manuscript is already present in reference [27] and any author can View it. 6- Answer : The remark (3. 1) is from our presenting and has been included because we needed it in our main results. 7- Answer : I think that the proof of Theorem No. (3. 2) is easy and clear and can be easily followed by researchers. 8- Answer : Because the Corollary (3.3) is a direct result of Theorem (3. 2) and its proof follows immediately from Theorem-3.2, via putting G = F = I X (Identity mapping). 9- Answer : All punctuation marks in the all sections of the manuscript have been checked and verified as directed by the respected reviewer. 10- Answer : The term "gratify" and the symbol ( & ) will be replaced in all chapters of the paper in the final version. 11- Answer : Our contributions and research objectives have been presented in the end of the introduction. 12- Answer : Yes, there are certainly many any practical applications for the topic of this paper in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems. 13- Answer : Yes, as it is known to all scientists and researchers, every metric space is T2 a topological space, as shown in the link attached below . https://math.uchicago.edu/~may/REU2013/REUPapers/Clarke.pdf 14 Answer : The applications of fixed point theory in the fields of mathematics and other branches of sciences have demonstrated in the introduction part. 15- Answer : All names of journals in the source list have been checked and verified as directed by the respected reviewer. Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Park C. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441496 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441496 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 02 Jan 2026 Choonkil Park , Hanyang University, Seoul, South Korea Not Approved VIEWS 0 https://doi.org/10.5256/f1000research.189957.r441496 The results are interesting. But the present form is not readable. The authors did not read the final version before submitting the paper. For example, (1) Definition 2.1 [26] A ; (2) Example 2.2 [26] Assume that ; Theorem 3.5 ... Continue reading READ ALL The results are interesting. But the present form is not readable. The authors did not read the final version before submitting the paper. For example, (1) Definition 2.1 [26] A ; (2) Example 2.2 [26] Assume that ; Theorem 3.5 If ~ are six self-maps in. The paper should be revised to be reviewed again. The present form contains a lot of errors in typos and grammatical errors. The paper should be revised to be reviewed again. 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? 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? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fixed point theory, Functional analysis, Nonlinear analysis, Fuzzy analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Park C. Reviewer Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441496 ) The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441496 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 16 Jan 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 16 Jan 2026 Author Response 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 ... Continue reading 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. We extend our deep thanks to the honored reviewer for his valuable observations, which aim to improve the research paper in order to enhance its rigor and scientific value and avoid typographical and grammatical errors. We would like to clarify the following: 1 - We would like to assure you that the manuscript was thoroughly reviewed before submission, but some typographical errors may have occurred during reprinting and were not fully noticed. These will be corrected in the final version before submission. 2 - We would like to assure you that the manuscript has been linguistically revised by the esteemed journal, and we confirm that all the linguistic revisions sent by the journal have been taken into account due to their great importance to this research paper. 3 - Also, we would like to assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. 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. We extend our deep thanks to the honored reviewer for his valuable observations, which aim to improve the research paper in order to enhance its rigor and scientific value and avoid typographical and grammatical errors. We would like to clarify the following: 1 - We would like to assure you that the manuscript was thoroughly reviewed before submission, but some typographical errors may have occurred during reprinting and were not fully noticed. These will be corrected in the final version before submission. 2 - We would like to assure you that the manuscript has been linguistically revised by the esteemed journal, and we confirm that all the linguistic revisions sent by the journal have been taken into account due to their great importance to this research paper. 