Special Issue "Recent Advances in Fractional Differential Equations and Inclusions"

A special issue of Foundations (ISSN 2673-9321). This special issue belongs to the section "Mathematical Sciences".

Deadline for manuscript submissions: 30 September 2022 | Viewed by 6549

Special Issue Editor

Special Issue Information

Dear Colleagues,

Fractional calculus is a generalization of classical calculus to an arbitrary real order and has evolved as an interesting and important area of research. Fractional differential equations have attracted much attention in literature because some real-world problems in physics, mechanics, engineering, game theory, stability, optimal control, and other fields can be described better with the help of fractional differential equations. Fractional differential equations and inclusions constitute a significant branch of nonlinear analysis.

This Special Issue invites papers that focus on recent and original research results of fractional differential equations and inclusions.

Prof. Dr. Sotiris K. Ntouyas
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Foundations is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fractional calculus
  • Fractional derivatives
  • Fractional integrals
  • Fractional differential equations
  • Fractional differential inclusions
  • Fractional boundary value problems
  • Fractional order nonlinear systems
  • Fractional integral equations
  • Fractional differences
  • Fractional inequalities

Published Papers (9 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Article
Simpson’s Type Inequalities for s-Convex Functions via a Generalized Proportional Fractional Integral
Foundations 2022, 2(3), 607-616; https://doi.org/10.3390/foundations2030041 - 25 Jul 2022
Viewed by 198
Abstract
In this paper, we give new Simpson’s type integral inequalities for the class of functions whose derivatives of absolute values are s-convex via generalized proportional fractional integrals. Some results in the literature are particular cases of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions
Foundations 2022, 2(2), 377-398; https://doi.org/10.3390/foundations2020026 - 22 Apr 2022
Viewed by 339
Abstract
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 [...] Read more.
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)
Article
Generalized Fractional Integrals Involving Product of a Generalized Mittag–Leffler Function and Two H-Functions
Foundations 2022, 2(1), 298-307; https://doi.org/10.3390/foundations2010021 - 11 Mar 2022
Viewed by 594
Abstract
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative [...] Read more.
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative examples for the efficiency of the general approach of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
A Note on a Coupled System of Hilfer Fractional Differential Inclusions
Foundations 2022, 2(1), 290-297; https://doi.org/10.3390/foundations2010020 - 03 Mar 2022
Cited by 1 | Viewed by 496
Abstract
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we [...] Read more.
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we avoid the applications of fixed point theorems as usual. An illustration of the results is given by a suitable example. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof
Foundations 2022, 2(1), 209-217; https://doi.org/10.3390/foundations2010013 - 07 Feb 2022
Viewed by 636
Abstract
In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Show Figures

Figure 1

Article
Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation
Foundations 2022, 2(1), 167-183; https://doi.org/10.3390/foundations2010010 - 25 Jan 2022
Cited by 2 | Viewed by 808
Abstract
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and [...] Read more.
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions
Foundations 2022, 2(1), 151-166; https://doi.org/10.3390/foundations2010009 - 21 Jan 2022
Cited by 1 | Viewed by 815
Abstract
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a [...] Read more.
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient requirements for at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method
Foundations 2022, 2(1), 6-19; https://doi.org/10.3390/foundations2010002 - 28 Dec 2021
Viewed by 655
Abstract
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also [...] Read more.
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Show Figures

Figure 1

Article
Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem
Foundations 2021, 1(2), 286-303; https://doi.org/10.3390/foundations1020021 - 03 Dec 2021
Viewed by 883
Abstract
The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems [...] Read more.
The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of βG, ψG contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Show Figures

Figure 1

Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Nonlocal $\psi$-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions
Authors: Sotiris K. Ntouyas; Bashir Ahmad; Jessada Tariboon
Affiliation: Department of Mathematics
Abstract: In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving $\psi$-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multi-point, fractional integral multi-order and derivative multi-order 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'ski\u{i} and Schaefer fixed point theorems as well as Leray-Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multi-valued analogue of the problem at hand. The existence results for convex and non-convex valued multifunctions are respectively proved by means of Leray-Schauder nonlinear alternative for multi valued maps, and Covitz-Nadler's fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented.

Title: Solvability of a parametric fractional order integral equation utilizing advance Darbo G-contraction theorem
Authors: Vishal Nikam; Dhananjay Gopal; Rabha W. Ibrahim
Affiliation: Institute of Electrical and Electronics Engineers
Abstract: The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed point theorems. The intention of this work is to prove fixed point theorems for the class of β−G, ψ−G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process some recent results of Jleli et al. have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann-Liouville formula. Moreover, we introduce numerical solutions of the class by utilizing the set of fixed points.

