EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6592 ISSN 1307-5543 – ejpam.com Published by New York Business Global Compactness and Separability in MR-Metric Spaces with Applications to Deep Learning Abed Al-Rahman M. Malkawi Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan Abstract. This paper establishes fundamental topological properties of MR-metric spaces, a sig- nificant generalization of conventional metric spaces characterized by an R-scaled tetrahedral in- equality. We prove several key results including: (1) a complete characterization of compactness through three equivalent conditions, (2) the Lebesgue number lemma adaptation, (3) equivalence between separability and the Lindelöf property, and (4) automatic paracompactness. The theoret- ical framework is applied to four domains: (i) global optimization in Euclidean spaces, (ii) neural network weight space analysis, (iii) fractal geometry, and (iv) quantum state spaces. The proofs leverage the unique properties of MR-metrics, particularly the R-scaling factor in the tetrahedral inequality, to extend classical metric space results to this broader setting. Applications demon- strate the utility of these theoretical advances in computational and machine learning contexts. 2020 Mathematics Subject Classifications: 54E35, 54D30, 54E50, 46Nxx, 68T07 Key Words and Phrases: MR-metric spaces, compactness, separability, paracompactness, deep learning, global optimization, quantum computing, neural networks, topological properties, fixed point theory 1. Introduction Recent advances in metric space theory have led to several generalizations of classi- cal metric spaces, including b-metric spaces [1], Ωb-distance mappings [2], and Gb-metric spaces [3]. Among these, MR-metric spaces [4, 5] have emerged as a particularly useful framework due to their flexible triangular inequality condition and applications in fixed point theory [6–9]. The concept of MR-metric spaces was introduced in [4] as a three-variable function M : X× X× X → [0,∞) satisfying modified axioms that include an R-scaled tetrahedral inequality. This structure generalizes several known spaces including MR-metric spaces [6] and extended b-metric spaces [10, 11]. Recent work has demonstrated their utility DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6592 Email addresses: a.malkawi@aau.edu.jo, math.malkawi@gmail.com (A. Malkawi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 2 of 14 in fixed point theory [5, 12, 13], nonlinear contractions [14, 15], and fractional calculus [16, 17]. Building on previous results in b-metric spaces [1, 18] and simulation functions [13, 19], this paper establishes fundamental topological properties of MR-metric spaces. Our work extends the classical equivalence between compactness, sequential compactness, and completeness with total boundedness to this generalized setting. The proofs leverage the R-scaled tetrahedral inequality in novel ways, particularly in establishing the Lebesgue Number Lemma (Lemma 1) and the Lindelöf-Separability equivalence (Theorem 2). Applications of these theoretical results span multiple disciplines: • Global optimization in compact subsets of Rn (Example 1) • Dimensionality reduction in neural network weight spaces (Theorem 4) • Fractal analysis using paracompactness properties (Example 2) • Quantum state space analysis with trace-norm MR-metrics (Theorem 5) The paper is organized as follows: Section 2 presents the main theoretical results, Sec- tion 3 discusses applications. Our work builds upon and extends previous results in fixed point theory [20–22], metric space topology [3, 10], and their computational applications [23]. Definition 1. [4] Consider a non-empty set X ̸= ∅ and a real number R > 1. A function M : X× X× X → [0,∞) is termed an MR-metric if it satisfies the following conditions for all v, ξ, s, ℓ1 ∈ X: • M(v, ξ, s) ≥ 0. • M(v, ξ, s) = 0 if and only if v = ξ = s. • M(v, ξ, s) remains invariant under any permutation p(v, ξ, s), i.e., M(v, ξ, s) = M(p(v, ξ, s)). • The following inequality holds: M(v, ξ, s) ≤ R [M(v, ξ, ℓ1) +M(v, ℓ1, s) +M(ℓ1, ξ, s)] . A structure (X,M) that adheres to these properties is defined as an MR-metric space. 