Integral inequalities play a crucial role in various areas of numerical analysis, particularly n the development of numerical integration formulas and numerical methods for differential equations. In the context of numerical methods for differential equations, these inequalities can be beneficial in approximating integrals during finite volume schemes over control volumes. This study introduces new integral and derivative operators in qq-calculus, which serve as generalizations of existing operators. The fundamental properties and connections of these new operators with the existing ones are discussed in detail. To demonstrate the significance and validity of these new concepts, several novel integral inequalities for convex functions are derived. These inequalities can be helpful in finding error bounds for numerical integration formulas, thus enhancing the accuracy and reliability of numerical analysis techniques. The findings of this research contribute to the ongoing development of qq-calculus and its applications in the field of numerical analysis, particularly in the areas of numerical integration and numerical methods for differential equations.

In this manuscript, we explore the concept of strong-coupled fixed points in the context of intuitionistic fuzzy metric spaces (IFMS). Our approach is grounded in the idea of intuitionistic fuzzy contractive couplings (IFCCs), which provide a framework for understanding fixed points in fuzzy settings. We begin by introducing a novel formulation of coupling, which combines the principles of coupled fuzzy contractions with cyclic mappings. This combination leads to a more generalized and effective method of identifying strong-coupled fixed points, extending previous results in fuzzy metric spaces. A key contribution to this paper is the proof of the existence of a unique strong-coupled fixed point. We establish this result through rigorous theoretical analysis and provide a corollary that strengthens the foundation of our work. Several non-trivial examples are presented to demonstrate the applicability of the theory and the robustness of the strong-coupled fixed point in various scenarios. Additionally, we present a practical application of our findings: the construction of a strong-coupled fractal set within the framework of intuitionistic fuzzy metric spaces. This is achieved by applying an intuitionistic iterated function system (IIFS), which is based on a family of intuitionistic fuzzy contractive couplings. The fractal generation process is illustrated through several examples, demonstrating the theoretical results in action. To further solidify the applicability of our approach, we introduce an intuitionistic fuzzy version of the Hausdorff distance between compact sets, a crucial tool in measuring the "closeness" of sets within the intuitionistic fuzzy context. Several examples are provided to clarify the fractal generation process, showing how the intuitionistic fuzzy metrics and couplings contribute to the creation of self-similar fractals. This work not only enhances the understanding of fixed points in intuitionistic fuzzy spaces but also provides new insights into their application in fractal geometry, offering both theoretical advancements and practical tools for future research in this area.

In this research, we generalize the fixed-point theorem for fuzzy Fisher contraction in fuzzy double controlled metric space, a newly developed mathematical structure that expands ordinary metric spaces and by integrating control functions. The interplay of fuzzy Fisher contraction and fuzzy double controlled metric space provides a new paradigm for investigating contraction mappings and their applications in fractal theory. Using fuzzy Fisher contraction in this generalized fuzzy metric spaces framework, we define a novel class of fractal set known as fuzzy double controlled Fisher fractals and discuss the fuzzy double controlled Fisher iterated function system that is the generalization classical iterative function system in fuzzy double controlled metric space. We develop the Collage theorem for fuzzy double controlled Fisher fractals, which is a useful tool for approximation in fractal generation. This theorem extends the usual collage theorem by incorporating the flexibility of fuzzy double controlled metric conditions, yielding more generalized approximation results. In addition, we focus on non-trivial cases including the graphical behavior of fuzzy double controlled Fisher contraction.

This study uses Raina’s function to obtain a new coordinated pq-integral identity. Using this identity, we construct several new pq-Simpson’s type inequalities for generalized convex functions on coordinates. Setting p1=p2=1 in these inequalities yields well-known quantum Simpson’s type inequalities for coordinated generalized convex functions. Our results have important implications for the creation of post quantum mathematical frameworks.

In this work, the authors introduce the concept of neutrosophic semi-metric spaces and prove several common fixed-point theorems for countable and uncountable family of mappings via an implicit relation of contractive and integral type by utilizing locally integrable functions. These results improve and generalize the several results in the existing literature. Further, the authors present some non-trivial examples to support our main results. Mathematics Subject Classification: 46S40, 47H10, 54H25.

