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17 pages, 358 KiB

Open AccessArticle

A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces

by Abdellatif Boutiara, Sina Etemad, Sabri T. M. Thabet, Sotiris K. Ntouyas, Shahram Rezapour and Jessada Tariboon

Symmetry 2023, 15(5), 1041; https://doi.org/10.3390/sym15051041 - 8 May 2023

Cited by 25 | Viewed by 2247

Abstract

In this paper, we study a coupled fully hybrid system of

(k, Φ)

–Hilfer fractional differential equations equipped with non-symmetric

(k, Φ)

–Riemann-Liouville (

RL

) integral conditions. To prove the existence and uniqueness results, we use [...] Read more.

In this paper, we study a coupled fully hybrid system of

(k, Φ)

–Hilfer fractional differential equations equipped with non-symmetric

(k, Φ)

–Riemann-Liouville (

RL

) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (

GBS

). Moreover, the Ulam–Hyers (

UH

) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)

26 pages, 422 KiB

Open AccessArticle

Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus

by Muhammad Bilal Khan, Gustavo Santos-García, Hatim Ghazi Zaini, Savin Treanță and Mohamed S. Soliman

Mathematics 2022, 10(4), 534; https://doi.org/10.3390/math10040534 - 9 Feb 2022

Cited by 12 | Viewed by 3090

Abstract

In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by

φ

-cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results. Full article

(This article belongs to the Special Issue Fuzzy Sets and Soft Computing)

12 pages, 793 KiB

Open AccessArticle

Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

by Pshtiwan Othman Mohammed, Thabet Abdeljawad, Shengda Zeng and Artion Kashuri

Symmetry 2020, 12(9), 1485; https://doi.org/10.3390/sym12091485 - 9 Sep 2020

Cited by 37 | Viewed by 2962

Abstract

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the

\overset{˘}{σ}

-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a

\overset{˘}{σ}

-convex function with respect to an increasing function involving the

φ

-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

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