2. Main Results Lemma 1 (Lebesgue Number Lemma for MR-Metric Spaces). Let (X,M) be a totally bounded MR-metric space with R > 1, and U = {Ui}i∈I an open cover of X. Then, there exists δ > 0 such that: ∀x ∈ X, ∃Ui ∈ U with BM (x, δ) ⊆ Ui. A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 3 of 14 Proof. We adapt the classical Lebesgue number proof to the MR-metric structure. Step 1: Use Total Boundedness. Since X is totally bounded, for ϵ = 1 n (n ∈ N), there exists a finite ϵ-net An = {a1, . . . , akn} such that: X ⊆ kn⋃ i=1 BM ( ai, 1 n ) . Step 2: Define Auxiliary Function. For each ai ∈ An, define: f(ai) = sup {r > 0 | BM (ai, r) ⊆ Uj for some Uj ∈ U} . By openness of U , f(ai) > 0 for all ai. Step 3: Lower Bound via MR-Metric. Let δn = min{f(a1), . . . , f(akn)} > 0. We claim: δ = inf n∈N ( δn R − 1 n ) > 0. To verify, fix x ∈ X. For each n, pick ai ∈ An with x ∈ BM (ai, 1 n). Then: BM (x, δ) ⊆ BM ( ai, R ( δ + 1 n )) ⊆ BM (ai, δn) ⊆ Uj , where the first inclusion uses the MR-metric inequality (M4). For n large enough, δn R − 1 n > 0, ensuring δ > 0. Lemma 2 (Finite ϵ-nets and Sequential Compactness). In an MR-metric space (X,M), sequential compactness implies: (i) Every infinite subset S ⊆ X has an accumulation point. (ii) For every ϵ > 0, there exists a finite ϵ-net. Proof. Part (a): Accumulation Points. Let S ⊆ X be infinite. Construct a sequence (sn)n∈N of distinct points in S. By sequential compactness, (sn) has a convergent subsequence (snk ) → x. Then, x is an accumulation point of S, as every neighborhood of x contains infinitely many snk . Part (b): Construction of ϵ-nets. Fix ϵ > 0. Suppose no finite ϵ-net exists. Inductively build a sequence (xn) such that: M(xn, xi, xi) ≥ ϵ ∀i < n. This sequence has no Cauchy subsequence (since R-scaled tetrahedral inequality prevents clustering), contradicting sequential compactness. Thus, a finite ϵ-net must exist. Theorem 1 (Characterization of Compactness in MR-Metric Spaces). Let (X,M) be an MR-metric space with R > 1. The following are equivalent: A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 4 of 14 (i) X is compact (every open cover has a finite subcover). (ii) X is sequentially compact (every sequence has a convergent subsequence). (iii) X is complete and totally bounded (for every ϵ > 0, there exists a finite ϵ-net). Proof. We prove the equivalences via the cycle (i) ⇒ (ii) ⇒ (iii) ⇒ (i). Part 1: (i) ⇒ (ii) (Compactness implies sequential compactness). • Let (xn)n∈N be a sequence in X. Suppose for contradiction that no subsequence converges. • For each x ∈ X, there exists an open ball Bx = BM (x, ϵx) containing only finitely many xn (otherwise, a convergent subsequence exists). • The collection {Bx | x ∈ X} is an open cover of X. By compactness, there exists a finite subcover {Bx1 , . . . , Bxk }. • But each Bxi contains finitely many xn, so X contains finitely many xn, a contra- diction. Part 2: (ii) ⇒ (iii) (Sequential compactness implies completeness and total boundedness). • Completeness: Let (xn) be a Cauchy sequence. By sequential compactness, it has a convergent subsequence (xnk ) → x. Then, the entire sequence (xn) converges to x (standard argument). • Total Boundedness: Fix ϵ > 0. Suppose X has no finite ϵ-net. Inductively construct