Axioms

Research

35 pages, 394 KiB

Open AccessArticle

Multivariate Approximation Using Symmetrized and Perturbed Hyperbolic Tangent-Activated Multidimensional Convolution-Type Operators

by George A. Anastassiou

Axioms 2024, 13(11), 779; https://doi.org/10.3390/axioms13110779 - 11 Nov 2024

Viewed by 687

Abstract

In this article, we introduce, for the first time, multivariate symmetrized and perturbed hyperbolic tangent-activated convolution-type operators in three forms. We present their approximation properties, that is, their quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with [...] Read more.

In this article, we introduce, for the first time, multivariate symmetrized and perturbed hyperbolic tangent-activated convolution-type operators in three forms. We present their approximation properties, that is, their quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with the multivariate global smoothness preservation of these operators. We present, in detail, the related multivariate iterative approximation, as well as, multivariate simultaneous approximation, and their combinations. Using differentiability in our research, we produce higher rates of approximation, and multivariate simultaneous global smoothness preservation is also achieved. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

20 pages, 338 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

by Roger Arnau, Enrique A. Sánchez Pérez and Sergi Sanjuan

Axioms 2024, 13(11), 760; https://doi.org/10.3390/axioms13110760 - 2 Nov 2024

Viewed by 816

Abstract

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells [...] Read more.

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space

M^{#}

defined by the real-valued Lipschitz functions on

M .

However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of

M^{#}

for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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33 pages, 404 KiB

Open AccessArticle

Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

by Hanan Alohali, Valer-Daniel Breaz, Omar Mutab Alsalami, Luminita-Ioana Cotirla and Ahmed Alamer

Axioms 2024, 13(10), 684; https://doi.org/10.3390/axioms13100684 - 2 Oct 2024

Viewed by 688

Abstract

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We [...] Read more.

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex FNVMs) and examine the idea of C-ƛ-pre-invex FNVMs in this paper. Furthermore, using C-ƛ-pre-invex FNVMs, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

17 pages, 294 KiB

Open AccessArticle

New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components

by Can Kızılateş, Wei-Shih Du, Nazlıhan Terzioğlu and Ren-Chuen Chen

Axioms 2024, 13(10), 677; https://doi.org/10.3390/axioms13100677 - 30 Sep 2024

Viewed by 795

Abstract

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain [...] Read more.

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

25 pages, 4236 KiB

Open AccessArticle

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

by Zizhao Zhou, Ahmad Aziz Al Ahmadi, Alina Alb Lupas and Khalil Hadi Hakami

Axioms 2024, 13(10), 666; https://doi.org/10.3390/axioms13100666 - 26 Sep 2024

Cited by 1 | Viewed by 958

Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and [...] Read more.

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

19 pages, 345 KiB

Open AccessArticle

Research on New Interval-Valued Fractional Integrals with Exponential Kernel and Their Applications

by Abdulrahman F. Aljohani, Ali Althobaiti and Saad Althobaiti

Axioms 2024, 13(9), 616; https://doi.org/10.3390/axioms13090616 - 11 Sep 2024

Viewed by 706

Abstract

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (

H H

),

H H

–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (

L R I V H

convex mappings) with an exponential function in [...] Read more.

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (

H H

),

H H

–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (

L R I V H

convex mappings) with an exponential function in the kernel. We use fractional operators to develop several generalizations, capturing unique outcomes that are currently under investigation, while also introducing a new operator. Generally, we propose two methods that, in conjunction with more generalized fractional integral operators with an exponential function in the kernel, can address certain novel generalizations of increasing mappings under the assumption of

L R I V

convexity, yielding some noteworthy results. The results produced by applying the suggested scheme show that the computational effects are extremely accurate, flexible, efficient, and simple to implement in order to explore the path of upcoming intricate waveform and circuit theory research. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

20 pages, 376 KiB

Open AccessArticle

Geometric Characterization of Validity of the Lyapunov Convexity Theorem in the Plane for Two Controls under a Pointwise State Constraint

by Clara Carlota, Mário Lopes and António Ornelas

Axioms 2024, 13(9), 611; https://doi.org/10.3390/axioms13090611 - 9 Sep 2024

Viewed by 1301

Abstract

This paper concerns control BVPs, driven by ODEs

x^{'} (t) = u (t)

, using controls

u^{0} (\cdot)

& u^{1} (\cdot)

in

L^{1} ((a, b), R^{2})

. We ask these two controls to satisfy a [...] Read more.

This paper concerns control BVPs, driven by ODEs

x^{'} (t) = u (t)

, using controls

u^{0} (\cdot)

& u^{1} (\cdot)

in

L^{1} ((a, b), R^{2})

. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write

(u^{1} - u^{0}) (\cdot) = v (\cdot) (1, f (\cdot))

for some

f (\cdot)

. Given a relaxed non bang-bang solution

\bar{x} (\cdot) \in W^{1, 1} ([a, b], R^{2})

, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution

\hat{x} (\cdot)

having lower first-coordinate

{\hat{x}}_{1} (\cdot) \leq {\bar{x}}_{1} (\cdot)

? Being the answer always yes in dimension

d = 1

, hence without

f (\cdot)

, as proved by Amar and Cellina, for

d = 2

the problem is to find out which functions

f (\cdot)

“are good”, namely “allow such 1-lower bang-bang solution

\hat{x} (\cdot)

to exist”. The aim of this paper is to characterize “goodness of

f (\cdot)

” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given

f (\cdot)

is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

31 pages, 1687 KiB

Open AccessArticle

Some Classical Inequalities Associated with Generic Identity and Applications

by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin, Hüseyin Budak and Silvestru Sever Dragomir

Axioms 2024, 13(8), 533; https://doi.org/10.3390/axioms13080533 - 6 Aug 2024

Cited by 1 | Viewed by 944

Abstract

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s [...] Read more.

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of

γ

and parameter

ξ

. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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27 pages, 1378 KiB

Open AccessArticle

Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications

by Miguel Vivas Cortez, Ali Althobaiti, Abdulrahman F. Aljohani and Saad Althobaiti

Axioms 2024, 13(7), 471; https://doi.org/10.3390/axioms13070471 - 12 Jul 2024

Viewed by 878

Abstract

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up [...] Read more.

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down (

U

·

D

) relations and over newly defined class

U

·

D

-ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of

U

·

D

-relations, recent findings have been developed using fuzzy Aumman’s, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

21 pages, 487 KiB

Open AccessArticle

Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions

by Loredana Ciurdariu and Eugenia Grecu

Axioms 2024, 13(6), 413; https://doi.org/10.3390/axioms13060413 - 19 Jun 2024

Cited by 3 | Viewed by 884

Abstract

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four [...] Read more.

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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14 pages, 268 KiB

Open AccessArticle

Weak Sharp Type Solutions for Some Variational Integral Inequalities

by Savin Treanţă and Tareq Saeed

Axioms 2024, 13(4), 225; https://doi.org/10.3390/axioms13040225 - 28 Mar 2024

Cited by 1 | Viewed by 1058

Abstract

Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided. [...] Read more.

Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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