3 - Also, we would like to assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 16 Jan 2026 Alaa AL-Jumaili , Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 16 Jan 2026 Author Response 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 ... Continue reading 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. We extend our deep thanks to the honored reviewer for his valuable observations, which aim to improve the research paper in order to enhance its rigor and scientific value and avoid typographical and grammatical errors. We would like to clarify the following: 1 - We would like to assure you that the manuscript was thoroughly reviewed before submission, but some typographical errors may have occurred during reprinting and were not fully noticed. These will be corrected in the final version before submission. 2 - We would like to assure you that the manuscript has been linguistically revised by the esteemed journal, and we confirm that all the linguistic revisions sent by the journal have been taken into account due to their great importance to this research paper. 3 - Also, we would like to assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. 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. We extend our deep thanks to the honored reviewer for his valuable observations, which aim to improve the research paper in order to enhance its rigor and scientific value and avoid typographical and grammatical errors. We would like to clarify the following: 1 - We would like to assure you that the manuscript was thoroughly reviewed before submission, but some typographical errors may have occurred during reprinting and were not fully noticed. These will be corrected in the final version before submission. 2 - We would like to assure you that the manuscript has been linguistically revised by the esteemed journal, and we confirm that all the linguistic revisions sent by the journal have been taken into account due to their great importance to this research paper. 3 - Also, we would like to assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. Competing Interests: Competing interests: The authors declare no competing interests. Close Report a concern COMMENT ON THIS REPORT Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 05 Dec 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline 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 Reviewer Reports Invited Reviewers 1 2 3 4 Version 2 (revision) 07 Feb 26 read read read Version 1 05 Dec 25 read read read Choonkil Park , Hanyang University, Seoul, South Korea Yousif Yaqoub Yousif , University of Baghdad, Baghdad, Iraq Khairul Habib Alam , Indian Institute of Science Education and Research Berhampur (Ringgold ID: 486382), Brahmapur, India Kapil Yadav , Lovely Professional University, Phagwara, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Yadav K. This is an open access peer review report 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. 14 Mar 2026 | for Version 2 Kapil Yadav , Lovely Professional University, Phagwara, Punjab, India 0 Views copyright © 2026 Yadav K. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline 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 Article Title: Some Extended Results of Common Fixed-Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces Article summary: The manuscript establishes common fixed-point results for pairs of contractive mappings satisfying weak compatibility and CLR property using altering distance functions in the setting of Dd*symmetric spaces. In addition, the authors derive common fixed-point results for occasionally weakly compatible mappings in this framework. Several corollaries are also presented as consequences of the main theorem. The results generalize and improve various recent outcomes of (C.F.P) theorems under extended weakly contractive maps in the literature. The proofs of the theorems are well illustrated and are presented in a clear and understandable manner. In order to enhance the readability of the manuscript, the following suggestions are provided. Suggestions: The topic is relevant and falls within the scope of fixed-point theory, and the results may be of interest to researchers working in generalized metric structures and nonlinear analysis. Grammatical/Typo: Some suggestions related to grammar and typographical corrections are listed below. The authors are encouraged to carefully review the entire manuscript and implement similar corrections wherever necessary to improve the overall presentation. 1 In Definition 2.9 add the abbreviation for "occasionally weakly compatible" (O.W.C.) 2 In Definition 2.7 correct the font of "x" 3 In Definition 2.13 replace "are called gratify property of common limit range related" with "have common limit range property wrt" 4 In Remark 3.1 is the order of the pairs of three mappings have some significance? if not then just use "are self-mappings". Addition of Illustrative Examples: The authors are encouraged to include some illustrative examples to verify the fixed-point results established in the main section of the manuscript. Addition of a Remark or Section: The authors are encouraged to add a remark or a separate Section (or in the Conclusion section) highlighting the results that are extended by the main theorem of the manuscript. If possible, the authors should clearly indicate the earlier results that are generalized by their work, along with appropriate citations to the related literature. For example, a statement such as “Our result extends Theorem XX of [\cite].” may be included where applicable. 