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

Abduljabar K. Abbas; Ali A. Shihab; Alaa M.F. Al-Jumaili

doi:10.12688/f1000research.172242.1

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Research Article

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

[version 1; peer review: awaiting peer review]

Abduljabar K. Abbas¹, Ali A. Shihab², Alaa M.F. Al-Jumaili ³

PUBLISHED 05 Dec 2025

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

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Background

The Banach fixed point theory stated that a mapping T: X→X always has a unique fixed point in X. After witnessing the implementations of this theory in giving the existence and uniqueness solutions for many integral and differential equations, various extensions of Banach fixed point theory were carried out. Therefore, fixed point theory has been developed and diversified to encompass various 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

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. 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 (Concisely, O.W.C) maps that are more general than weakly compatible. Additionally, various common fixed point outcomes for the two pairs of self-maps were determined.

Conclusion

Our main results in this manuscript have been explored novel various outcomes related to 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 implementation in each advanced branches of science.

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: 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: © 2025 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 1; peer review: awaiting peer review]. F1000Research 2025, 14:1363 (https://doi.org/10.12688/f1000research.172242.1) First published: 05 Dec 2025, 14:1363 (https://doi.org/10.12688/f1000research.172242.1) Latest published: 05 Dec 2025, 14:1363 (https://doi.org/10.12688/f1000research.172242.1)

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, Jungck¹ extended the celebrated Banach contraction principle by exploiting the idea of commuting maps and established a common fixed point theory. Subsequently, in 1982, Sessa² 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 Moutawakil⁴ 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 Rhoades⁶ 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 2007⁷ 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 Bhatt⁹ 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, Eke¹² 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. Jumaili¹³ applied $D^{*}$ -metric space and offered various coincidence fixed point theorems for maps gratifying 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 Abed¹⁵ studied fixed point of set-valued contractions in ordered G-metric spaces. In addition, Nagaraju¹⁶ 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 2023²⁴ 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, additionally²⁵ they investigated and proved various outcomes of common and coincidence of fixed points in S-metric spaces. Subsequently, in 2024, Abed and Al-Jumaili²⁶ 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 space.

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 $X \neq \emptyset$ is a map $D_{d}^{*} : X \times X \times X \to [0, \infty)$ (s. t) $\forall x^{*}, y^{*}, z^{*} \in X$ , the next axioms are gratified:

\begin{matrix} (D_{d_{1}}^{*}) D_{d}^{*} (x^{*}, y^{*}, z^{*}) \geq 0, \forall x^{*}, y^{*}, z^{*} \in 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 . \end{matrix}

In this case, $D_{d}^{*}$ is called $D_{d}^{*}$ -Symmetric and $(X, D_{d}^{*})$ is called symmetric space ( $D_{d}^{*}$ -Sym-sp).

Example 2.2:

²⁶ Presume that $X = [0, 1]$ equipped with $D^{*}$ -Symmetric map described through $D_{d}^{*} (x^{*}, y^{*}, z^{*}) = {(x^{*} - y^{*})}^{2} + {(y^{*} - z^{*})}^{2} + {(z^{*} - x^{*})}^{2}, \forall x^{*}, y^{*}, z^{*} \in X$ . Here, $(D_{d}^{*}, X)$ are $D_{d}^{*}$ -Sym-sp.

Definition 2.3:

²⁶ Assume that $(X, D_{d}^{*})$ is a $D_{d}^{*}$ -Sym-sp, therefore a sequence ${x_{s}^{*}}$ is called $D_{d}^{*}$ -converges to $x^{*} \in X$ iff $D_{d}^{*} (x_{s}^{*}, x_{s}^{*}, x^{*}) = D_{d}^{*} (x^{*}, x^{*}, x_{s}^{*}) ⟶ 0 as s ⟶ \infty .$ (i. e), $\forall ε > 0$ , $\exists k \in ℕ$ (s. t), $\forall s \geq h ⟹ D_{d}^{*} (x^{*}, x^{*}, x_{s}^{*}) < ε$ .

