Special Issue "Fractional Differential Equations, Inclusions and Inequalities with Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 March 2020.

Special Issue Editor

Prof. Dr. Sotiris K. Ntouyas
E-Mail Website
Guest Editor
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Interests: initial and boundary value problems for differential equations and inclusions; inequalities
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

During the last few decades, fractional differential equations, inclusions, and inequalities have been studied extensively. As a matter of fact, fractional derivatives and integrals provide a much better tool for the description of memory and hereditary properties of various materials and processes than integer derivatives. Engineers and scientists have developed new precise models which involve fractional differential equations and inequalities. These models have been applied successfully, e.g., in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc.

The purpose of this Special Issue is to establish a collection of articles that reflect the latest mathematical developments in the field of fractional differential equations, inclusions, and inequalities with their applications.

Prof. Sotiris K. Ntouyas
Guest Editor

Manuscript Submission Information

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Keywords

  • Fractional differential equations
  • Fractional differential inclusions
  • Fractional inequalities
  • Boundary value problem
  • Existence

Published Papers (20 papers)

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Research

Open AccessArticle
Positive Solutions for a System of Fractional Integral Boundary Value Problems of Riemann–Liouville Type Involving Semipositone Nonlinearities
Mathematics 2019, 7(10), 970; https://doi.org/10.3390/math7100970 - 14 Oct 2019
Abstract
In this work by the index of fixed point and matrix theory, we discuss the positive solutions for the system of Riemann–Liouville type fractional boundary value problems D0+αu(t)+f1(t,u( [...] Read more.
In this work by the index of fixed point and matrix theory, we discuss the positive solutions for the system of Riemann–Liouville type fractional boundary value problems D 0 + α u ( t ) + f 1 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ( 0 , 1 ) , D 0 + α v ( t ) + f 2 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ( 0 , 1 ) , D 0 + α w ( t ) + f 3 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 0 ) = = u ( n 2 ) ( 0 ) = 0 , D 0 + p u ( t ) | t = 1 = 0 1 h ( t ) D 0 + q u ( t ) d t , v ( 0 ) = v ( 0 ) = = v ( n 2 ) ( 0 ) = 0 , D 0 + p v ( t ) | t = 1 = 0 1 h ( t ) D 0 + q v ( t ) d t , w ( 0 ) = w ( 0 ) = = w ( n 2 ) ( 0 ) = 0 , D 0 + p w ( t ) | t = 1 = 0 1 h ( t ) D 0 + q w ( t ) d t , where α ( n 1 , n ] with n N , n 3 , p , q R with p [ 1 , n 2 ] , q [ 0 , p ] , D 0 + α is the α order Riemann–Liouville type fractional derivative, and f i ( i = 1 , 2 , 3 ) C ( [ 0 , 1 ] × R + × R + × R + , R ) are semipositone nonlinearities. Full article
Open AccessArticle
Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems
Mathematics 2019, 7(10), 961; https://doi.org/10.3390/math7100961 - 13 Oct 2019
Abstract
In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results [...] Read more.
In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results generalize those obtained by Wei et al., (Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 2019, 87, 101–107.) to the fractional setting. Full article
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Open AccessArticle
A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations
Mathematics 2019, 7(10), 949; https://doi.org/10.3390/math7100949 - 12 Oct 2019
Abstract
The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to [...] Read more.
The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method. Full article
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Open AccessArticle
Stability Results for a Coupled System of Impulsive Fractional Differential Equations
Mathematics 2019, 7(10), 927; https://doi.org/10.3390/math7100927 - 06 Oct 2019
Abstract
In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s fixed point [...] Read more.
In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s fixed point theorem. The main theoretical results are well illustrated with the help of an example. Full article
Open AccessFeature PaperArticle
Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel
Mathematics 2019, 7(9), 845; https://doi.org/10.3390/math7090845 - 12 Sep 2019
Abstract
In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving [...] Read more.
In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented. Full article
Open AccessArticle
Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications
Mathematics 2019, 7(9), 807; https://doi.org/10.3390/math7090807 - 02 Sep 2019
Cited by 2
Abstract
In the present paper, we investigate some Hermite-Hadamard (HH) inequalities related to generalized Riemann-Liouville fractional integral (GRLFI) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex [...] Read more.
In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means. Full article
Open AccessArticle