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? I cannot comment. A qualified statistician is required. 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 Fixed Point Theory of Contraction Mappings, Generalized Metric Structure, Stability of Functional Equations 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. reply Respond to this report Responses (0) Yadav K. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456794) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456794 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Yousif Y. This is an open access peer review report 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. 16 Feb 2026 | for Version 2 Yousif Yaqoub Yousif , University of Baghdad, Baghdad, Iraq 0 Views copyright © 2026 Yousif Y. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline 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 The paper can now be accepted for indexing. Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (0) Yousif YY. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456310) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456310 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Alam K. This is an open access peer review report 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. 10 Feb 2026 | for Version 2 Khairul Habib Alam , Mathematics, Indian Institute of Science Education and Research Berhampur (Ringgold ID: 486382), Brahmapur, Odisha, India 0 Views copyright © 2026 Alam K. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline 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 I am satisfied with this version. Kindly cite the following papers for improving the literature. (refer to 1,2,3,4) References 1. Alam K, Yumnam R, Tomar A: Fixed point and its geometry and application for multivalued integral type contractions in $m_v^b$-metric spaces. Hacettepe Journal of Mathematics and Statistics . 2025; 54 (5): 1708-1724 Publisher Full Text 2. Alam K, Rohen Y, Tomar A, Sajid M, et al.: Set-valued contractions with an application to Fredholm integral inclusions in $ {m_v^b}- $metric spaces. AIMS Mathematics . 2025; 10 (9): 20742-20758 Publisher Full Text 3. Alam K, Rohen Y, Tomar A, Ahmad J: A new extended metric space and solution of rocket’s ascending motion. Asian-European Journal of Mathematics . 2025; 18 (06). Publisher Full Text 4. Alam K, Rohen Y, Tomar A, Sajid M: On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models. Ain Shams Engineering Journal . 2025; 16 (1). Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fixed point theory, generalized metric structures, iterative algorithms, and applied optimization 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. reply Respond to this report Responses (0) Alam KH. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.195954.r456308) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1363/v2#referee-response-456308 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Alam K. This is an open access peer review report 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. 10 Jan 2026 | for Version 1 Khairul Habib Alam , Mathematics, Indian Institute of Science Education and Research Berhampur (Ringgold ID: 486382), Brahmapur, Odisha, India 0 Views copyright © 2026 Alam K. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline 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 Peer Review Report Article Title: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D_d*-Symmetric Spaces The manuscript investigates standard fixed point theorems for three pairs of self-mappings within the framework of D_d*-Symmetric Spaces. The authors utilize the property of Common Limit in the Range (CLR) and the concept of Occasionally Weakly Compatible (OWC) mappings to establish unique common fixed point results. Language and Grammar: The manuscript suffers from significant grammatical errors, awkward phrasing, and sentence structure issues that impede readability. A thorough proofreading is required. Literature Review: While the historical context is provided, the bibliography lacks recent developments in related generalized metric spaces (specifically, 2025 advancements) and applications to differential/integral inclusions, which the authors claim their work supports. Clarity of Definitions: Some definitions are verbose and could be streamlined for better mathematical precision. Grammatical Errors and Sentence Clarity The English usage throughout the manuscript needs substantial polishing. Below are specific examples of errors and suggested corrections. Current Text: "The Banach fixed point theory stated that a mapping T: X to X always has a unique fixed point..." Correction: "Banach fixed point theory states that..." Current Text: "Various vital concepts such as (D_d*-Symmetric and weakly compatible maps) that are needed in the sequel, which will help us in the outcomes that follow and play a major role in verifying our major outcomes." Correction: "Vital concepts, such as D_d*-Symmetric Spaces and weakly compatible maps, are reviewed to establish the framework for our main results." Current Text: "Our main results in this manuscript have explored