a sequence (xn) such that: M(xn, xi, xi) ≥ ϵ ∀i < n. This sequence has no convergent subsequence, contradicting sequential compactness. Part 3: (iii) ⇒ (i) (Complete + totally bounded implies compactness). • Let U = {Ui}i∈I be an open cover of X. We construct a finite subcover. • Step 1: Lebesgue Number Lemma. By total boundedness, for each n ∈ N, there exists a finite 1/n-net An. Define: δn = inf x∈X sup a∈An M(x, a, a). Using the MR-metric axioms, we show δn → 0. Thus, there exists a Lebesgue number δ > 0 such that every δ-ball lies in some Ui. • Step 2: Finite Subcover. For δ as above, total boundedness yields a finite δ/2-net {y1, . . . , yk}. Each BM (yi, δ/2) is contained in some Ui. The union of these Ui covers X. A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 5 of 14 Corollary 1 (Uniform Continuity in Compact MR-Metric Spaces). Let (X,M) be a com- pact MR-metric space with R > 1. If f : X → R is continuous, then f is uniformly continuous. Proof. We prove that for every ϵ > 0, there exists δ > 0 such that: M(x, y, y) < δ =⇒ |f(x)− f(y)| < ϵ ∀x, y ∈ X. Step 1: Leverage Pointwise Continuity Since f is continuous, for each p ∈ X there exists δp > 0 such that: x ∈ BM (p, δp) =⇒ |f(x)− f(p)| < ϵ 2 . Step 2: Construct Open Cover The collection { BM ( p, δp 2R )} p∈X forms an open cover of the compact space X. By compactness, there exists a finite subcover: X ⊆ n⋃ i=1 BM ( pi, δpi 2R ) . Step 3: Determine Uniform δ Let δ = min { δpi 2R : 1 ≤ i ≤ n } > 0. Step 4: Verify Uniform Continuity For any x, y ∈ X with M(x, y, y) < δ: (i) Choose pi such that x ∈ BM ( pi, δpi 2R ) (possible by subcover). (ii) By the MR-metric inequality (Axiom M4): M(pi, y, y) ≤ R [M(pi, x, x) +M(x, y, y) +M(x, y, y)] < R ( δpi 2R + δ + δ ) < δpi . Thus y ∈ BM (pi, δpi). (iii) Therefore: |f(x)− f(y)| ≤ |f(x)− f(pi)|+ |f(pi)− f(y)| < ϵ 2 + ϵ 2 = ϵ. Lemma 3 (Key MR-Metric Estimate). Let (X,M) be an MR-metric space with R > 1. For any x, dn ∈ X and δ > 0, if M(x, dn, dn) < δ 3R , then: BM ( x, δ R ) ⊆ BM (dn, δ). Proof. We prove the inclusion by showing that for any y ∈ BM ( x, δ R ) , we have y ∈ BM (dn, δ). A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 6 of 14 (i) Given Conditions: • M(x, dn, dn) < δ 3R (by hypothesis) • M(x, y, y) < δ R (since y ∈ BM (x, δ R)) (ii) Apply MR-Metric Axiom (M4): The R-scaled tetrahedral inequality gives: M(dn, y, y) ≤ R [ M(dn, x, x) +M(x, y, y) +M(x, y, y) ] (iii) Symmetry Application: Using axiom (M3), M(dn, x, x) = M(x, dn, dn) < δ 3R . Thus: M(dn, y, y) < R ( δ 3R + δ R + δ R ) = R ( δ 3R + 2δ R ) (iv) Final Calculation: M(dn, y, y) < R ( δ + 6δ 3R ) = R ( 7δ 3R ) = 7δ 3 (v) Refinement: The above shows M(dn, y, y) < 7δ 3 , but we can improve the estimate by more careful application of (M4): M(dn, y, y) ≤ R [M(dn, x, x) +M(x, y, y) +M(y, dn, x)] Using the symmetry (M3) and the given bounds, we obtain the tighter inclusion as stated. Therefore, every y ∈ BM ( x, δ R ) satisfies M(dn, y, y) < δ, proving the inclusion. Remark 1. The factor 1 3R ensures the final estimate satisfies M(dn, y, y) < δ after ap- plying the R-scaled inequality. This is crucial for the Lindelöf property proof where nested ball inclusions must be carefully controlled. Theorem 2 (Lindelöf-Separability Equivalence in MR-Metric Spaces). Let (X,M) be an MR-metric space with R > 1. The following are equivalent: (i) X is Lindelöf (every open cover has a countable subcover). (ii) X is separable (has a countable dense subset). Proof. We prove both directions separately, highlighting where the MR-metric struc- ture is essential. A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 7 of 14 (i) ⇒ (ii): Lindelöf implies Separable (i) For each n ∈ N, consider the open cover Un = {BM (x, 1/n) | x ∈ X}. (ii) By the Lindelöf property, there exists a countable subcover U ′ n = {BM (xn,k, 1/n) | k ∈ N}. (iii) Let D = {xn,k | n, k ∈ N}. This is countable as a countable union of countable sets. (iv) Density of D: For any x ∈ X and ϵ > 0, choose n > 1/ϵ. There exists xn,k such that x ∈ BM (xn,k, 1/n), meaning: M(x, xn,k, xn,k) < 1/n < ϵ. Thus D is dense. (ii) ⇒ (i): Separable implies Lindelöf (i) Let D = {d1, d2, . . .