Remark 2.4:

Equivalent to Wilson axiom’s²⁷ in $D_{d}^{*}$ -Sym-space as

$(W_{1})$ Given ${x_{s}^{*}}$ , $x^{*}, y^{*} \in X, D_{d}^{*} (x_{s}^{*}, x^{*}, x^{*}) ⟶ 0$ with $D_{d}^{*} (x_{s}^{*}, y^{*}, y^{*}) ⟶ 0 ⟹ x^{*} = y^{*} .$

$(W_{2})$ Given ${x_{s}^{*}}$ & ${y_{s}^{*}}$ , such that $x^{*}, y^{*} \in X, D_{d}^{*} (x_{s}^{*}, x^{*}, x^{*}) ⟶ 0$ 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, $D_{d}^{*}$ -Sym-sp. For an arbitrary sequence ${x_{s}^{*}} in X$ , we have $lim_{s, r \to \infty} D_{d}^{*} (x_{s}^{*}, x_{r}^{*}, x_{r}^{*}) = 0$ , iff $lim_{s \to \infty} D_{d}^{*} (x_{s}^{*}, x_{s + 1}^{*}, x_{s + 1}^{*}) = 0$ .

The next Lemma is analogue of Lemma²⁷ in $(X, D_{d}^{*})$ :

Lemma 2.5:

Each $D_{d}^{*}$ -Symmetric $(X, D_{d}^{*})$ , describes a symmetric $d_{D_{d}^{*}}$ on $X$ via:

$d_{D_{d}^{*}} (x^{*}, y^{*}) = D_{d}^{*} (x^{*}, y^{*}, y^{*}) + D_{d}^{*} (y^{*}, x^{*}, x^{*}), \forall x^{*}, y^{*} \in X$ , that is:

$d_{D_{d}^{*}} (x^{*}, y^{*}) = 2 D_{d}^{*} (x^{*}, y^{*}, y^{*}), \forall x^{*}, y^{*} \in X .$

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 $x^{*} \in X$ , 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 $F, G : (X, D_{d}^{*}) \to (X, D_{d}^{*})$ are single-valued self-maps. If $F (x^{*}) = G (x^{*}) = x^{*}$ for $x^{*} \in X,$ consequently $x^{*}$ is called the common fixed point (C.F.P) of $F & G$ .

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, that is, $Fx = Gx,$ for $x \in X$ , so $FGx = GFx$ .

Definition 2.9

⁶: Self-maps $F$ and $G$ of $(X, D_{d}^{*})$ are called occasionally weakly compatible iff $\exists x \in X$ which is coincidence point of $F$ & $G$ which $F$ & $G$ 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 $μ : [0, \infty) \to [0, \infty)$ 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 $\exists {x_{s}^{*}}$ in $X$ (s. t) $lim_{s \to \infty} η x_{s}^{*} = lim_{s \to \infty} ξ x_{s}^{*} = ξ p$ for $p \in X$ .

Definition 2.13:

¹¹ The pairs $(ξ, η)$ & $(F, G)$ of self-maps in sym-sp $(X, d)$ are called gratify property of common limit range related to the maps $η$ & $G$ , indicated via $({C L R}_{(η, G)})$ , if $\exists {x_{s}^{*}}$ & ${y_{s}^{*}}$ (s. t) $lim_{s \to \infty} ξ x_{s}^{*} = lim_{s \to \infty} η x_{s}^{*} = lim_{s \to \infty} F y_{s}^{*} = lim_{s \to \infty} G y_{s}^{*} = t$ with $t = η p = Gq$ for some $t, p, q \in X$ .

Remark 2.14:

¹⁰ If $ξ = F$ & $η = G$ , 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

Our motivation for introducing this segment is to study and verify the uniqueness of various common fixed-point theorems for three pairs of self-maps gratifying $({C L R}_{(F F, GJ)})$ properties under influence of other enhanced categories of extended contractive maps in $D_{d}^{*}$ -Sym-sp. 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 $ℂ = {\begin{matrix} ψ / ψ : [0, \infty) \to [0, \infty) is lower semi continuous and \\ nondecreasing map where ψ (t) = 0 ⟺ t = 0 \end{matrix}}$

(ii) Assume $S, ℚ, F, F, G$ 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^{*})) \leq μ (L (x^{*}, y^{*}, z^{*})) - ψ (L (x^{*}, y^{*}, z^{*})), \forall x^{*}, y^{*}, z^{*} \in X

Wherever, $μ$ is altering distance map, $ψ \in ℂ$ with

L (x^{*}, y^{*}, z^{*}) = max {\begin{matrix} 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^{*}), \\ \frac{1}{3} {D_{d}^{*} (F F x^{*}, ℚ y^{*}, ℚ z^{*}) + D_{d}^{*} (GJ y^{*}, GJ y^{*}, S x^{*})} \end{matrix}}

Theorem 3.2:

Let $S, ℚ, F, F, G and J$ are three pairs of self-maps of a $D_{d}^{*}$ -Symmetric-space $(X, D_{d}^{*})$ gratify inequality (1) and the following conditions:

(i) If $(S, F F)$ & $(ℚ, GJ)$ gratify ${C L R}_{(F F, GJ)}$ property.

