Bandar Bin-Mohsin

This article is devoted to the derivation of new inequalities within the theory of fractal–fractional calculus. Its principal contribution is the construction of a new auxiliary identity of fractal–fractional type. By exploiting this identity, we establish a variety of Bullen–Simpson-type inequalities associated with functions whose local fractional derivatives possess generalized harmonic convexity. Moreover, by employing the generalized Hölder and power mean inequalities, we derive supplementary bounds that further enhance the scope of the analysis. The accuracy of the theoretical developments is illustrated through a representative example, together with graphical visualizations that substantiate the consistency of the proposed results. Finally, an application to certain special fractal means is presented in order to demonstrate the effectiveness and practical relevance of the obtained inequalities.

This work developed a class of new integral inequalities within the framework of fractal-fractional calculus. The core contribution was the derivation of a new fractal-fractional auxiliary identity, which provides a unified basis for establishing several Maclaurin-type inequalities. The obtained results were formulated for functions whose local fractional derivatives satisfy generalized harmonic convexity. Furthermore, complementary upper bounds were derived by means of the generalized Hölder inequality and the power mean inequality, thereby extending the scope and flexibility of the proposed estimates. The validity of the theoretical results was illustrated through a representative example and graphical visualizations, which confirm the consistency of the developed inequalities. A feed-forward artificial neural network was also implemented to provide a complementary computational validation of one of the derived estimates by learning the dependence of the upper bound on the Hölder conjugate parameters. An application to selected special fractal means was also included to demonstrate the usefulness and practical significance of the proposed approach.

The diverse decision values may fail to capture an accurate perspective when multiple decision-makers are part of the process. To address this challenge, this work introduces the diamond Pythagorean fuzzy set (Dia‑ PyFS ), an advancement over both the Pythagorean fuzzy ( PyFS ) set and the interval-valued Pythagorean fuzzy set ( IVPyFS ). Due to the established extension of intuitionistic fuzzy sets into Pythagorean fuzzy sets, the Dia‑ PyFS model, a broader form of the diamond intuitionistic fuzzy set model, demonstrates enhanced performance. We then introduce the Dia‑ PyFS as an extension of PyFS . The Diamond Pythagorean fuzzy values that define the elements within Dia‑ PyFS may share a common norm. For Dia‑ PyFS , we define fundamental algebraic and arithmetic operations, including union, intersection, addition, multiplication, and scalar multiplication, and analyze their primary properties. Additionally, we propose some new Dia‑ PyF weighted average and geometric aggregation operators as well as explore their unique properties. We also then propose several algebraic operations between Dia‑ PyFVs using general triangular ???? -norms and ???? -conorms. To transform input values represented by Dia‑ PyFs into a single output value, we also introduce specific weighted aggregation operators based on these algebraic methods. Additionally, the Dia‑ PyFS framework builds upon the “combinative distance-based assess” ( CODAS ) methodology, which relies on both Euclidean and Hamming distances. To illustrate the applicability of this new approach, the feasibility and suitability of the Dia‑ PyFS set approach for choosing the best options are demonstrated by the summary and comparative analysis of the produced reports.

This work focuses on the study of Milne’s inequality in the framework of Katugampola fractional multiplicative integrals, inspired by recent progress in non-Newtonian fractional calculus. We develop a new integral identity related to these operators and employ it to derive Milne-type inequalities for the class of multiplicative differentiable [ ]-convex mappings. Our analysis is further enriched through the application of Hölders inequality and the power mean inequality, which allow us to establish tighter bounds and extended results. To illustrate the validity and applicability of our theoretical approach, we include numerical examples that demonstrate the consistency and importance of the obtained estimates. Finally, we highlight several potential research directions intended to promote future investigations into this evolving domain of mathematical analysis.

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.

The notion of parameter mappings is about creating and managing a structured relationship between parameters across different systems or processes. This concept is vital in ensuring that data remains consistent, correctly interpreted, and accurately transformed as it moves through different parts of a system or between different systems. In this paper, the concept of reference parameter mappings is introduced to handle reference parameters that will help the decision makers. To overcome the uncertainty by giving direct value to reference parameters without any rule, a new class of fuzzy sets is presented which is known as ( q ₁ , q ₂ )‐linear Diophantine fuzzy set (( q ₁ , q ₂ ) L D F S ), where q ₁ and q ₂ are reference parameter mappings. Because the q ₁ and q ₂ can reflect a wider variety of reference parameters than LDFSs and q ‐rung LDFSs, ( q ₁ , q ₂ ) L D F S s can provide more ambiguous conditions. There is symmetry in the values of both the membership grades function and the nonmembership grades function. Furthermore, when discussing the symmetry between two or more objects, the evolution of a (( q ₁ , q ₂ ) L D F S s via q ₁ and q ₂ is more adaptable than the diffused concept of a q ‐rung orthopair fuzzy sets or a LDFSs. The primary benefit of ( q ₁ , q ₂ ) L D F S s , which are useful in a variety of decision‐making situations, is that they are able to characterize a greater number of uncertainties with respect to reference parameter mappings q ₁ and q ₂ than LDFSs. Next, we propose several geometric and averaging operators for a ( q ₁ , q ₂ ) linear Diophantine fuzzy numbers, based on established operating rules. In the latter half of the paper, different ranking algorithms based on proposed aggregation operators are presented to address a realistic assessment of the patient’s high blood pressure conditions is conducted to demonstrate the viability and value of the suggested strategies.

In this study, a new family of robust multiobjective fractional variational control problems was introduced and investigated. To this end, a dual model was associated with the class of problems. Further, by considering variants of convexity for the involved functionals (determined by curvilinear integrals that did not depend on the path), we provided some characterization and equivalence results for the considered models. Moreover, an illustrative numerical example was formulated.