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21 pages, 674 KiB

Open AccessArticle

Properties and Applications of Symmetric Quantum Calculus

by Miguel Vivas-Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Silvestru Sever Dragomir and Ahmed M. Zidan

Fractal Fract. 2024, 8(2), 107; https://doi.org/10.3390/fractalfract8020107 - 12 Feb 2024

Cited by 7 | Viewed by 2763

Abstract

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and [...] Read more.

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)

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17 pages, 2267 KiB

Open AccessArticle

Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme

by Alicia Cordero, Renso V. Rojas-Hiciano, Juan R. Torregrosa and Maria P. Vassileva

Fractal Fract. 2023, 7(6), 459; https://doi.org/10.3390/fractalfract7060459 - 3 Jun 2023

Cited by 6 | Viewed by 1435

Abstract

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the [...] Read more.

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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17 pages, 345 KiB

Open AccessArticle

Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions

by Waqar Afzal, Evgeniy Yu. Prosviryakov, Sheza M. El-Deeb and Yahya Almalki

Symmetry 2023, 15(4), 831; https://doi.org/10.3390/sym15040831 - 30 Mar 2023

Cited by 19 | Viewed by 2006

Abstract

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn [...] Read more.

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard (

H . H

), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types. Full article

(This article belongs to the Special Issue Symmetry in CFD: Convection, Diffusion and Dynamics)

15 pages, 727 KiB

Open AccessArticle

Performance of a New Sixth-Order Class of Iterative Schemes for Solving Non-Linear Systems of Equations

by Marlon Moscoso-Martínez, Francisco I. Chicharro, Alicia Cordero and Juan R. Torregrosa

Mathematics 2023, 11(6), 1374; https://doi.org/10.3390/math11061374 - 12 Mar 2023

Cited by 2 | Viewed by 1755

Abstract

This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous [...] Read more.

This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous step and employing a divided difference operator, resulting in a triparametric scheme with a convergence order of four. The convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family. We performed dynamic and numerical development to analyze the stability of the sixth-order family. Previous studies for scalar functions allow us to isolate those elements of the family with stable performance for solving practical problems. In this regard, we present dynamical planes showing the complexity of the family. In addition, the numerical properties of the class are analyzed with several test problems. Full article

(This article belongs to the Special Issue Numerical Analysis and Modeling)

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15 pages, 858 KiB

Open AccessArticle

On Ostrowski–Mercer’s Type Fractional Inequalities for Convex Functions and Applications

by Soubhagya Kumar Sahoo, Artion Kashuri, Munirah Aljuaid, Soumyarani Mishra and Manuel De La Sen

Fractal Fract. 2023, 7(3), 215; https://doi.org/10.3390/fractalfract7030215 - 25 Feb 2023

Cited by 8 | Viewed by 1998

Abstract

This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main findings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special [...] Read more.

This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main findings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special means, q-digamma functions and modified Bessel functions, some applications of the acquired results were obtained. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)

24 pages, 491 KiB

Open AccessArticle

New Hermite–Hadamard and Ostrowski-Type Inequalities for Newly Introduced Co-Ordinated Convexity with Respect to a Pair of Functions

by Muhammad Aamir Ali, Fongchan Wannalookkhee, Hüseyin Budak, Sina Etemad and Shahram Rezapour

Mathematics 2022, 10(19), 3469; https://doi.org/10.3390/math10193469 - 23 Sep 2022

Cited by 3 | Viewed by 1308

Abstract

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls [...] Read more.

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls under the umbrella of convexity. However, it is a significant theory that affects practically all areas of mathematics. In this paper, we introduce the notions of

(g, h)

-convexity or convexity with respect to a pair of functions on co-ordinates and discuss its fundamental properties. Moreover, we establish some novel Hermite–Hadamard- and Ostrowski-type inequalities for newly introduced co-ordinated convexity. Additionally, it is presented that the newly introduced notion of the convexity and given inequalities are generalizations of existing studies in the literature. Lastly, we look at various mathematical examples and graphs to confirm the validity of the newly found inequalities. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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21 pages, 350 KiB

Open AccessArticle

Some New Mathematical Integral Inequalities Pertaining to Generalized Harmonic Convexity with Applications

by Muhammad Tariq, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh and Kamsing Nonlaopon

Mathematics 2022, 10(18), 3286; https://doi.org/10.3390/math10183286 - 10 Sep 2022

Viewed by 1387

Abstract

The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and [...] Read more.