(ii) If $(S, F F)$ & $(ℚ, GJ)$ are weakly compatible maps.

In this case, the maps $S, ℚ, F F$ & $GJ$ have a unique (C. F. P) in $X$ . Additionally, $S, ℚ, F, F, G$ & $J$ contain a unique(C. F. P) provided that the pairs of maps $(F, F), (S, F), (S, F), (G, J), (ℚ, G) & (ℚ, J)$ are commuting.

Proof:

Because $(S, F F)$ & $(ℚ, GJ)$ gratify $({C L R}_{(F F, GJ)})$ property, we can discover two sequences ${x_{s}^{*}}$ & ${y_{s}^{*}}$ in $X$ (s. t), $lim_{s \to \infty} S x_{s}^{*} = lim_{s \to \infty} F F x_{s}^{*} = lim_{s \to \infty} ℚ y_{s}^{*} = lim_{s \to \infty} GJ y_{s}^{*} = t$ with $t = F Fp = GJq$ for some $t, p, q \in X$ .

Firstly verify $F Fp = S p$ . From inequality (1) and $x^{*} = p, y^{*} = z^{*} = y_{s}^{*}$ , get

(2)

μ (D_{d}^{*} (S p, S p, ℚ y^{*})) \leq μ (L (p, p, y_{s}^{*})) - ψ (L (p, p, y_{s}^{*}))

Such that

L (p, p, y_{s}^{*}) = max {\begin{matrix} 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}^{*}), \\ \frac{1}{3} {D_{d}^{*} (F Fp, F Fp, ℚ y_{s}^{*}) + D_{d}^{*} (GJ y_{s}^{*}, GJ y_{s}^{*}, S p)} \end{matrix}}

Consequently

lim_{s \to \infty} L (p, p, y_{s}^{*}) = max {\begin{matrix} D_{d}^{*} (t, t, t), D^{*} (t, t, S p), D_{d}^{*} (t, t, t), \\ \frac{1}{3} {D_{d}^{*} (t, t, t) + D_{d}^{*} (t, t, S p)} \end{matrix}}

= max {0, D_{d}^{*} (t, t, S p), 0, \frac{1}{3} {D_{d}^{*} (t, t, S p)}} = D_{d}^{*} (S p, S p, t), because D_{d}^{*} is symmetric .

Let $s \to \infty$ in inequality (2), $μ (D_{d}^{*} (S p, S p, t)) \leq μ (D_{d}^{*} (S p, S p, t)) - ψ (D_{d}^{*} (S p, S p, t))$ 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 $S$ & $F F$ .

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)) \leq μ (L (x_{s}^{*}, x_{s}^{*}, q)) - ψ (L (x_{s}^{*}, x_{s}^{*}, q))

Where

L (x_{s}^{*}, x_{s}^{*}, q) = max {\begin{matrix} 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), \\ \frac{1}{3} {D_{d}^{*} (F F x_{s}^{*}, F F x_{s}^{*}, ℚ q) + D_{d}^{*} (GJq, GJq, S x_{s}^{*})} \end{matrix}}

Therefore,

\begin{matrix} lim_{s \to \infty} L (x_{s}^{*}, x_{s}^{*}, q) = max {\begin{matrix} D_{d}^{*} (t, t, t), D^{*} (t, t, t), D_{d}^{*} (t, t, ℚ q), \\ \frac{1}{3} {D_{d}^{*} (t, t, ℚ q) + D_{d}^{*} (t, t, t)} \end{matrix}} \\ = max {0, 0, D^{*} (t, t, ℚ q), \frac{1}{3} {D_{d}^{*} (t, t, ℚ q) + 0}} = D_{d}^{*} (t, t, ℚ q) . \end{matrix}

Selecting limit as $s \to \infty$ in inequality (3),

μ (D_{d}^{*} (t, t, ℚ q)) \leq 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 $ℚ$ & $GJ$ . 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

\begin{matrix} μ (D_{d}^{*} (S t, S t, ℚ q)) \leq μ (L (t, t, q)) - ψ (L (t, t, q)) \\ Or μ (D_{d}^{*} (S t, S t, t)) \leq μ (L (t, t, q)) - ψ (L (t, t, q)) \end{matrix}