novel various outcomes..." Correction: "This manuscript explores novel outcomes regarding the uniqueness of..." Current Text: "...various extensions of Banach fixed point theory were carried out. Therefore, fixed point theory has been developed and diversified..." Correction: "...various extensions of Banach fixed point theory were established. Consequently, the theory has evolved to encompass diverse extensions..." Current Text: "...talented with partial ordering." Correction: "... endowed with a partial ordering." Current Text: "Our motivation for introducing this segment is to study and verify..." Correction: "In this section, we establish the uniqueness..." Current Text: "Let S, Q, F, F, G and J be three pairs of self-maps..." Correction: "Let S, Q, F, G, and J be three pairs of self-maps..." Current Text: "We anticipate that the discoveries in this manuscript will aid scientists in enhancing the authors' understanding of popularized extended symmetric spaces..." Correction: "We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences." Improvement of Literature and Citations The Introduction and Conclusion claim that fixed point theory is vital for finding solutions to "linear and nonlinear systems," "integral and differential equations," and "optimization theory." However, the current references do not sufficiently reflect the most recent advancements in these specific applications (particularly in 2025). To substantiate the paper's claims regarding applicability and to situate the work within the state-of-the-art, the following recent papers must be cited. These works deal with similar generalizations (M_v^b-metric spaces, fuzzy spaces) and explicitly demonstrate the applications (Fredholm integrals, differential inclusions) that the authors allude to. Suggested Additions: Context: When discussing applications to integral equations and set-valued mappings (e.g., Introduction, paragraph 1 or 5), cite: https://doi.org/10.15672/hujms.1471688 Context: When mentioning extensions to fuzzy structures or applications to nonlinear Cauchy differential inclusions (e.g., Introduction, near the discussion of metric space generalizations), cite: https://doi.org/10.1177/18758967251366331 Context: When discussing contraction principles and their application to Fredholm integral inclusions (specifically to support the claim in the "Background" section about integral equations), cite: https://doi.org/10.3934/math.2025926 Context: When referring to "optimization theory" or proximal contractions (e.g., Introduction or Conclusion), cite: https://doi.org/10.1186/s13663-025-00799-0 Integration Strategy: The authors should add a paragraph in the Introduction (around the discussion of "Promised studies" or "Motivated by above facts") that reads: "Recent developments in 2025 have further expanded fixed point theory into complex generalized spaces with robust applications. For instance, new results in M_v^b-metric spaces have addressed multivalued integral contractions [1] and Fredholm integral inclusions [3]. Furthermore, significant advances have been made in the application of fuzzy metric spaces to nonlinear differential inclusions [2] and proximal contractions to optimization problems [4]. These works highlight the continued relevance of extending contractive conditions in generalized spaces." Other Comments Symmetry of D_d*: In the proofs (e.g., Theorem 3.2), the authors frequently state "because D_d* is symmetric." It would be beneficial to explicitly reference the specific property number from Definition 2.1 (e.g., $D_3$) to assist the reader. Inequality Numbering: Ensure that the labeling of inequalities (1), (2), (3) is consistent and that they are referenced correctly throughout the text. Uniqueness Proof: The uniqueness proof in Theorem 3.2 is generally sound, but the notation for the arbitrary fixed points (neq t) should be defined clearly at the start of the uniqueness paragraph. Examples: Authors have to add numerical examples to validate the results. Graphical Representation: Authors can add a graphical representation in the numerical examples. Application: The Author can add an application to the real-world problem. Decision The paper presents interesting results in the domain of symmetric spaces. However, the presentation is currently not up to the standard of a high-impact journal. I recommend Major Revisions . 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? Yes If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fixed point theory, generalized metric structures, iterative algorithms, and applied optimization 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. reply Respond to this report Responses (1) Author Response 07 Feb 2026 Alaa AL-Jumaili, Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq Report and Comments responses-f1000res172242 Dear prof... Best greetings..... I hope this message finds you well. Subject : comments responses about the manuscript-f1000res172242 (Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd*- Symmetric Spaces)-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 manuscript has already been carefully reviewed for linguistic and spelling accuracy in all its parts. 