Title: FIXED POINTS OF MULTIVALUED MAPPINGS USEFUL IN THE THEORY OF DIFFERENTIAL AND RANDOM DIFFERENTIAL INCLUSIONS
Authors: Lech Górniewicz
Affiliation: Faculty of Mathematics and Computer Science ,Nicolaus Copernicus University ,87-100 Toruń , POLAND
Abstract: Fixed point theory is very useful in nonlinear analysis , diferential equations , differential and random differential inclusions .It is well known that different types of fixed points implies the existence of specific solutions of the respective problem concrning differential equations or inclusions (see:[7,8,9,10,11,12,21,25,28,29,30] ). There are several classifications of fixed points for single valued mappings.Recall that in 1949 M.K. Fort [19 ] introduced the notion of essential fixed points. In 1965 F.E . Browder [13,14 ] introduced the notions of ejective and repulsive fixed points .In 1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappings of subsets of the Euclidean space Rn . For more information see also: [ 16, 19, 18 , 120, 22, 20,23,25,26, 27] . Note that for multivalued mappings these problems were considered only in a few papers (see:[ 1,2,3, 4 .5,6,7,15,21,24,32]);always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s). In this paper are studied ejective , repulsive and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) . Let as remark that the class of MANR-s is much larger as the class of ANR-s (see: [26]) . In order to study the above notions we generalize the fixed point index from the case of ANR-s onto the case of ANMR-s.Next using the above fixed point index we are able to prove several new results concerning repulsive ejective and essential fixed points of admissible multivalued mappings . Moreover the random case is mentioned. For possible applications to differential and random differential inclusions see: [7,,8,9,10,11,12]

Title: Analytical and Qualitative Study of Some Families ofFODEs Via Differential Transform Method
Authors: Kamal Shah; Neelma .
Affiliation: Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan 3
Abstract: This research work, we develop qualitative theory along with analytical solutions to some families of fractional order differential equations (FODEs). To achieve the concerned goal, we apply differential transform method (DTM) and few fixed point results of Banach and Schauder. Also by some proper examples we demonstrate the results

Title: Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions
Authors: Jagan Mohan Jonnalagadda; Gopal N S
Affiliation: Birla Institute of Technology and Science Pilani
Abstract: In this work, we consider the following two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions: \begin{equation*} \label{BVP N} \begin{cases} -\big{(}\nabla^{\kappa}_{\rho(a)} u\big{)}(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\ u(a) = g_1(u),\quad u(b) = g_2(u). \end{cases} \end{equation*} First, we derive the associated Green's function and some of its properties. Using the Guo--Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient conditions on the existence of at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results.

Title: Analytical study of a ϕ- fractional order quadratic functional integral equation
Authors: Ahmed El-Sayed; Hind Hashem; Shorouk Al-issa
Affiliation: Faculty of Science, Alexandria University, 21544 Alexandria, Egypt
Abstract: Here we shall study the existence of continuous solutions of a ϕ−fractional-orders quadratic functional integral equation, establish some properties of the solutions, prove the existence of maximal and minimal solutions of that quadratic integral equation. Finally, we introduce some particular cases.

Title: Maximum and anti-maximum principles for boundary value problems for Riemann-Liouville di erential equations in neighborhoods of simple eigen- values
Authors: Paul Eloe
Affiliation: Department of Mathematics, University of Dayton300 College Park,Dayton, OH 45469-2316,USA,Phone:(937) 229-2016

Title: On fractional Lyapunov functions of nonlinear dynamicsystems and Mittag-Leffler stability
Authors: Attiq ul Rehman; Ram Singh; Praveen Agarwal
Affiliation: Anand ICE
Abstract: In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is proposed. The global stability of the epidemic model at an equilibrium state is established

Title: GENERALIZED FRACTIONAL INTEGRALS INVOLVING PRODUCTOF A GENERALIZED MITTAG-LEFFLER FUNCTION AND TWOH-FUNCTIONS
Authors: Prakash Singh; Shilpi Jain; PRAVEEN AGARWAL
Affiliation: Poornima College of Engineering
Abstract: The objective of this research is to obtain generalized fractional integral formulas concerning a generalized Mittag-Lefer function and two H-functions. These integral formulas are described in the expressions of the generalized multivariate H functions.

Title: Detailed Error Analysis for the Fractional Adam's Method for the Caputo Hadamard Derivative
Authors: Charles Wing Ho Green and Yubin Yan
Affiliation: University of Chester

Back to TopTop