Research

32 pages, 466 KiB

Open AccessArticle

On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

by Mieczysław Cichoń, Hussein A. H. Salem and Wafa Shammakh

Mathematics 2023, 11(13), 2875; https://doi.org/10.3390/math11132875 - 27 Jun 2023

Viewed by 684

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking [...] Read more.

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology. Full article

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

13 pages, 272 KiB

Open AccessArticle

On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

by Chenkuan Li, Reza Saadati, Joshua Beaudin and Andrii Hrytsenko

Mathematics 2023, 11(12), 2752; https://doi.org/10.3390/math11122752 - 17 Jun 2023

Viewed by 588

Abstract

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert [...] Read more.

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA. Full article

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

19 pages, 339 KiB

Open AccessArticle

Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions

by Vassili N. Kolokoltsov

Mathematics 2023, 11(11), 2566; https://doi.org/10.3390/math11112566 - 03 Jun 2023

Viewed by 698

Abstract

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace [...] Read more.

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here, we derive the corresponding limiting equations in the case of position-depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo–Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalizations are analyzed. Full article

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

22 pages, 3024 KiB

Open AccessArticle

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

by Ashish Rayal, Bhagawati Prasad Joshi, Mukesh Pandey and Delfim F. M. Torres

Mathematics 2023, 11(11), 2503; https://doi.org/10.3390/math11112503 - 29 May 2023

Cited by 1 | Viewed by 1024

Abstract

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits [...] Read more.

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science. Full article

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

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17 pages, 351 KiB

Open AccessArticle

Sandwich-Type Theorems for a Family of Non-Bazilevič Functions Involving a q-Analog Integral Operator

by Sarem H. Hadi, Maslina Darus, Firas Ghanim and Alina Alb Lupaş

Mathematics 2023, 11(11), 2479; https://doi.org/10.3390/math11112479 - 28 May 2023

Cited by 3 | Viewed by 851

Abstract

This article presents a new q-analog integral operator, which generalizes the q-Srivastava–Attiya operator. Using this q-analog operator, we define a family of analytic non-Bazilevič functions, denoted as [...] Read more.

This article presents a new q-analog integral operator, which generalizes the q-Srivastava–Attiya operator. Using this q-analog operator, we define a family of analytic non-Bazilevič functions, denoted as

T_{q, τ + 1, u}^{μ} (ϑ, λ, M, N)

. Furthermore, we investigate the differential subordination properties of univalent functions using q-calculus, which includes the best dominance, best subordination, and sandwich-type properties. Our results are proven using specialized techniques in differential subordination theory. Full article

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

27 pages, 489 KiB

Open AccessArticle

q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions

by Wuyang Wang, Khansa Hina Khalid, Akbar Zada, Sana Ben Moussa and Jun Ye

Mathematics 2023, 11(9), 2132; https://doi.org/10.3390/math11092132 - 02 May 2023

Cited by 3 | Viewed by 878

Abstract

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation [...] Read more.

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer’s fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples. Full article

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

11 pages, 1045 KiB

Open AccessArticle

Fractional Stochastic Search Algorithms: Modelling Complex Systems via AI

by Bodo Herzog

Mathematics 2023, 11(9), 2061; https://doi.org/10.3390/math11092061 - 26 Apr 2023

Cited by 1 | Viewed by 850

Abstract

The aim of this article is to establish a stochastic search algorithm for neural networks based on the fractional stochastic processes

{B_{t}^{H}, t \geq 0}

with the Hurst parameter

H \in (0, 1)

. We [...] Read more.