Such that

L (t, t, q) = max {\begin{matrix} D_{d}^{*} (F Ft, F Ft, GJq), D_{d}^{*} (F Ft, F Ft, S t), D_{d}^{*} (GJq, GJq, ℚ q), \\ \frac{1}{3} {D_{d}^{*} (F Ft, F Ft, ℚ q) + D_{d}^{*} (GJq, GJq, S t)} \end{matrix}}

Consequently

\begin{matrix} max {D_{d}^{*} (S t, S t, t), D_{d}^{*} (S t, S t, S t), D_{d}^{*} (t, t, t), \frac{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, \frac{2}{3} {D_{d}^{*} (S t, S t, t)}}, because D_{d}^{*} is symmetric . \\ = D_{d}^{*} (S t, S t, t) . \end{matrix}

Thus, $μ (D_{d}^{*} (S t, S t, t)) \leq μ (D_{d}^{*} (S t, S t, t)) - ψ (D_{d}^{*} (S t, S t, t)) ⟹$ $ψ (D_{d}^{*} (S t, S t, t)) = 0$ , 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) $x^{*} = p, y^{*} = z^{*} = t$ , obtain

\begin{matrix} μ (D_{d}^{*} (S p, S p, ℚ t)) \leq μ (L (p, p, t)) - ψ (L (p, p, t)) \\ Orμ (D_{d}^{*} (t, t, ℚ t)) \leq μ (L (p, p, t)) - ψ (L (p, p, t)) \end{matrix}

Such that

\begin{matrix} L (p, p, t) = max {\begin{matrix} D_{d}^{*} (F Fp, F Fp, GJt), D_{d}^{*} (F Fp, F Fp, S p), D_{d}^{*} (GJt, GJt, ℚ t), \\ \frac{1}{3} {D_{d}^{*} (F Fp, F Fp, ℚ t) + D_{d}^{*} (GJt, GJt, S p)} \end{matrix}} \\ = max {D_{d}^{*} (t, t, ℚ t), D_{d}^{*} (t, t, t), D^{*} (ℚ t, ℚ t, ℚ t), \frac{1}{3} {D_{d}^{*} (t, t, ℚ t) + D_{d}^{*} (ℚ t, ℚ t, t)}} \\ \begin{matrix} = max {D_{d}^{*} (t, t, ℚ t), 0, 0, \frac{1}{3} {D_{d}^{*} (t, t, ℚ t) + D_{d}^{*} (ℚ t, ℚ t, t)}} \\ = max {D_{d}^{*} (t, t, ℚ t), \frac{2}{3} {D_{d}^{*} (t, t, ℚ t)}}, because D_{d}^{*} is symmetric . \\ = D_{d}^{*} (t, t, ℚ t) . \end{matrix} \end{matrix}

Consequently, $μ (D_{d}^{*} (t, t, ℚ t)) \leq μ (D_{d}^{*} (t, t, ℚ t)) - ψ (D_{d}^{*} (t, t, ℚ t)) ⟹ (D_{d}^{*} (t, t, ℚ t)) = 0 .$ 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 $(n \neq t)$ is different (C. F. P) for $S, ℚ, F F$ & $JG$ . Consequently, we have $S n = F Fn = ℚ n = GJn = n$ . Now, from inequality (1) with $x^{*} = t, y^{*} = z^{*} = n$ , obtain

\begin{matrix} μ (D_{d}^{*} (S t, S t, ℚ n)) \leq μ (L (t, t, n)) - ψ (L (t, t, n)) \\ Or μ (D_{d}^{*} (t, t, n)) \leq μ (L (t, t, n)) - ψ (L (t, t, n)), \end{matrix}

Such that

\begin{matrix} L (t, t, n) = max {\begin{matrix} D_{d}^{*} (F Ft, F Ft, GJn), D_{d}^{*} (F Ft, F Ft, S t), D_{d}^{*} (GJn, GJn, ℚ n), \\ \frac{1}{3} {D_{d}^{*} (F Ft, F Ft, ℚ n) + D_{d}^{*} (GJn, GJn, S t)} \end{matrix}} \\ = max {D_{d}^{*} (t, t, n), D_{d}^{*} (t, t, t), D_{d}^{*} (n, n, n), \frac{1}{3} {D_{d}^{*} (t, t, n) + D_{d}^{*} (n, n, t)}} \\ \begin{matrix} = max {D_{d}^{*} (t, t, n), 0, 0, \frac{1}{3} {D_{d}^{*} (t, t, n) + D_{d}^{*} (n, n, t)}} \\ = max {D_{d}^{*} (t, t, n), \frac{2}{3} {D_{d}^{*} (t, t, n)}} = D_{d}^{*} (t, t, n), since D_{d}^{*} is symmetric . \end{matrix} \end{matrix}

Therefore,

μ (D_{d}^{*} (t, t, n)) \leq 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 (C. F. P) in $X$ .