The aim of this paper is to obtain optimum fuzzy soft constants through multi-criteria decision-making approaches. TOPSIS and VIKOR are utilized for this purpose and results are compared with those obtained through Bonferroni mean. The hesitant fuzzy soft set is taken as initial data in decision-making methods. Hesitant fuzzy Bonferroni means and distance measures for TOPSIS and VIKOR are calculated in the structure of hesitant fuzzy set and hesitant fuzzy soft set. OFSCs are chosen from the constants which rank the alternatives in the decision-making process. By using the system of linear differential equations based on OFSCs, the future approach of people with respect to their decisions is analyzed and is observed through phase portrait and a line graph of that system of differential equations. To explain the proposed idea, explanative examples for two techniques are also given. These examples illustrate that if two persons select the two different alternatives, they do not favor each other after that decision.

The main focus of this article is to derive some new counterparts to Simpson’s and Newton’s type inequalities involve a class of generalized coordinated convex mappings. This class contains several new and known classes of convexity as special cases. For further demonstration, we deploy the concept of right quantum derivatives to develop two new identities involving Raina’s function. Moreover, by implementing these auxiliary results together with generalized convexity, we acquire a Holder-type inequality. We also acquire some applications of our main findings by making use of suitable substitutions in Raina’s function.

In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results.

In this study, first we establish a p,q-integral identity involving the second p,q-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.

In this research, we give a generalized version of the quantum Montgomery identity using the quantum integral. We establish some new inequalities of Ostrowski type by means of newly derived identity. Moreover, we consider the special cases of the newly obtained results and prove several new and known Ostrowski and midpoint inequalities.

In this paper, we prove two identities involving quantum derivatives, quantum integrals, and certain parameters. Using the newly proved identities, we prove new inequalities of Simpson’s and Newton’s type for quantum differentiable convex functions under certain assumptions. Moreover, we discuss the special cases of our main results and obtain some new and existing Simpson’s type inequalities, Newton’s type inequalities, midpoint type inequalities and trapezoidal type inequalities.

Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and symmetry, has various applications for quantum calculus. Inequalities has a strong association with convex and symmetric convex functions. In this study, first we establish a p,q-integral identity involving the second p,q-derivative and then we used this result to prove some new trapezoidal type inequalities for twice p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of some existing results in the field of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

Let G=G1×G2×⋯×Gm be the strong product of simple, finite connected graphs, and let ϕ:ℕ⟶0,∞ be an increasing function. We consider the action of generalized maximal operator MGϕ on ℓp spaces. We determine the exact value of ℓp-quasi-norm of MGϕ for the case when G is strong product of complete graphs, where 0

In this research, we offer two new quantum integral equalities for recently defined q(epsilon 2)-integral and derivative, the derived equalities then used to prove quantum integral inequalities of Simpson's and Newton's type for preinvex functions. We also considered the special cases of established results and offer several new and existing results inside the literature of Simpson's and Newton's type inequalities.

In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for p,q-differentiable preinvex functions in the context of p,q-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

The core aim of this study is to propose a novel computational procedure, namely, Elzaki transform iterative method to work out two-dimensional nonlinear time-fractional Zakharov–Kuznetsov equation numerically. We execute the suggested iterative procedure on two models and results are presented graphically in the form of surface plot and absolute error is compared with the VIM and HPM to show that the method is more powerful than VIM and HPM and deduce that the offered numerical pattern is more efficient in simulating linear and nonlinear fractional order models.

In this paper, we prove some new Simpson's type inequalities for partial $ (p, q) $-differentiable convex functions of two variables in the context of $ (p, q) $-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.

In this research, we introduce the notions of $(p,q)$ -derivative and integral for interval-valued functions and discuss their fundamental properties. After that, we prove some new inequalities of Hermite–Hadamard type for interval-valued convex functions employing the newly defined integral and derivative. Moreover, we find the estimates for the newly proved inequalities of Hermite–Hadamard type. It is also shown that the results proved in this study are the generalization of some already proved research in the field of Hermite–Hadamard inequalities.

In this work, we introduce the notion of interval-valued coordinated convexity and demonstrate Hermite–Hadamard type inequalities for interval-valued convex functions on the co-ordinates in a rectangle from the plane. Moreover, we prove Hermite–Hadamard inequalities for the product of interval-valued convex functions on coordinates. Our results generalize several other well-known inequalities given in the existing literature on this subject.

The aim of this paper is to initiat formal study of hypersoft sets. We first, present basic operations like union, intersection and difference of hypersoft sets; basic ingrediants for topological structures on the collection of hypersoft sets. Moreover we introduce hypersoft points in different envorinments like fuzzy hypersoft set, intuitionistic fuzzy hypersoft set, neutrosophic hypersoft, plithogenic hypersoft set, and give some basic properties of hypersoft points in these envorinments. We expect that this will constitue an appropriate framework of hypersoft functions and the study of hypersoft function spaces. Examples are provided to explain the newly defined concepts.