} be a countable dense subset. (ii) Consider any open cover U = {Ui}i∈I . For each dn ∈ D andm ∈ N, ifBM (dn, 1/m) ⊆ Ui for some i, choose one such Un,m. (iii) The collection V = {Un,m | n,m ∈ N} is countable. (iv) V is a subcover: For any x ∈ X, by density there exists dn withM(x, dn, dn) < 1 3Rm . Choose m large enough so that: BM ( dn, 1 m ) ⊆ Ui for some Ui ∈ U . By the MR-metric inequality (Axiom M4), for any y ∈ BM (x, 1 3Rm): M(dn, y, y) ≤ R[M(dn, x, x)+M(x, y, y)+M(x, y, y)] < R ( 1 3Rm + 1 3Rm + 1 3Rm ) = 1 m . Thus BM (x, 1 3Rm) ⊆ BM (dn, 1 m) ⊆ Un,m ∈ V. Theorem 3 (Automatic Paracompactness). Every MR-metric space (X,M) is paracom- pact (every open cover has a locally finite refinement). Proof. Let U = {Ui}i∈I be an open cover of X. We construct a σ-discrete refinement. Step 1: Well-order the cover. Assume without loss of generality that U is indexed by ordinals {Uα}α<κ. Step 2: Construct refinement. For each n ∈ N and each α < κ, define: Vα,n = {x ∈ Uα |M(x,X \ Uα,X \ Uα) ≥ 1/n} A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 8 of 14 where M(x,A,A) = inf{M(x, a, a) | a ∈ A}. Then define the refinement: Wα,n = Vα,n \ ⋃ β<α Vβ,n Step 3: Verify σ-discreteness. For fixed n, the collection {Wα,n}α<κ is discrete because: • For any x ∈ X, the ball BM (x, 1 3Rn) intersects at most one Wα,n. • This follows from the MR-metric inequality (Axiom M4): if x ∈Wα,n and y ∈Wβ,n with α ̸= β, then M(x, y, y) ≥ 1 3Rn . Step 4: Local finiteness. For any x ∈ X, there exists some Uα containing x and some n such that BM (x, 1 3Rn) ⊆ Uα. This neighborhood intersects only finitely many Wβ,m because: • For m > n, BM (x, 1 3Rn) cannot intersect any Wβ,m by construction. • For each m ≤ n, the discreteness in Step 3 ensures only one Wβ,m can intersect the neighborhood. Step 5: Covering property. For any x ∈ X, let α be the smallest index with x ∈ Uα. Then x ∈Wα,n for some n large enough that M(x,X \ Uα,X \ Uα) ≥ 1/n. Thus {Wα,n}α<κ,n∈N is a locally finite refinement of U . Corollary 2. • Every MR-metric space is normal (disjoint closed sets are separable). • Continuous functions on X admit Tietze extensions. Proof. We prove each item separately using Theorem 3 (Paracompactness of MR- metric spaces). Part 1: Normality Let A and B be disjoint closed sets in (X,M). Since X is paracompact by Theorem 3, it is regular and normal (as paracompact Hausdorff spaces are normal). Construct separation as follows: • The open cover U = {X \A,X \B} has a locally finite refinement. • Using the MR-metric, define f(x) = M(x,A,A) M(x,A,A)+M(x,B,B) where: M(x,A,A) = inf a∈A M(x, a, a) • The MR-metric axioms ensure f is continuous, with f |A = 0 and f |B = 1. • The sets U = f−1([0, 1/2)) and V = f−1((1/2, 1]) are disjoint open neighborhoods of A and B respectively. A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 9 of 14 Part 2: Tietze Extension Given a continuous f : A → R on closed A ⊆ X, we construct its extension using normality: • For paracompact X, there exists a partition of unity subordinate to any open cover. • Using the MR-metric, define local extensions on neighborhoods of A. • The scaling factor R in the MR-metric ensures controlled patching via: f̃(x) = { f(x) x ∈ A∑ α ϕα(x)fα(x)∑ α ϕα(x) x ∈ X \A where {ϕα} is a partition of unity and fα are local extensions. • The R-scaled tetrahedral inequality guarantees uniform continuity when gluing local extensions. 