Next, we shall verify $S, ℚ, F, F, G, J$ have unique (C. F. P).

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 $F t & Ft$ are (C. F. Ps) of $F F$ and $S$ . Therefore, via the uniqueness of (C. F. P), we obtain $F t = Ft = t$ .

Likewise, because $(G, J), (ℚ, G) & (ℚ, J)$ are commuting, obtain $Gt = G (ℚ t) = (ℚ G) t$ with $Gt = G (GJt) = GJ (Gt)$ .

Moreover, $Jt = J (ℚ t) = ℚ (Jt)$ & $Jt = J (GJt) = GJ (Jt)$ . This explains $Gt$ and $Jt$ are (C. F. P) of $GJ$ and $ℚ$ . Consequently, via the uniqueness of (C. F. P), obtain $Gt = Jt = t$ . As a result, $S t = ℚ t = F t = Ft = Gt = Jt = t$ , verifying $t$ is unique (C.F.P) to $S, ℚ, F, F, G, J .$

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 $μ (D_{d}^{*} (S x^{*}, ℚ y^{*}, ℚ z^{*})) \leq μ (L (x^{*}, y^{*}, z^{*})) - ψ (L (x^{*}, y^{*}, z^{*})) \forall x^{*}, y^{*}, z^{*} \in X .$ (s. t), $μ$ alters the distance map with $ψ : ℝ^{+} \to ℝ^{+}$ , is a lower semi-continuous and nondecreasing map (s. t) $ψ (t) = 0$ iff $t = 0$ with

L (x^{*}, y^{*}, z^{*}) = max {\begin{matrix} D_{d}^{*} (F x^{*}, J y^{*}, J z^{*}), D_{d}^{*} (F x^{*}, F x^{*}, S x^{*}), D_{d}^{*} (J y^{*}, ℚ y^{*}, ℚ z^{*}), \\ \frac{1}{3} {D_{d}^{*} (F x^{*}, ℚ y^{*}, ℚ z^{*}) + D_{d}^{*} (J y^{*}, J y^{*}, S x^{*})} \end{matrix}}

(ii) If $(S, F)$ & $(ℚ, J)$ satisfy $({C L R}_{(F, J)})$ property.

(iii) If $(S, F)$ and $(ℚ, J)$ are (W. C. M). In this case, $S, ℚ, F & J$ 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 $X \neq \emptyset$ , with $ξ$ & $η$ are (O. W. C) maps. If $ξ$ and $η$ include a unique point of coincidence $w^{*} = ξ t = η t$ for $t \in X$ , then $w^{*}$ is unique (C. F. P) of $ξ$ and $η$ .

Theorem 3.5:

If $S, ℚ, F, F, G & J$ are three pairs of self-maps in $D_{d}^{*}$ -symmetric $(X, D_{d}^{*})$ satisfies the following conditions:

(i) If $μ (D_{d}^{*} (S x^{*}, ℚ y^{*}, ℚ z^{*})) \leq μ (M (x^{*}, y^{*}, z^{*})) - ψ (M (x^{*}, y^{*}, z^{*})), \forall x^{*}, y^{*}, z^{*} \in X$ , (s. t), $μ$ μ alters the distance map with $ψ : ℝ^{+} \to ℝ^{+}$ , is a lower semi-continuous and nondecreasing map (s. t) $ψ (t) = 0$ iff $t = 0$ with

M (x^{*}, y^{*}, z^{*}) = max {\begin{matrix} 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^{*}) \end{matrix}}

(ii) If $(S, F F)$ & $(ℚ, GJ)$ 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 $S, ℚ, F, F, G & J$ include unique (C.F.P), provided the pairs of maps $(F, F), (S, F), (S, F), (G, J), (ℚ, G) & (ℚ, J)$ are commuting.

Proof:

Assuming that $(S, F F)$ & $(ℚ, GJ)$ are (O. W. C), we can discover $p$ & $q$ in $X$ (s. t) $S p = F Fp = p^{1}$ & $S (F F) p = (F F) S p$ with $ℚ q = GJq = q^{1}$ & $ℚ (GJ) q = (GJ) ℚ q$ .