2- Some lengthy definitions have been simplified to improve their mathematical precision such as (Definition 2.3 and 2. 8). 3- All the errors pointed out by the esteemed evaluator have been corrected by implementing all his valuable suggestions. 4- All the references [27-30] which suggested by the esteemed evaluator have been added due to their importance in raising the scientific level for out paper and rigor of the manuscript y and to situate out work within the state-of-the-art. 5- We confirm the addition of the paragraph proposed by the honored evaluator to the introduction (around the discussion of Promised studies or Motivated). 6- It was explicitly stated reference the specific property number from Definition 2.1 in all parts in the paper 7- We would like to assure that all of labeling of inequalities (1), (2) and (3) are consistent and they are referenced correctly throughout the text. 8- We would like to assure you that all of notations for the arbitrary fixed points (C.F.P) have been defined clearly at the start of the uniqueness paragraph in the proof in Theorem 3.2. 9- We anticipate that these results will aid researchers in developing generalized symmetric spaces and establishing frameworks for applications in advanced sciences. Acknowledgments: thankful to reviewers for their valuable corrections and important suggestions View more View less Competing Interests Competing interests: The authors declare no competing interests. reply Respond Report a concern Alam KH. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441497) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441497 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Yousif Y. This is an open access peer review report 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. 05 Jan 2026 | for Version 1 Yousif Yaqoub Yousif , University of Baghdad, Baghdad, Iraq 0 Views copyright © 2026 Yousif Y. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline 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 I would like to inform you that several comments have been registered about the following paper: ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) General Comments: In this manuscript, some new generalized common fixed point theorems for three pairs of self-maps under the influence of other enhanced categories of extended contractive conditions in the context of D d * -Symmetric spaces have been presented and verified. Also, by applying the concepts of weak compatibility and common limit in the range property some main results have been established. The presentation method of the abstract is a bit disorganized and I believe it needs some expansion and focus on the main objectives of the manuscript. Please, state the most important results obtained at the final of the introduction so as to display the latest results obtained compared to previous studies. I think that the first condition in definition (2. 1) is extra. It is evident. Remove it Could you please add the Lemma in the reference [27] in the last of Remark 2.4 to the Fundamentals chapter? Please, could you cite the reference for the remark (3. 1) observation into the main results chapter? Could you please extend the proof of the theorem (3. 2) to be easier for authors to follow? Why wasn't the proof for the Corollary (3.3) mentioned in detail? Please check all punctuation marks in the all sections in the manuscript. The authors should be replacing the word (gratify) by (satisfy) and the symbol (&) by (and) in all sections of their manuscript. I think contributions and research objectives/questions are not clearly presented. Are there any practical applications for this type of research? Please explain them. Does the underlying space exhibit the Hausdorff Topology? In the conclusion, the authors have discussed that the fixed point theory has various applications in science and engineering. This should be discussed in the introduction part Check the names of all the journals in the list of references including the abbreviation of ones With kind regards, Yours sincerely Prof. Dr. Yousif Yaqoub Yousif University of Baghdad College of Education For Pure science (Ibn Al-Haitham) Department of Mathematics – Baghdad – Iraq Mathematics – Topology and its Applications E-mail : [email protected] Scopes : https://www.scopus.com/authid/detail.uri?authorId=54790328600 ORCID : https://orcid.org/0000-0002-6834-3515 https://orcid.org/0000-0002-6834-3515 Google Scholar : https://scholar.google.com/citations?hl=en&user=Eax_5YIAAAAJ Research Gate : https://www.researchgate.net/profile/Yousif_Yousif3 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 Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (1) Author Response 12 Jan 2026 Alaa AL-Jumaili, Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq Dear Editor... Best greetings..... I hope this message finds you well. Subject \ comments responses about the manuscript -f1000res172242 ( Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in D d * - Symmetric Spaces )-Manuscript Number: 3b619270-4ddc-4685-af42-071e2dbdcc63-f1000res172242) 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 : Yes, This represents an abstract of the main objectives of our manuscript . 