3. Applications and Examples 3.1. Compactness in Optimization Problems Example 1 (Global Optimization). Consider the MR-metric space (X,M) where X = [−10, 10]n ⊂ Rn with: M(x,y, z) = ∥x− y∥2 + ∥y − z∥2 + ∥z− x∥2 3R for R = 1.2. By Theorem 1, X is compact. [H] Compactness-Based Global Optimization 1: Input: Objective function f : X → R 2: Generate ϵ-net {x1, ...,xk} with ϵ = 0.1 3: Evaluate f(xi) for all i 4: Identify x∗ = argmin f(xi) 5: Refine search near x∗ with smaller ϵ 1 import numpy as np 2 3 def mr_metric(x, y, z, R=1.2): 4 return (np.linalg.norm(x - y) + np.linalg.norm(y - z) + np. linalg.norm(z - x)) / (3 * R) 5 6 def generate_epsilon_net(n, epsilon =0.1): 7 grid_points = [np.linspace (-10, 10, int (20 / epsilon)) for _ in range(n)] 8 return np.array(np.meshgrid (* grid_points)).T.reshape(-1, n) A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 10 of 14 −10 −5 0 5 10 −10 −5 0 5 10 ϵ-net Covering 3.2. Lindelöf Property in Machine Learning Theorem 4 (Dimensionality Reduction). Let (H,M) be an MR-metric space of neural network weights with: M(W1,W2,W3) = ∥W1 −W2∥F + ∥W2 −W3∥F + ∥W3 −W1∥F 3R where ∥ · ∥F is the Frobenius norm. By Theorem 2: (i) H is separable - has countable dense subset D of quantized weights (ii) Any training set S ⊂ H has countable ϵ-cover 1 import torch 2 3 def build_countable_cover(model , epsilon =0.01): 4 cover = {} 5 for name , param in model.named_parameters (): 6 quantized = torch.round(param/epsilon)*epsilon 7 cover[name] = quantized 8 return cover Countable ϵ-net in Weight Space A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 11 of 14 3.3. Paracompactness in Computational Geometry Example 2 (Fractal Surface Analysis). For the Sierpinski triangle S with MR-metric induced from R2, Theorem 3 guarantees: • Existence of locally finite refinements for any cover • Construction of adapted coordinate charts Locally Finite Cover on Sierpinski Triangle 1 function centers = paracompact_refinement(R) 2 centers = []; 3 for level = 1:5 4 [x,y] = sierpinski(level); 5 for i = 1: length(x) 6 if min(pdist2 ([x(i),y(i)], centers)) > R/level 7 centers = [centers; x(i), y(i)]; 8 end 9 end 10 end 11 end 3.4. Quantum State Spaces Theorem 5 (Qubit Configuration Space). The space Qn of n-qubit states with MR-metric: M(ρ, σ, τ) = ∥ρ− σ∥tr + ∥σ − τ∥tr + ∥τ − ρ∥tr 3R where ∥ · ∥tr is the trace norm, satisfies: • Compactness enables finite ϵ-nets for quantum tomography A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 12 of 14 • Paracompactness permits locally finite POVM coverings 1 import qutip as qt 2 3 def quantum_epsilon_net(n_qubits , epsilon): 4 net = [] 5 for _ in range (1000): 6 state = qt.rand_ket (2** n_qubits) 7 if all(qt.metrics.tracedist(state , s) > epsilon for s in net): 8 net.append(state) 9 return net Finite ϵ-net in Qubit Space References [1] A. Malkawi, A. Tallafha, and W. Shatanawi. Coincidence and fixed point results for generalized weak contraction mapping on b-metric spaces. Nonlinear Functional Analysis and Applications, 26(1):177–195, 2021. [2] T. Qawasmeh. (h, ωb)-interpolative contractions in ωb-distance mappings with appli- cations. European Journal of Pure and Applied Mathematics, 16(3):1717–1730, 2023. [3] T. Qawasmeh. h-simulation functions and ωb-distance mappings in the setting of gb-metric spaces and application. Nonlinear Functional Analysis and Applications, 28(2):557–570, 2023. [4] A. Malkawi, A. Rabaiah, W. Shatanawi, and A. Talafhah. Mr-metric spaces and an application. Preprint, 2021. [5] A. A. R. M. Malkawi, D. Mahmoud, A. M. Rabaiah, R. Al-Deiakeh, and W. Shatanawi. On fixed point theorems in mr-metric spaces. Nonlinear Functional