The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and contemporary times because of their applications in various branches of sciences. In this work, we reveal the idea of a new version of generalized harmonic convexity i.e., an m–polynomial p–harmonic s–type convex function. We discuss this new idea by employing some examples and demonstrating some interesting algebraic properties. Furthermore, this work leads us to establish some new generalized Hermite–Hadamard- and generalized Ostrowski-type integral identities. Additionally, employing Hölder’s inequality and the power-mean inequality, we present some refinements of the H–H (Hermite–Hadamard) inequality and Ostrowski inequalities. Finally, we investigate some applications to special means involving the established results. These new results yield us some generalizations of the prior results in the literature. We believe that the methodology and concept examined in this paper will further inspire interested researchers. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

9 pages, 266 KiB

Open AccessArticle

Ćirić-Type Operators and Common Fixed Point Theorems

by Claudia Luminiţa Mihiţ, Ghiocel Moţ and Gabriela Petruşel

Mathematics 2022, 10(11), 1947; https://doi.org/10.3390/math10111947 - 6 Jun 2022

Cited by 1 | Viewed by 1831

Abstract

In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we [...] Read more.

In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we establish existence, uniqueness and approximation results for the common fixed point. In the same framework, the second problem is to study various stability properties. More precisely, we will obtain sufficient conditions assuring that the common fixed point problem is well-posed and has the Ulam–Hyers stability, as well as the Ostrowski property for the considered problem. Some examples and applications are finally given in order to illustrate the abstract theorems proposed in the first part of the paper. Our results extend and complement some theorems in the recent literature. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)

23 pages, 858 KiB

Open AccessArticle

New Generalized Class of Convex Functions and Some Related Integral Inequalities

by Artion Kashuri, Ravi P. Agarwal, Pshtiwan Othman Mohammed, Kamsing Nonlaopon, Khadijah M. Abualnaja and Yasser S. Hamed

Symmetry 2022, 14(4), 722; https://doi.org/10.3390/sym14040722 - 2 Apr 2022

Cited by 9 | Viewed by 2334

Abstract

There is a strong correlation between convexity and symmetry concepts. In this study, we investigated the new generic class of functions called the

(n, m)

–generalized convex and studied its basic algebraic properties. The Hermite–Hadamard inequality for the [...] Read more.

There is a strong correlation between convexity and symmetry concepts. In this study, we investigated the new generic class of functions called the

(n, m)

–generalized convex and studied its basic algebraic properties. The Hermite–Hadamard inequality for the

(n, m)

–generalized convex function, for the products of two functions and of this type, were proven. Moreover, this class of functions was applied to several known identities; midpoint-type inequalities of Ostrowski and Simpson were derived. Our results are extensions of many previous contributions related to integral inequalities via different convexities. Full article

(This article belongs to the Special Issue Symmetry in the Mathematical Inequalities)

20 pages, 327 KiB

Open AccessArticle

q₁q₂-Ostrowski-Type Integral Inequalities Involving Property of Generalized Higher-Order Strongly n-Polynomial Preinvexity

by Humaira Kalsoom and Miguel Vivas-Cortez

Symmetry 2022, 14(4), 717; https://doi.org/10.3390/sym14040717 - 1 Apr 2022

Cited by 7 | Viewed by 2137

Abstract

Quantum calculus has numerous applications in mathematics. This novel class of functions may be used to produce a variety of conclusions in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. It can drive additional research in a variety of [...] Read more.

