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7 pages, 293 KiB

Open AccessEditorial

Advances in Boundary Value Problems for Fractional Differential Equations

by Rodica Luca

Fractal Fract. 2023, 7(5), 406; https://doi.org/10.3390/fractalfract7050406 - 17 May 2023

Cited by 4 | Viewed by 1477

Abstract

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image [...] Read more.

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image processing, aerodynamics, transport dynamics, thermodynamics, viscoelasticity, hydrology, statistical mechanics, electromagnetics, astrophysics, cosmology, and rheology [...] Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

Research

Jump to: Editorial

20 pages, 394 KiB

Open AccessArticle

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2022, 6(10), 610; https://doi.org/10.3390/fractalfract6100610 - 19 Oct 2022

Cited by 4 | Viewed by 1900

Abstract

In this paper, we study the existence of positive solutions for a system of fractional differential equations with

ρ

-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary [...] Read more.

In this paper, we study the existence of positive solutions for a system of fractional differential equations with

ρ

-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

21 pages, 346 KiB

Open AccessArticle

Solvability Criterion for Fractional q-Integro-Difference System with Riemann-Stieltjes Integrals Conditions

by Changlong Yu, Si Wang, Jufang Wang and Jing Li

Fractal Fract. 2022, 6(10), 554; https://doi.org/10.3390/fractalfract6100554 - 29 Sep 2022

Cited by 8 | Viewed by 1298

Abstract

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system [...] Read more.

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system with the nonlocal boundary conditions involving diverse fractional q-derivatives and Riemann-Stieltjes q-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii’s fixed point theorem, Schaefer’s fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

16 pages, 364 KiB

Open AccessArticle

Analytic Resolving Families for Equations with the Dzhrbashyan–Nersesyan Fractional Derivative

by Vladimir E. Fedorov, Marina V. Plekhanova and Elizaveta M. Izhberdeeva

Fractal Fract. 2022, 6(10), 541; https://doi.org/10.3390/fractalfract6100541 - 25 Sep 2022

Cited by 8 | Viewed by 1448

Abstract

In this paper, a criterion for generating an analytic family of operators, which resolves a linear equation solved with respect to the Dzhrbashyan–Nersesyan fractional derivative, via a linear closed operator is obtained. The properties of the resolving families are investigated and applied to [...] Read more.

In this paper, a criterion for generating an analytic family of operators, which resolves a linear equation solved with respect to the Dzhrbashyan–Nersesyan fractional derivative, via a linear closed operator is obtained. The properties of the resolving families are investigated and applied to prove the existence of a unique solution for the corresponding initial value problem of the inhomogeneous equation with the Dzhrbashyan–Nersesyan fractional derivative. A solution is presented explicitly using resolving families of operators. A theorem on perturbations of operators from the found class of generators of resolving families is proved. The obtained results are used for a study of an initial-boundary value problem to a model of the viscoelastic Oldroyd fluid dynamics. Thus, the Dzhrbashyan–Nersesyan initial value problem is investigated in the essentially infinite-dimensional case. The use of the proved abstract results to study initial-boundary value problems for a system of partial differential equations is demonstrated. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

16 pages, 360 KiB

Open AccessArticle

Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

by Chandra Bose Sindhu Varun Bose and Ramalingam Udhayakumar

Fractal Fract. 2022, 6(9), 532; https://doi.org/10.3390/fractalfract6090532 - 19 Sep 2022

Cited by 18 | Viewed by 1698

Abstract

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli’s fixed point theorem, we prove the primary results. In addition, the application is [...] Read more.

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli’s fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

12 pages, 689 KiB

Open AccessArticle

Study on Infinitely Many Solutions for a Class of Fredholm Fractional Integro-Differential System

by Dongping Li, Yankai Li and Fangqi Chen

Fractal Fract. 2022, 6(9), 467; https://doi.org/10.3390/fractalfract6090467 - 26 Aug 2022

Cited by 3 | Viewed by 1312

Abstract

This paper deals with a class of nonlinear fractional Sturm–Liouville boundary value problems. Each sub equation in the system is a fractional partial equation including the second kinds of Fredholm integral equation and the p-Laplacian operator, simultaneously. Infinitely many solutions are derived [...] Read more.