Ostrowski Type Inequalities Involving ψ-Hilfer Fractional Integrals
Mathematics 2019, 7(9), 770; https://doi.org/10.3390/math7090770 - 21 Aug 2019
Abstract
In this study we introduce several new Ostrowski-type inequalities for both left and right sided fractional integrals of a function g with respect to another function ψ. Our results generalized the ones presented previously by Farid. Furthermore, two illustrative examples are presented [...] Read more.
In this study we introduce several new Ostrowski-type inequalities for both left and right sided fractional integrals of a function g with respect to another function ψ . Our results generalized the ones presented previously by Farid. Furthermore, two illustrative examples are presented to support our results. Full article
Open AccessArticle
Existence Theory for a Fractional q-Integro-Difference Equation with q-Integral Boundary Conditions of Different Orders
Mathematics 2019, 7(8), 659; https://doi.org/10.3390/math7080659 - 24 Jul 2019
Abstract
In this paper, we study the existence of solutions for a new class of fractional q-integro-difference equations involving Riemann-Liouville q-derivatives and a q-integral of different orders, supplemented with boundary conditions containing q-integrals of different orders. The first existence result [...] Read more.
In this paper, we study the existence of solutions for a new class of fractional q-integro-difference equations involving Riemann-Liouville q-derivatives and a q-integral of different orders, supplemented with boundary conditions containing q-integrals of different orders. The first existence result is obtained by means of Krasnoselskii’s fixed point theorem, while the second one relies on a Leray-Schauder nonlinear alternative. The uniqueness result is derived via the Banach contraction mapping principle. Finally, illustrative examples are presented to show the validity of the obtained results. The paper concludes with some interesting observations. Full article
Open AccessArticle
Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative
Mathematics 2019, 7(7), 630; https://doi.org/10.3390/math7070630 - 15 Jul 2019
Abstract
In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ-Riesz-Caputo fractional derivative. A new definition of ψ-Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear [...] Read more.
In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results. Full article
Open AccessArticle
The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral
Mathematics 2019, 7(6), 533; https://doi.org/10.3390/math7060533 - 11 Jun 2019
Abstract
In this paper, we establish sufficient conditions for the existence of solutions for a nonlinear Langevin equation based on Liouville-Caputo-type generalized fractional differential operators of different orders, supplemented with nonlocal boundary conditions involving a generalized integral operator. The modern techniques of functional analysis [...] Read more.
In this paper, we establish sufficient conditions for the existence of solutions for a nonlinear Langevin equation based on Liouville-Caputo-type generalized fractional differential operators of different orders, supplemented with nonlocal boundary conditions involving a generalized integral operator. The modern techniques of functional analysis are employed to obtain the desired results. The paper concludes with illustrative examples. Full article
Open AccessArticle
Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem
Mathematics 2019, 7(6), 526; https://doi.org/10.3390/math7060526 - 10 Jun 2019
Abstract
Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term [...] Read more.
Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term of the first order derivative ( a C D α x ) ( t ) = f ( t , x ( t ) , x ( t ) ) , a < t < b , 1 < α < 2 , x ( a ) = 0 , x ( b ) = μ x ( η ) , a < η < b , μ > λ , where λ = b a η a and a C D α denotes the Caputo’s fractional derivative, and f : [ a , b ] × R × R R is a continuous function satisfying the certain conditions. Full article
Open AccessArticle
Existence Result and Uniqueness for Some Fractional Problem
Mathematics 2019, 7(6), 516; https://doi.org/10.3390/math7060516 - 05 Jun 2019
Abstract
In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the [...] Read more.
In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the main results, some examples are presented. Full article
Open AccessArticle
On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders
Mathematics 2019, 7(5), 471; https://doi.org/10.3390/math7050471 - 24 May 2019
Abstract
In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional [...] Read more.
In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in n + 1 fractional differences and m + 1 fractional sums. Finally, we present an illustrative example. Full article
Open AccessArticle
Stability, Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Differential Equations
Mathematics 2019, 7(4), 354; https://doi.org/10.3390/math7040354 - 16 Apr 2019
Cited by 2
Abstract
The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the [...] Read more.
The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. Obtained results are supported by examples and illustrated in the last section. Full article