Quantum calculus has numerous applications in mathematics. This novel class of functions may be used to produce a variety of conclusions in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. It can drive additional research in a variety of pure and applied fields. This article’s main objective is to introduce and study a new class of preinvex functions, which is called higher-order generalized strongly n-polynomial preinvex function. We derive a new

q_{1} q_{2}

-integral identity for mixed partial

q_{1} q_{2}

-differentiable functions. Because of the nature of generalized convexity theory, there is a strong link between preinvexity and symmetry. Utilizing this as an auxiliary result, we derive some estimates of upper bound for functions whose mixed partial

q_{1} q_{2}

-differentiable functions are higher-order generalized strongly n-polynomial preinvex functions on co-ordinates. Our results are the generalizations of the results in earlier papers. Quantum inequalities of this type and the techniques used to solve them have applications in a wide range of fields where symmetry is important. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

19 pages, 1403 KiB

Open AccessArticle

The Dynamical Analysis of a Biparametric Family of Six-Order Ostrowski-Type Method under the Möbius Conjugacy Map

by Xiaofeng Wang and Xiaohe Chen

Fractal Fract. 2022, 6(3), 174; https://doi.org/10.3390/fractalfract6030174 - 21 Mar 2022

Cited by 5 | Viewed by 1952

Abstract

In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter [...] Read more.

In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter spaces related to the critical points and dynamic planes are used to visualize their dynamic properties. Eventually, we find the most stable member of the biparametric family of six-order Ostrowski-type methods. Some test equations are examined for supporting the theoretical results. Full article

(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)

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14 pages, 305 KiB

Open AccessArticle

Refinements of Ostrowski Type Integral Inequalities Involving Atangana–Baleanu Fractional Integral Operator

by Hijaz Ahmad, Muhammad Tariq, Soubhagya Kumar Sahoo, Sameh Askar, Ahmed E. Abouelregal and Khaled Mohamed Khedher

Symmetry 2021, 13(11), 2059; https://doi.org/10.3390/sym13112059 - 1 Nov 2021

Cited by 26 | Viewed by 2230

Abstract

In this article, first, we deduce an equality involving the Atangana–Baleanu (

AB

)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality [...] Read more.

In this article, first, we deduce an equality involving the Atangana–Baleanu (

AB

)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of

| Υ |

. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus. Full article

(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications II)

18 pages, 350 KiB

Open AccessArticle

New Ostrowski-Type Fractional Integral Inequalities via Generalized Exponential-Type Convex Functions and Applications

by Soubhagya Kumar Sahoo, Muhammad Tariq, Hijaz Ahmad, Jamshed Nasir, Hassen Aydi and Aiman Mukheimer

Symmetry 2021, 13(8), 1429; https://doi.org/10.3390/sym13081429 - 4 Aug 2021

Cited by 28 | Viewed by 2827

Abstract

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for [...] Read more.

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature. Full article

(This article belongs to the Special Issue Symmetry in the Mathematical Inequalities)

20 pages, 292 KiB

Open AccessArticle

Fractional Weighted Ostrowski-Type Inequalities and Their Applications

by Artion Kashuri, Badreddine Meftah, Pshtiwan Othman Mohammed, Alina Alb Lupaş, Bahaaeldin Abdalla, Y. S. Hamed and Thabet Abdeljawad

Symmetry 2021, 13(6), 968; https://doi.org/10.3390/sym13060968 - 29 May 2021

Cited by 21 | Viewed by 2748

Abstract

An important area in the field of applied and pure mathematics is the integral inequality. As it is known, inequalities aim to develop different mathematical methods. Nowadays, we need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. [...] Read more.

An important area in the field of applied and pure mathematics is the integral inequality. As it is known, inequalities aim to develop different mathematical methods. Nowadays, we need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition and its properties. Furthermore, there is a strong correlation between convexity and symmetry concepts. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the last few years. In this study, by using a new identity, we establish some new fractional weighted Ostrowski-type inequalities for differentiable quasi-convex functions. Further, further results for functions with a bounded first derivative are given. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtain. The obtained results generalize and refine certain known results. Full article

(This article belongs to the Section Mathematics)

20 pages, 271 KiB

Open AccessArticle

Multiple Diamond-Alpha Integral in General Form and Their Properties, Applications

by Zhong-Xuan Mao, Ya-Ru Zhu, Jun-Ping Hou, Chun-Ping Ma and Shi-Pu Liu

Mathematics 2021, 9(10), 1123; https://doi.org/10.3390/math9101123 - 15 May 2021

Cited by 3 | Viewed by 1921

Abstract

In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it [...] Read more.

In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper. Full article

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