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Fractional Differential Equations, Inclusions and Inequalities with Applications
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Fractional Differential Equations, Inclusions and Inequalities with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 March 2020) | Viewed by 85641

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Sotiris K. Ntouyas

grade E-Mail Website
Guest Editor
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Interests: initial value problems; boundary value problems; inequalities
Special Issues, Collections and Topics 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

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Keywords

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

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Published Papers (33 papers)

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Research

10 pages, 256 KiB  
Article
Some Fractional Hermite–Hadamard Type Inequalities for Interval-Valued Functions
by Fangfang Shi, Guoju Ye, Dafang Zhao and Wei Liu
Mathematics 2020, 8(4), 534; https://doi.org/10.3390/math8040534 - 4 Apr 2020
Cited by 12 | Viewed by 2074
Abstract
In this paper, firstly we prove the relationship between interval h-convex functions and interval harmonically h-convex functions. Secondly, several new Hermite–Hadamard type inequalities for interval h-convex functions via interval Riemann–Liouville type fractional integrals are established. Finally, we obtain some new [...] Read more.
In this paper, firstly we prove the relationship between interval h-convex functions and interval harmonically h-convex functions. Secondly, several new Hermite–Hadamard type inequalities for interval h-convex functions via interval Riemann–Liouville type fractional integrals are established. Finally, we obtain some new fractional Hadamard–Hermite type inequalities for interval harmonically h-convex functions by using the above relationship. Also we discuss the importance of our results and some special cases. Our results extend and improve some previously known results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
14 pages, 292 KiB  
Article
Certain Hadamard Proportional Fractional Integral Inequalities
by Gauhar Rahman, Kottakkaran Sooppy Nisar and Thabet Abdeljawad
Mathematics 2020, 8(4), 504; https://doi.org/10.3390/math8040504 - 2 Apr 2020
Cited by 24 | Viewed by 2150
Abstract
In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of [...] Read more.
In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of n ( n ∈ N ) positive functions by utilizing Hadamard proportional fractional integral operator. The inequalities presented in this paper are more general than the inequalities existing in the literature. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
13 pages, 267 KiB  
Article
On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations
by Jiraporn Reunsumrit and Thanin Sitthiwirattham
Mathematics 2020, 8(4), 476; https://doi.org/10.3390/math8040476 - 31 Mar 2020
Cited by 1 | Viewed by 1711
Abstract
In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in [...] Read more.
In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
15 pages, 280 KiB  
Article
Some Fractional Dynamic Inequalities of Hardy’s Type via Conformable Calculus
by Samir Saker, Mohammed Kenawy, Ghada AlNemer and Mohammed Zakarya
Mathematics 2020, 8(3), 434; https://doi.org/10.3390/math8030434 - 16 Mar 2020
Cited by 28 | Viewed by 2733
Abstract
In this article, we prove some new fractional dynamic inequalities on time scales via conformable calculus. By using chain rule and Hölder’s inequality on timescales we establish the main results. When α = 1 we obtain some well-known time-scale inequalities due to Hardy, [...] Read more.
In this article, we prove some new fractional dynamic inequalities on time scales via conformable calculus. By using chain rule and Hölder’s inequality on timescales we establish the main results. When α = 1 we obtain some well-known time-scale inequalities due to Hardy, Copson, Bennett and Leindler inequalities. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
12 pages, 273 KiB  
Article
Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay
by Ekaterina Madamlieva, Mihail Konstantinov, Marian Milev and Milena Petkova
Mathematics 2020, 8(3), 364; https://doi.org/10.3390/math8030364 - 6 Mar 2020
Cited by 8 | Viewed by 2173
Abstract
The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the [...] Read more.