Open AccessArticle
Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative
Mathematics 2019, 7(4), 333; https://doi.org/10.3390/math7040333 - 05 Apr 2019
Cited by 1
Abstract
In this paper, the Hyers–Ulam stability of linear Caputo–Fabrizio fractional differential equation is established using the Laplace transform method. We also derive a generalized Hyers–Ulam stability result via the Gronwall inequality. In addition, we establish existence and uniqueness of solutions for nonlinear Caputo–Fabrizio [...] Read more.
In this paper, the Hyers–Ulam stability of linear Caputo–Fabrizio fractional differential equation is established using the Laplace transform method. We also derive a generalized Hyers–Ulam stability result via the Gronwall inequality. In addition, we establish existence and uniqueness of solutions for nonlinear Caputo–Fabrizio fractional differential equations using the generalized Banach fixed point theorem and Schaefer’s fixed point theorem. Finally, two examples are given to illustrate our main results. Full article
Open AccessArticle
Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points
Mathematics 2019, 7(4), 331; https://doi.org/10.3390/math7040331 - 04 Apr 2019
Cited by 2
Abstract
This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some [...] Read more.
This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results. Full article
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Open AccessArticle
A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line
Mathematics 2019, 7(3), 256; https://doi.org/10.3390/math7030256 - 12 Mar 2019
Cited by 1
Abstract
In this article, we propose a coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line and study its existence result by using Schauder’s fixed point theorem. An example is provided to illustrate the results. Full article
Open AccessArticle
Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann-Stieltjes Integro-Multipoint Boundary Conditions
Mathematics 2019, 7(3), 249; https://doi.org/10.3390/math7030249 - 11 Mar 2019
Cited by 1
Abstract
We discuss the existence and uniqueness of solutions for a Caputo-type fractional order boundary value problem equipped with non-conjugate Riemann-Stieltjes integro-multipoint boundary conditions on an arbitrary domain. Modern tools of functional analysis are applied to obtain the main results. Examples are constructed for [...] Read more.
We discuss the existence and uniqueness of solutions for a Caputo-type fractional order boundary value problem equipped with non-conjugate Riemann-Stieltjes integro-multipoint boundary conditions on an arbitrary domain. Modern tools of functional analysis are applied to obtain the main results. Examples are constructed for the illustration of the derived results. We also investigate different kinds of Ulam stability, such as Ulam-Hyers stability, generalized Ulam-Hyers stability, and Ulam-Hyers-Rassias stability for the problem at hand. Full article
Open AccessArticle
On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions
Mathematics 2019, 7(3), 223; https://doi.org/10.3390/math7030223 - 27 Feb 2019
Abstract
This manuscript is devoted to establishing existence theory of solutions to a nonlinear coupled system of fractional order differential equations (FODEs) under integral boundary conditions (IBCs). For uniqueness and existence we use the Perov-type fixed point theorem. Further, to investigate multiplicity results of [...] Read more.
This manuscript is devoted to establishing existence theory of solutions to a nonlinear coupled system of fractional order differential equations (FODEs) under integral boundary conditions (IBCs). For uniqueness and existence we use the Perov-type fixed point theorem. Further, to investigate multiplicity results of the concerned problem, we utilize Krasnoselskii’s fixed-point theorems of cone type and its various forms. Stability analysis is an important aspect of existence theory as well as required during numerical simulations and optimization of FODEs. Therefore by using techniques of functional analysis, we establish conditions for Hyers-Ulam (HU) stability results for the solution of the proposed problem. The whole analysis is justified by providing suitable examples to illustrate our established results. Full article
Open AccessArticle
Nonlocal Fractional Evolution Inclusions of Order α ∈ (1,2)
Mathematics 2019, 7(2), 209; https://doi.org/10.3390/math7020209 - 24 Feb 2019
Abstract
This paper studies the existence of mild solutions and the compactness of a set of mild solutions to a nonlocal problem of fractional evolution inclusions of order α(1,2). The main tools of our study include the [...] Read more.
This paper studies the existence of mild solutions and the compactness of a set of mild solutions to a nonlocal problem of fractional evolution inclusions of order α ( 1 , 2 ) . The main tools of our study include the concepts of fractional calculus, multivalued analysis, the cosine family, method of measure of noncompactness, and fixed-point theorem. As an application, we apply the obtained results to a control problem. Full article
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