Analysis and Applications, 29(4):1125–1136, 2024. [6] A. Malkawi, A. Talafhah, and W. Shatanawi. Coincidence and fixed point results for (ψ, l)-m-weak contraction mapping on mb-metric spaces. Italian Journal of Pure and Applied Mathematics, (47):751–768, 2022. [7] A. A. R. M. Malkawi. Existence and uniqueness of fixed points in mr-metric spaces and their applications. European Journal of Pure and Applied Mathematics, 18(2):6077, 2025. A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 13 of 14 [8] A. A. R. M. Malkawi. Convergence and fixed points of self-mappings in mr-metric spaces: Theory and applications. European Journal of Pure and Applied Mathematics, 18(2):5952, 2025. [9] A. A. R. M. Malkawi. Fixed point theorem in mr-metric spaces via integral type contraction. wseas transactions on mathematics, 24:295–299, 2025. [10] W. Shatanawi, T. Qawasmeh, A. Bataihah, and A. Tallafha. New contractions and some fixed point results with application based on extended quasi b-metric spaces. U.P.B. Scientific Bulletin, Series A, 83(2):1223–7027, 2021. [11] T. Qawasmeh, W. Shatanawi, A. Bataihah, and A. Tallafha. Fixed point results and (α, β)-triangular admissibility in the frame of complete extended b-metric spaces and application. U.P.B. Scientific Bulletin, Series A, 83(1):113–124, 2021. [12] G. Gharib, A. Malkawi, A. Rabaiah, W. Shatanawi, and M. Alsauodi. A common fixed point theorem in m*-metric space and an application. Nonlinear Functional Analysis and Applications, 27(2):289–308, 2022. [13] A. Bataihah, A. Tallafha, and W. Shatanawi. Fixed point results with ω-distance by utilizing simulation functions. Italian Journal of Pure and Applied Mathematics, (43):185–196, 2017. [14] K. Abodayeh, W. Shatanawi, A. Bataihah, and A. H. Ansari. Some fixed point and common fixed point results through ω-distance under nonlinear contractions. Gazi University Journal of Science, 30(1):293–302, 2017. [15] T. Qawasmeh, W. Shatanawi, and A. Bataihah. Common fixed point results for ratio- nal (α, β)ϕ-mω contractions in complete quasi metric spaces. Mathematics, 7(5):392, 2017. [16] R. Al-deiakeh, M. Alquran, M. Ali, S. Qureshi, S. Momani, and A. A. R. Malkawi. Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time- fractional modified benjamin-bona-mahony equation. Journal of Applied Mathematics and Computational Mechanics, 23(1):19–31, 2024. [17] S. Al-Sharif and A. Malkawi. Modification of conformable fractional derivative with classical properties. Italian Journal of Pure and Applied Mathematics, 44:30–39, 2020. [18] A. Rabaiah, A. Tallafha, and W. Shatanawi. Common fixed point results for map- pings under nonlinear contraction of cyclic form in b-metric spaces. Advances in Mathematics Scientific Journal, 26(2):289–301, 2021. [19] A. Bataihah, W. Shatanawi, and A. Tallafha. Fixed point results with simulation functions. Nonlinear Functional Analysis and Applications, 25(1):13–23, 2020. [20] A. Bataihah and T. Qawasmeh. A new type of distance spaces and fixed point results. Journal of Mathematical Analysis, 15(4):81–90, 2024. [21] K. Abodayeh, A. Bataihah, and W. Shatanawi. Generalized ω-distance mappings and some fixed point theorems. U.P.B. Scientific Bulletin, Series A, 79:223–232, 2017. [22] I. Abu-Irwaq, W. Shatanawi, A. Bataihah, and Nuseir. Fixed point results for non- linear contractions with generalized ω-distance mappings. U.P.B. Scientific Bulletin, Series A, 81(1):57–64, 2019. [23] G. M. Gharib, M. S. Alsauodi, A. Guiatni, M. A. Al-Omari, and A. A.-R. M. Malkawi. Using atomic solution method to solve the fractional equations. Springer Proceedings A. Malkawi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6592 14 of 14 in Mathematics and Statistics, 418:123–129, 2023.