The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
13 pages, 260 KiB  
Article
Nonlinear Integro-Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals with Nonlocal Boundary Data
by Bashir Ahmad, Abrar Broom, Ahmed Alsaedi and Sotiris K. Ntouyas
Mathematics 2020, 8(3), 336; https://doi.org/10.3390/math8030336 - 3 Mar 2020
Cited by 22 | Viewed by 2615
Abstract
In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann–Liouville fractional derivatives and Riemann–Liouville fractional integrals of different orders. Our results rely on the standard tools [...] Read more.
In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann–Liouville fractional derivatives and Riemann–Liouville fractional integrals of different orders. Our results rely on the standard tools of functional analysis. Examples are constructed to demonstrate the application of the derived results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
11 pages, 268 KiB  
Article
Certain Fractional Proportional Integral Inequalities via Convex Functions
by Gauhar Rahman, Kottakkaran Sooppy Nisar, Thabet Abdeljawad and Samee Ullah
Mathematics 2020, 8(2), 222; https://doi.org/10.3390/math8020222 - 9 Feb 2020
Cited by 18 | Viewed by 2340
Abstract
The goal of this article is to establish some fractional proportional integral inequalities for convex functions by employing proportional fractional integral operators. In addition, we establish some classical integral inequalities as the special cases of our main findings. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
12 pages, 295 KiB  
Article
Fractional q-Difference Inclusions in Banach Spaces
by Badr Alqahtani, Saïd Abbas, Mouffak Benchohra and Sara Salem Alzaid
Mathematics 2020, 8(1), 91; https://doi.org/10.3390/math8010091 - 6 Jan 2020
Cited by 6 | Viewed by 2322
Abstract
In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Mönch’s fixed point theorems). Finally we give [...] Read more.
In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Mönch’s fixed point theorems). Finally we give an illustrative example in the last section. We initiate the study of fractional q-difference inclusions on infinite dimensional Banach spaces. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
11 pages, 266 KiB  
Article
Integral Inequalities for s-Convexity via Generalized Fractional Integrals on Fractal Sets
by Ohud Almutairi and Adem Kılıçman
Mathematics 2020, 8(1), 53; https://doi.org/10.3390/math8010053 - 1 Jan 2020
Cited by 11 | Viewed by 2255
Abstract
In this study, we establish new integral inequalities of the Hermite–Hadamard type for s-convexity via the Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann–Liouville into a single form. We show that the new integral inequalities of Hermite–Hadamard type can [...] Read more.
In this study, we establish new integral inequalities of the Hermite–Hadamard type for s-convexity via the Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann–Liouville into a single form. We show that the new integral inequalities of Hermite–Hadamard type can be obtained via the Riemann–Liouville fractional integral. Finally, we give some applications to special means. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
13 pages, 293 KiB  
Article
On Neutral Functional Differential Inclusions involving Hadamard Fractional Derivatives
by Bashir Ahmad, Ahmed Alsaedi, Sotiris K. Ntouyas and Hamed H. Al-Sulami
Mathematics 2019, 7(11), 1084; https://doi.org/10.3390/math7111084 - 10 Nov 2019
Cited by 4 | Viewed by 1845
Abstract
We prove the existence of solutions for neutral functional differential inclusions involving Hadamard fractional derivatives by applying several fixed point theorems for multivalued maps. We also construct examples for illustrating the obtained results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
16 pages, 805 KiB  
Article
Generalized Integral Inequalities for Hermite–Hadamard-Type Inequalities via s-Convexity on Fractal Sets
by Ohud Almutairi and Adem Kılıçman
Mathematics 2019, 7(11), 1065; https://doi.org/10.3390/math7111065 - 6 Nov 2019
Cited by 9 | Viewed by 2075
Abstract
In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a [...] Read more.
In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ . We also prove certain inequalities involving Riemann–Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
19 pages, 845 KiB  
Article
A Mollification Regularization Method for the Inverse Source Problem for a Time Fractional Diffusion Equation