The aim of this article is to establish a stochastic search algorithm for neural networks based on the fractional stochastic processes

{B_{t}^{H}, t \geq 0}

with the Hurst parameter

H \in (0, 1)

. We define and discuss the properties of fractional stochastic processes,

{B_{t}^{H}, t \geq 0}

, which generalize a standard Brownian motion. Fractional stochastic processes capture useful yet different properties in order to simulate real-world phenomena. This approach provides new insights to stochastic gradient descent (SGD) algorithms in machine learning. We exhibit convergence properties for fractional stochastic processes. Full article

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

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13 pages, 308 KiB

Open AccessArticle

Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator

by Adriana Cătaş, Emilia-Rodica Borşa and Sheza M. El-Deeb

Mathematics 2023, 11(7), 1742; https://doi.org/10.3390/math11071742 - 05 Apr 2023

Viewed by 800

Abstract

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the [...] Read more.

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator

J_{μ, p}

. Full article

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

13 pages, 2486 KiB

Open AccessArticle

Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

by A. E. Matouk, T. N. Abdelhameed, D. K. Almutairi, M. A. Abdelkawy and M. A. E. Herzallah

Mathematics 2023, 11(3), 591; https://doi.org/10.3390/math11030591 - 22 Jan 2023

Cited by 3 | Viewed by 1064

Abstract

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for [...] Read more.

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction. Full article

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

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18 pages, 432 KiB

Open AccessArticle

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Muchenedi Hari Kishor, Sharifah E. Alhazmi and Shrideh Al-Omari

Mathematics 2023, 11(1), 159; https://doi.org/10.3390/math11010159 - 28 Dec 2022

Cited by 7 | Viewed by 1322

Abstract

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors [...] Read more.

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions. Full article

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

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16 pages, 616 KiB

Open AccessArticle

Poincaré Map for Discontinuous Fractional Differential Equations

by Ivana Eliašová and Michal Fečkan

Mathematics 2022, 10(23), 4476; https://doi.org/10.3390/math10234476 - 27 Nov 2022

Cited by 1 | Viewed by 895

Abstract

We work with a perturbed fractional differential equation with discontinuous right-hand sides where a discontinuity function crosses a discontinuity boundary transversally. The corresponding Poincaré map in a neighbourhood of a periodic orbit of an unperturbed equation is found. Then, bifurcations of periodic boundary [...] Read more.

We work with a perturbed fractional differential equation with discontinuous right-hand sides where a discontinuity function crosses a discontinuity boundary transversally. The corresponding Poincaré map in a neighbourhood of a periodic orbit of an unperturbed equation is found. Then, bifurcations of periodic boundary solutions are analysed together with a concrete example. Full article

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

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18 pages, 364 KiB

Open AccessArticle

A Generalized Approach of the Gilpin–Ayala Model with Fractional Derivatives under Numerical Simulation

by Manel Amdouni, Jehad Alzabut, Mohammad Esmael Samei, Weerawat Sudsutad and Chatthai Thaiprayoon

Mathematics 2022, 10(19), 3655; https://doi.org/10.3390/math10193655 - 05 Oct 2022

Cited by 5 | Viewed by 1339

Abstract

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we [...] Read more.

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we showed that the classical nonlinear fractional model is bounded. The Banach contraction mapping principle was used to prove that the model has a unique positive asymptotical periodic solution. We provide an example and numerical simulation to inspect the correctness and availability of our essential outcomes. Full article

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

16 pages, 306 KiB

Open AccessArticle

Riemann-Liouville Fractional Inclusions for Convex Functions Using Interval Valued Setting

by Vuk Stojiljković, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby and Stojan Radenović

Mathematics 2022, 10(19), 3491; https://doi.org/10.3390/math10193491 - 24 Sep 2022

Cited by 13 | Viewed by 1193

Abstract

In this work, various fractional convex inequalities of the Hermite–Hadamard type in the interval analysis setting have been established, and new inequalities have been derived thereon. Recently defined p interval-valued convexity is utilized to obtain many new fractional Hermite–Hadamard type convex inequalities. The [...] Read more.