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)) \leq μ (M (p, p, q)) - ψ (M (p, p, q))

Such that

\begin{matrix} M (p, p, q) = max {\begin{matrix} 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) \end{matrix}} \\ = max {\begin{matrix} 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) \end{matrix}} \\ \begin{matrix} = 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), because D_{d}^{*} is symmetric . \end{matrix} \end{matrix}

Consequently, $μ (D_{d}^{*} (S p, S p, ℚ q)) \leq μ (D_{d}^{*} (S p, S p, ℚ q)) - ψ (D_{d}^{*} (S p, S p, ℚ q))$ 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 $S$ & $F F$ include unique (C.F.P), namely, $t$ . Likewise, it can be verified $ℚ$ & $GJ$ 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)

\begin{matrix} μ (D_{d}^{*} (S t, S t, ℚ t^{1})) \leq μ (M (t, t, t^{1})) - ψ (M (t, t, t^{1})) \\ Or μ (D_{d}^{*} (t, t, t^{1})) \leq μ (M (t, t, t^{1})) - ψ (M (t, t, t^{1})) \end{matrix}

Where

\begin{matrix} M (t, t, t^{1}) = max {\begin{matrix} 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) \end{matrix}} \\ = max {\begin{matrix} 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) \end{matrix}} \\ \begin{matrix} = max {D_{d}^{*} (t, t^{1}, t^{1}), 0, 0, D_{d}^{*} (t^{1}, t, t)} \\ = D_{d}^{*} (t, t, t^{1}), because D_{d}^{*} is symmetric . \end{matrix} \end{matrix}

Therefore, $μ (D_{d}^{*} (t, t, t^{1})) \leq μ (D_{d}^{*} (t, t, t^{1})) - ψ (D_{d}^{*} (t, t, t^{1})) ⟹ ψ (D_{d}^{*} (t, t, t^{1})) = 0$ , as a result $D_{d}^{*} (t, t, t^{1}) = 0$ , that is, $t = t^{1}$ . Consequently, $S, ℚ, F F & GJ$ 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 $S, ℚ, F, F, G, J$ .

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 $μ (D_{d}^{*} (S x^{*}, ℚ y^{*}, ℚ z^{*})) \leq μ (M (x^{*}, y^{*}, z^{*})) - ψ (M (x^{*}, y^{*}, z^{*})), \forall x^{*}, y^{*}, z^{*} \in X$ , (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 {\begin{matrix} 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^{*}) \end{matrix}}

(ii) If $(S, F)$ and $(ℚ, J)$ are (O. W. C) maps. In this case, maps $S, ℚ, F$ & $J$ have a unique (C.F.P) 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 $D_{d}^{*}$ -sym-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 the discoveries in this manuscript will aid scientists in enhancing the authors on popularized extended symmetric-sp to elevate a universal framework for their practical implementation in all advanced branches of science.

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

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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.

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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 1; peer review: awaiting peer review]. F1000Research 2025, 14:1363 (https://doi.org/10.12688/f1000research.172242.1)

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[1] 1. Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976; 83: 261–263. Publisher Full Text

[2] 2. Sessa S: on a weak commutativity condition of mappings in fixed point considerations. Publ. Inst. Math. 1982; 32: 149–153.

[3] 3. Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986; 9: 771–779. Publisher Full Text

[4] 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] 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] 6. Jungck G, Rhoades BE: fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory. 2006; 7(2): 287–296.

[7] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 27. Wilson WA: On Semi-Metric Spaces. Am. J. Math. 1931; 53(2): 361–373. Publisher Full Text

[28] 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] 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] 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

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

Abstract

Background

Methods and Results

Conclusion

Keywords

1. Introduction

2. Materials and methods

Definition 2.1:

Example 2.2:

Definition 2.3:

Remark 2.4:

Lemma 2.5:

Definition 2.6:

Definition 2.7

Definition 2.8:

Definition 2.9

Remark 2.10:

Definition 2.11:

Definition 2.12:

Definition 2.13:

Remark 2.14:

3. Main results of new extended categories of common fixed point theorems

Remark 3.1:

(1)

Theorem 3.2:

Proof:

(2)

(3)

Corollary 3.3:

Proof:

Lemma 3.4:

Theorem 3.5:

Proof:

(4)

(5)

Corollary 3.6:

Proof:

4. Conclusion

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