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14 pages, 313 KiB

Open AccessArticle

New Generalized Hermite–Hadamard–Mercer’s Type Inequalities Using (k, ψ)-Proportional Fractional Integral Operator

by Henok Desalegn Desta, Eze R. Nwaeze, Tadesse Abdi and Jebessa B. Mijena

Foundations 2023, 3(1), 49-62; https://doi.org/10.3390/foundations3010005 - 11 Jan 2023

Cited by 2 | Viewed by 1563

In this paper, by using Jensen–Mercer’s inequality we obtain Hermite–Hadamard–Mercer’s type inequalities for a convex function employing left-sided

(k, ψ)

-proportional fractional integral operators involving continuous strictly increasing function. Our findings are a generalization of some results that existed in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)

35 pages, 424 KiB

Open AccessReview

A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Fractal Fract. 2022, 6(5), 273; https://doi.org/10.3390/fractalfract6050273 - 18 May 2022

Cited by 11 | Viewed by 2016

Abstract

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville,

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, [...] Read more.

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

22 pages, 375 KiB

Open AccessArticle

Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Foundations 2022, 2(2), 377-398; https://doi.org/10.3390/foundations2020026 - 22 Apr 2022

Cited by 6 | Viewed by 2084

Abstract

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the

ψ

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the

ψ

-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given problem is first converted into an equivalent fixed point problem, which is then solved by means of the standard fixed point theorems. The Banach contraction mapping principle is used to establish the existence of a unique solution, while the Krasnosel’skiĭ and Schaefer fixed point theorems as well as the Leray–Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multivalued analogue of the problem at hand. The existence results for convex- and nonconvex-valued multifunctions are respectively proved by means of the Leray–Schauder nonlinear alternative for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)

19 pages, 877 KiB

Open AccessArticle

(k, ψ)-Proportional Fractional Integral Pólya–Szegö- and Grüss-Type Inequalities

by Tariq A. Aljaaidi, Deepak B. Pachpatte, Mohammed S. Abdo, Thongchai Botmart, Hijaz Ahmad, Mohammed A. Almalahi and Saleh S. Redhwan

Fractal Fract. 2021, 5(4), 172; https://doi.org/10.3390/fractalfract5040172 - 18 Oct 2021

Cited by 16 | Viewed by 2258

Abstract

The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators [...] Read more.

ψ

. Then, we state and prove some of its properties and special cases. With the help of this generalized operator, we investigate some Pólya–Szegö- and Grüss-type fractional integral inequalities. The functions used in this work are bounded by two positive functions to obtain Pólya–Szegö- and Grüss-type k-fractional integral inequalities in a new sense. Moreover, we discuss some new special cases of the Pólya–Szegö- and Grüss-type inequalities through this work. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications)

18 pages, 315 KiB

Open AccessArticle

Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function

by Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom and Yu-Ming Chu

Mathematics 2019, 7(12), 1225; https://doi.org/10.3390/math7121225 - 11 Dec 2019

Cited by 106 | Viewed by 4549

Abstract

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function

Ψ .

The authors prove several inequalities for newly defined GPF-integral with respect to another function

Ψ .

Our [...] Read more.

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function

Ψ .

The authors prove several inequalities for newly defined GPF-integral with respect to another function

Ψ .

Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function

Ψ

and the proportionality index

ς

. Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems. Full article

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