In this work, various fractional convex inequalities of the Hermite–Hadamard type in the interval analysis setting have been established, and new inequalities have been derived thereon. Recently defined p interval-valued convexity is utilized to obtain many new fractional Hermite–Hadamard type convex inequalities. The derived results have been supplemented with suitable numerical examples. Our results generalize some recently reported results in the literature. Full article

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

13 pages, 1101 KiB

Open AccessArticle

Fractional-Order Multivariable Adaptive Control Based on a Nonlinear Scalar Update Law

by Fang Yan, Xiaorong Hou and Tingting Tian

Mathematics 2022, 10(18), 3385; https://doi.org/10.3390/math10183385 - 18 Sep 2022

Cited by 2 | Viewed by 1284

Abstract

This paper proposes a new fractional-order model reference adaptive control (FOMRAC) framework for a fractional-order multivariable system with parameter uncertainty. The designed FOMRAC scheme depends on a fractional-order nonlinear scalar update law. Specifically, the scalar update law does not change as the input–output [...] Read more.

This paper proposes a new fractional-order model reference adaptive control (FOMRAC) framework for a fractional-order multivariable system with parameter uncertainty. The designed FOMRAC scheme depends on a fractional-order nonlinear scalar update law. Specifically, the scalar update law does not change as the input–output dimension changes. The main advantage of the proposed adaptive controller is that only one parameter online update is needed such that the computational burden in the existing FOMRAC can be relieved. Furthermore, we show that all signals in this adaptive scheme are bounded and the mean value of the squared norm of the error converges to zero. Two illustrative numerical examples are presented to demonstrate the efficiency of the proposed control scheme. Full article

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

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12 pages, 537 KiB

Open AccessArticle

New Extensions of the Parameterized Inequalities Based on Riemann–Liouville Fractional Integrals

by Hasan Kara, Hüseyin Budak and Fatih Hezenci

Mathematics 2022, 10(18), 3374; https://doi.org/10.3390/math10183374 - 16 Sep 2022

Cited by 6 | Viewed by 995

Abstract

In this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann–Liouville fractional integral operators and limited second derivative mappings. These established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. [...] Read more.

In this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann–Liouville fractional integral operators and limited second derivative mappings. These established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. Thus, a generalization of many inequalities and new results were obtained. Moreover, some examples of obtained inequalities are given for better understanding by the reader. Furthermore, the theoretical results are supported by graphs in order to illustrate the accuracy of each of the inequalities obtained according to the specific choices of the parameter. Full article

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

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15 pages, 308 KiB

Open AccessArticle

On New Fractional Version of Generalized Hermite-Hadamard Inequalities

by Abd-Allah Hyder, Areej A. Almoneef, Hüseyin Budak and Mohamed A. Barakat

Mathematics 2022, 10(18), 3337; https://doi.org/10.3390/math10183337 - 15 Sep 2022

Cited by 8 | Viewed by 1172

Abstract

In this study, we establish a novel version of Hermite-Hadamard inequalities through neoteric generalized Riemann-Liouville fractional integrals (RLFIs). For functions with the convex absolute values of derivatives, we create a variety of midpoint and trapezoid form inequalities, including the generalized RLFIs. Moreover, multiple [...] Read more.

In this study, we establish a novel version of Hermite-Hadamard inequalities through neoteric generalized Riemann-Liouville fractional integrals (RLFIs). For functions with the convex absolute values of derivatives, we create a variety of midpoint and trapezoid form inequalities, including the generalized RLFIs. Moreover, multiple fractional inequalities can be produced as special cases of the findings of this study. Full article

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

20 pages, 825 KiB

Open AccessArticle

Some New Quantum Hermite–Hadamard Inequalities for Co-Ordinated Convex Functions

by Fongchan Wannalookkhee, Kamsing Nonlaopon, Sotiris K. Ntouyas, Mehmet Zeki Sarikaya, Hüseyin Budak and Muhammad Aamir Ali

Mathematics 2022, 10(12), 1962; https://doi.org/10.3390/math10121962 - 07 Jun 2022

Cited by 4 | Viewed by 1023

Abstract

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via

q_{1}, q_{2}

-integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for [...] Read more.