by Le Dinh Long, Yong Zhou, Tran Thanh Binh and Nguyen Can
Mathematics 2019, 7(11), 1048; https://doi.org/10.3390/math7111048 - 4 Nov 2019
Cited by 7 | Viewed by 2842
Abstract
We consider a time-fractional diffusion equation for an inverse problem to determine an unknown source term, whereby the input data is obtained at a certain time. In general, the inverse problems are ill-posed in the sense of Hadamard. Therefore, in this study, we [...] Read more.
We consider a time-fractional diffusion equation for an inverse problem to determine an unknown source term, whereby the input data is obtained at a certain time. In general, the inverse problems are ill-posed in the sense of Hadamard. Therefore, in this study, we propose a mollification regularization method to solve this problem. In the theoretical results, the error estimate between the exact and regularized solutions is given by a priori and a posteriori parameter choice rules. Besides, the proposed regularized methods have been verified by a numerical experiment. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
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13 pages, 275 KiB  
Article
On a Generalized Langevin Type Nonlocal Fractional Integral Multivalued Problem
by Ahmed Alsaedi, Bashir Ahmad, Madeaha Alghanmi and Sotiris K. Ntouyas
Mathematics 2019, 7(11), 1015; https://doi.org/10.3390/math7111015 - 25 Oct 2019
Cited by 8 | Viewed by 1823
Abstract
We establish sufficient criteria for the existence of solutions for a nonlinear generalized Langevin-type nonlocal fractional-order integral multivalued problem. The convex and non-convex cases for the multivalued map involved in the given problem are considered. Our results rely on Leray–Schauder nonlinear alternative for [...] Read more.
We establish sufficient criteria for the existence of solutions for a nonlinear generalized Langevin-type nonlocal fractional-order integral multivalued problem. The convex and non-convex cases for the multivalued map involved in the given problem are considered. Our results rely on Leray–Schauder nonlinear alternative for multivalued maps and Covitz and Nadler’s fixed point theorem. Illustrative examples for the main results are included. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
19 pages, 298 KiB  
Article
Positive Solutions for a System of Fractional Integral Boundary Value Problems of Riemann–Liouville Type Involving Semipositone Nonlinearities
by Youzheng Ding, Jiafa Xu and Zhengqing Fu
Mathematics 2019, 7(10), 970; https://doi.org/10.3390/math7100970 - 14 Oct 2019
Cited by 5 | Viewed by 2170
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 [...] 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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
15 pages, 3384 KiB  
Article
Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems
by Imed Bachar, Habib Mâagli and Hassan Eltayeb
Mathematics 2019, 7(10), 961; https://doi.org/10.3390/math7100961 - 13 Oct 2019
Cited by 3 | Viewed by 2141
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
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21 pages, 341 KiB  
Article
A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations
by Hassan Eltayeb, Said Mesloub, Yahya T. Abdalla and Adem Kılıçman
Mathematics 2019, 7(10), 949; https://doi.org/10.3390/math7100949 - 12 Oct 2019
Cited by 7 | Viewed by 2194
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
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29 pages, 906 KiB  
Article
Stability Results for a Coupled System of Impulsive Fractional Differential Equations
by Akbar Zada, Shaheen Fatima, Zeeshan Ali, Jiafa Xu and Yujun Cui
Mathematics 2019, 7(10), 927; https://doi.org/10.3390/math7100927 - 6 Oct 2019
Cited by 18 | Viewed by 2436
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
12 pages, 268 KiB  
Article
Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel
by Xia Wu, JinRong Wang and Jialu Zhang
Mathematics 2019, 7(9), 845; https://doi.org/10.3390/math7090845 - 12 Sep 2019
Cited by 24 | Viewed by 3270
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
18 pages, 280 KiB  
Article
Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications
by Saima Rashid, Thabet Abdeljawad, Fahd Jarad and Muhammad Aslam Noor
Mathematics 2019, 7(9), 807; https://doi.org/10.3390/math7090807 - 2 Sep 2019
Cited by 49 | Viewed by 3138
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
10 pages, 760 KiB  
Article