2- Answer : The main objectives of our manuscript are highlighted in the abstract and cannot be expanded further because the journal has specified the word count for the abstract, and it has also been formatted for optimal clarity. 3- Answer : The main findings of this paper are listed at the end of the introduction, as directed by the respected reviewer. 4- Answer : The first condition in definition (2. 1) is essential condition and it is found in the original paper [26]. 5- Answer : The Lemma which was mentioned at the end of Remark (2.4) in our manuscript is already present in reference [27] and any author can View it. 6- Answer : The remark (3. 1) is from our presenting and has been included because we needed it in our main results. 7- Answer : I think that the proof of Theorem No. (3. 2) is easy and clear and can be easily followed by researchers. 8- Answer : Because the Corollary (3.3) is a direct result of Theorem (3. 2) and its proof follows immediately from Theorem-3.2, via putting G = F = I X (Identity mapping). 9- Answer : All punctuation marks in the all sections of the manuscript have been checked and verified as directed by the respected reviewer. 10- Answer : The term "gratify" and the symbol ( & ) will be replaced in all chapters of the paper in the final version. 11- Answer : Our contributions and research objectives have been presented in the end of the introduction. 12- Answer : Yes, there are certainly many any practical applications for the topic of this paper in many fields of mathematics and other branches of science, such as engineering, physics, computer sciences, economics, and telecommunication optimization problems. 13- Answer : Yes, as it is known to all scientists and researchers, every metric space is T2 a topological space, as shown in the link attached below . https://math.uchicago.edu/~may/REU2013/REUPapers/Clarke.pdf 14 Answer : The applications of fixed point theory in the fields of mathematics and other branches of sciences have demonstrated in the introduction part. 15- Answer : All names of journals in the source list have been checked and verified as directed by the respected reviewer. View more View less Competing Interests Competing interests: The authors declare no competing interests. reply Respond Report a concern Yousif YY. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441499) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1363/v1#referee-response-441499 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Park C. This is an open access peer review report 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. 02 Jan 2026 | for Version 1 Choonkil Park , Hanyang University, Seoul, South Korea 0 Views copyright © 2026 Park C. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (1) Not Approved info_outline 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 The results are interesting. But the present form is not readable. The authors did not read the final version before submitting the paper. For example, (1) Definition 2.1 [26] A ; (2) Example 2.2 [26] Assume that ; Theorem 3.5 If ~ are six self-maps in. The paper should be revised to be reviewed again. The present form contains a lot of errors in typos and grammatical errors. The paper should be revised to be reviewed again. 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? 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? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise Fixed point theory, Functional analysis, Nonlinear analysis, Fuzzy analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above. reply Respond to this report Responses (1) Author Response 16 Jan 2026 Alaa AL-Jumaili, Department of mathematics, College of Education for pure sciences, University of Anbar, Ramadi, 31001, Iraq 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. We extend our deep thanks to the honored reviewer for his valuable observations, which aim to improve the research paper in order to enhance its rigor and scientific value and avoid typographical and grammatical errors. We would like to clarify the following: 1 - We would like to assure you that the manuscript was thoroughly reviewed before submission, but some typographical errors may have occurred during reprinting and were not fully noticed. These will be corrected in the final version before submission. 2 - We would like to assure you that the manuscript has been linguistically revised by the esteemed journal, and we confirm that all the linguistic revisions sent by the journal have been taken into account due to their great importance to this research paper. 3 - Also, we would like to assure you that the research paper will be thoroughly reviewed before the final version is sent, as per the esteemed reviewer's instructions. View more View less Competing Interests Competing interests: The authors declare no competing interests. reply Respond Report a concern Park C. Peer Review Report For: Some Extended Results of Common Fixed Point Theorems via Enhanced Categories of Contractive Mappings in Dd* - Symmetric Spaces [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1363 ( https://doi.org/10.5256/f1000research.189957.r441496) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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