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via

q_{1}, q_{2}

-integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of

q_{1}, q_{2} \in (0, 1)

. We hope that the readers show their interest in these results. Full article

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

9 pages, 456 KiB

Open AccessArticle

Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators

by Kamsing Nonlaopon, Pshtiwan Othman Mohammed, Y. S. Hamed, Rebwar Salih Muhammad, Aram Bahroz Brzo and Hassen Aydi

Mathematics 2022, 10(10), 1753; https://doi.org/10.3390/math10101753 - 20 May 2022

Cited by 2 | Viewed by 1578

Abstract

In this paper, first, we intend to determine the relationship between the sign of

({Δ_{c_{0}}}^{β} y) (c_{0} + 1),

for

1 < β < 2

, and [...] Read more.

In this paper, first, we intend to determine the relationship between the sign of

({Δ_{c_{0}}}^{β} y) (c_{0} + 1),

for

1 < β < 2

, and

(Δ y) (c_{0} + 1) > 0

, in the case we assume that

({Δ_{c_{0}}}^{β} y) (c_{0} + 1)

is negative. After that, by considering the set

D_{ℓ + 1, θ} \subseteq D_{ℓ, θ}

, which are subsets of

(1, 2)

, we will extend our previous result to make the relationship between the sign of

({Δ_{c_{0}}}^{β} y) (z)

and

(Δ y) (z) > 0

(the monotonicity of y), where

({Δ_{c_{0}}}^{β} y) (z)

will be assumed to be negative for each

z \in N_{c_{0}}^{T} : = {c_{0}, c_{0} + 1, c_{0} + 2, \dots, T}

and some

T \in N_{c_{0}} : = {c_{0}, c_{0} + 1, c_{0} + 2, \dots}

. The last part of this work is devoted to see the possibility of information reduction regarding the monotonicity of y despite the non-positivity of

({Δ_{c_{0}}}^{β} y) (z)

by means of numerical simulation. Full article

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

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20 pages, 341 KiB

Open AccessArticle

Some (p, q)-Integral Inequalities of Hermite–Hadamard Inequalities for (p, q)-Differentiable Convex Functions

by Waewta Luangboon, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas and Hüseyin Budak

Mathematics 2022, 10(5), 826; https://doi.org/10.3390/math10050826 - 04 Mar 2022

Cited by 4 | Viewed by 1857

Abstract

In this paper, we establish a new

{(p, q)}^{b}

-integral identity involving the first-order

{(p, q)}^{b}

-derivative. Then, we use this result to prove some new

{(p, q)}^{b}

-integral inequalities related [...] Read more.

In this paper, we establish a new

{(p, q)}^{b}

-integral identity involving the first-order

{(p, q)}^{b}

-derivative. Then, we use this result to prove some new

{(p, q)}^{b}

-integral inequalities related to Hermite–Hadamard inequalities for

{(p, q)}^{b}

-differentiable convex functions. Furthermore, our main results are used to study some special cases of various integral inequalities. The newly presented results are proven to be generalizations of some integral inequalities of already published results. Finally, some examples are given to illustrate the investigated results. Full article

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

26 pages, 467 KiB

Open AccessEditor’s ChoiceArticle

A Theoretical Analysis of a Fractional Multi-Dimensional System of Boundary Value Problems on the Methylpropane Graph via Fixed Point Technique

by Shahram Rezapour, Chernet Tuge Deressa, Azhar Hussain, Sina Etemad, Reny George and Bashir Ahmad

Mathematics 2022, 10(4), 568; https://doi.org/10.3390/math10040568 - 12 Feb 2022

Cited by 18 | Viewed by 1794

Abstract

Few studies have investigated the existence and uniqueness of solutions for fractional differential equations on star graphs until now. The published papers on the topic are based on the assumption of existence of one junction node and some boundary nodes as the origin [...] Read more.