Ostrowski Type Inequalities Involving ψ-Hilfer Fractional Integrals
by Yasemin Basci and Dumitru Baleanu
Mathematics 2019, 7(9), 770; https://doi.org/10.3390/math7090770 - 21 Aug 2019
Cited by 22 | Viewed by 2623
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
15 pages, 277 KiB  
Article
Existence Theory for a Fractional q-Integro-Difference Equation with q-Integral Boundary Conditions of Different Orders
by Sina Etemad, Sotiris K. Ntouyas and Bashir Ahmad
Mathematics 2019, 7(8), 659; https://doi.org/10.3390/math7080659 - 24 Jul 2019
Cited by 56 | Viewed by 3026
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
15 pages, 291 KiB  
Article
Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative
by Dandan Yang and Chuanzhi Bai
Mathematics 2019, 7(7), 630; https://doi.org/10.3390/math7070630 - 15 Jul 2019
Cited by 5 | Viewed by 2705
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
10 pages, 278 KiB  
Article
The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral
by Bashir Ahmad, Madeaha Alghanmi, Ahmed Alsaedi, Hari M. Srivastava and Sotiris K. Ntouyas
Mathematics 2019, 7(6), 533; https://doi.org/10.3390/math7060533 - 11 Jun 2019
Cited by 31 | Viewed by 2882
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
11 pages, 271 KiB  
Article
Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem
by Ehsan Pourhadi, Reza Saadati and Sotiris K. Ntouyas
Mathematics 2019, 7(6), 526; https://doi.org/10.3390/math7060526 - 10 Jun 2019
Cited by 3 | Viewed by 2169
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
12 pages, 256 KiB  
Article
Existence Result and Uniqueness for Some Fractional Problem
by Guotao Wang, Abdeljabbar Ghanmi, Samah Horrigue and Samar Madian
Mathematics 2019, 7(6), 516; https://doi.org/10.3390/math7060516 - 5 Jun 2019
Cited by 11 | Viewed by 3097
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
16 pages, 320 KiB  
Article
On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders
by Saowaluck Chasreechai and Thanin Sitthiwirattham
Mathematics 2019, 7(5), 471; https://doi.org/10.3390/math7050471 - 24 May 2019
Cited by 7 | Viewed by 2293
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
12 pages, 255 KiB  
Article
Stability, Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Differential Equations
by Nazim I Mahmudov and Areen Al-Khateeb
Mathematics 2019, 7(4), 354; https://doi.org/10.3390/math7040354 - 16 Apr 2019
Cited by 9 | Viewed by 2919
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
14 pages, 292 KiB  
Article
Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative
by Kui Liu, Michal Fečkan, D. O’Regan and JinRong Wang
Mathematics 2019, 7(4), 333; https://doi.org/10.3390/math7040333 - 5 Apr 2019
Cited by 43 | Viewed by 4603
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
13 pages, 314 KiB  
Article
Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points
by Yu Chen and JinRong Wang
Mathematics 2019, 7(4), 331; https://doi.org/10.3390/math7040331 - 4 Apr 2019
Cited by 7 | Viewed by 2448
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
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22 pages, 323 KiB  
Article
A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line
by Jarunee Soontharanon, Saowaluck Chasreechai and Thanin Sitthiwirattham
Mathematics 2019, 7(3), 256; https://doi.org/10.3390/math7030256 - 12 Mar 2019
Cited by 7 | Viewed by 2107
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
14 pages, 274 KiB  
Article
Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann-Stieltjes Integro-Multipoint Boundary Conditions
by Bashir Ahmad, Ymnah Alruwaily, Ahmed Alsaedi and Sotiris K. Ntouyas
Mathematics 2019, 7(3), 249; https://doi.org/10.3390/math7030249 - 11 Mar 2019
Cited by 21 | Viewed by 2655 | Correction
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
20 pages, 356 KiB  
Article
On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions
by Kamal Shah, Poom Kumam and Inam Ullah
Mathematics 2019, 7(3), 223; https://doi.org/10.3390/math7030223 - 27 Feb 2019
Cited by 4 | Viewed by 2555
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
17 pages, 309 KiB  
Article
Nonlocal Fractional Evolution Inclusions of Order α ∈ (1,2)
by Jia Wei He, Yong Liang, Bashir Ahmad and Yong Zhou
Mathematics 2019, 7(2), 209; https://doi.org/10.3390/math7020209 - 24 Feb 2019
Cited by 65 | Viewed by 3328
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
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)
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