Few studies have investigated the existence and uniqueness of solutions for fractional differential equations on star graphs until now. The published papers on the topic are based on the assumption of existence of one junction node and some boundary nodes as the origin on a star graph. These structures are special cases and do not cover more general non-star graph structures. In this paper, we state a labeling method for graph vertices, and then we prove the existence results for solutions to a new family of fractional boundary value problems (FBVPs) on the methylpropane graph. We design the chemical compound of the methylpropane graph with vertices specified by 0 or 1, and on every edge of the graph, we consider fractional differential equations. We prove the existence of solutions for the proposed FBVPs by means of the Krasnoselskii’s and Scheafer’s fixed point theorems, and further, we study the Ulam–Hyers type stability for the given multi-dimensional system. Finally, we provide an illustrative example to examine our results. Full article

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

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14 pages, 325 KiB

Open AccessArticle

Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in q-Calculus

by Dafang Zhao, Ghazala Gulshan, Muhammad Aamir Ali and Kamsing Nonlaopon

Mathematics 2022, 10(3), 444; https://doi.org/10.3390/math10030444 - 29 Jan 2022

Cited by 6 | Viewed by 1823

Abstract

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter

m \in [0, 1]

. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities [...] Read more.

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter

m \in [0, 1]

. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable

(α, m)

-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field. Full article

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

22 pages, 899 KiB

Open AccessArticle

Some New Hermite-Hadamard-Fejér Fractional Type Inequalities for h-Convex and Harmonically h-Convex Interval-Valued Functions

by Humaira Kalsoom, Muhammad Amer Latif, Zareen A. Khan and Miguel Vivas-Cortez

Mathematics 2022, 10(1), 74; https://doi.org/10.3390/math10010074 - 26 Dec 2021

Cited by 23 | Viewed by 2541

Abstract

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function

\overset{˘}{Υ}

using an another function

ϑ (\dot{ζ})

. As an additional observation, it is noted that the new class of weighted interval-valued [...] Read more.

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function

\overset{˘}{Υ}

using an another function

ϑ (\dot{ζ})

. As an additional observation, it is noted that the new class of weighted interval-valued fractional integrals of a function

\overset{˘}{Υ}

by employing an additional function

ϑ (\dot{ζ})

characterizes a variety of new classes as special cases, which is a generalization of the previous class. Secondly, we prove a new version of the Hermite-Hadamard-Fejér type inequality for h-convex interval-valued functions using weighted interval-valued fractional integrals of a function

\overset{˘}{Υ}

according to another function

ϑ (\dot{ζ}) .

Finally, by using weighted interval-valued fractional integrals of a function

\overset{˘}{Υ}

according to another function

ϑ (\dot{ζ})

, we are establishing a new Hermite-Hadamard-Fejér type inequality for harmonically h-convex interval-valued functions that is not previously known. Moreover, some examples are provided to demonstrate our results. Full article

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

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14 pages, 309 KiB

Open AccessArticle

On Some Generalized Simpson’s and Newton’s Inequalities for (α, m)-Convex Functions in q-Calculus

by Ifra Bashir Sial, Sun Mei, Muhammad Aamir Ali and Kamsing Nonlaopon

Mathematics 2021, 9(24), 3266; https://doi.org/10.3390/math9243266 - 16 Dec 2021

Cited by 21 | Viewed by 1723

Abstract

In this paper, we first establish two right-quantum integral equalities involving a right-quantum derivative and a parameter

m \in [0, 1]

. Then, we prove modified versions of Simpson’s and Newton’s type inequalities using established equalities for right-quantum differentiable [...] Read more.

In this paper, we first establish two right-quantum integral equalities involving a right-quantum derivative and a parameter

m \in [0, 1]

. Then, we prove modified versions of Simpson’s and Newton’s type inequalities using established equalities for right-quantum differentiable

(α, m)

-convex functions. The newly developed inequalities are also proven to be expansions of comparable inequalities found in the literature. Full article

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

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