This paper deals with a class of nonlinear fractional Sturm–Liouville boundary value problems. Each sub equation in the system is a fractional partial equation including the second kinds of Fredholm integral equation and the p-Laplacian operator, simultaneously. Infinitely many solutions are derived due to perfect involvements of fractional calculus theory and variational methods with some simpler and more easily verified assumptions. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

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18 pages, 614 KiB

Open AccessArticle

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

by Daliang Zhao and Yongyang Liu

Fractal Fract. 2022, 6(8), 424; https://doi.org/10.3390/fractalfract6080424 - 30 Jul 2022

Cited by 4 | Viewed by 1374

Abstract

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool [...] Read more.

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of

C_{0}

-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

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15 pages, 338 KiB

Open AccessArticle

On a System of Riemann–Liouville Fractional Boundary Value Problems with ϱ-Laplacian Operators and Positive Parameters

by Johnny Henderson, Rodica Luca and Alexandru Tudorache

Fractal Fract. 2022, 6(6), 299; https://doi.org/10.3390/fractalfract6060299 - 29 May 2022

Cited by 3 | Viewed by 1737

Abstract

In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with

ϱ

-Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply [...] Read more.

In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with

ϱ

-Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder fixed point theorem in the proof of the existence result. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

11 pages, 300 KiB

Open AccessArticle

Boundary Value Problem for Fractional q-Difference Equations with Integral Conditions in Banach Spaces

by Nadia Allouch, John R. Graef and Samira Hamani

Fractal Fract. 2022, 6(5), 237; https://doi.org/10.3390/fractalfract6050237 - 25 Apr 2022

Cited by 14 | Viewed by 2010

Abstract

The authors investigate the existence of solutions to a class of boundary value problems for fractional q-difference equations in a Banach space that involves a q-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch’s [...] Read more.

The authors investigate the existence of solutions to a class of boundary value problems for fractional q-difference equations in a Banach space that involves a q-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch’s fixed point theorem and the technique of measures of noncompactness. This approach has proved to be an interesting and useful approach to studying such problems. Some basic concepts from the fractional q-calculus are introduced, including q-derivatives and q-integrals. An example of the main result is included as well as some suggestions for future research. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

26 pages, 437 KiB

Open AccessArticle

Topological Structure of the Solution Sets for Impulsive Fractional Neutral Differential Inclusions with Delay and Generated by a Non-Compact Demi Group

by Zainab Alsheekhhussain, Ahmed Gamal Ibrahim and Akbar Ali

Fractal Fract. 2022, 6(4), 188; https://doi.org/10.3390/fractalfract6040188 - 28 Mar 2022

Cited by 3 | Viewed by 1833

Abstract

In this paper, we give an affirmative answer to a question about the sufficient conditions which ensure that the set of mild solutions for a fractional impulsive neutral differential inclusion with state-dependent delay, generated by a non-compact semi-group, are not empty compact and [...] Read more.

In this paper, we give an affirmative answer to a question about the sufficient conditions which ensure that the set of mild solutions for a fractional impulsive neutral differential inclusion with state-dependent delay, generated by a non-compact semi-group, are not empty compact and an

R_{δ}

-set. This means that the solution set may not be a singleton, but it has the same homology group as a one-point space from the point of view of algebraic topology. In fact, we demonstrate that the solution set is an intersection of a decreasing sequence of non-empty compact and contractible sets. Up to now, proving that the solution set for fractional impulsive neutral semilinear differential inclusions in the presence of impulses and delay and generated by a non-compact semigroup is an

R_{δ}

-set has not been considered in the literature. Since fractional differential equations have many applications in various fields such as physics and engineering, the aim of our work is important. Two illustrative examples are given to clarify the wide applicability of our results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

11 pages, 560 KiB

Open AccessArticle

The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations

by Jun-Sheng Duan, Li-Xia Jing and Ming Li

Fractal Fract. 2022, 6(3), 148; https://doi.org/10.3390/fractalfract6030148 - 9 Mar 2022

Cited by 6 | Viewed by 2199

Abstract

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval

0 \leq x \leq 1

was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary [...] Read more.

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval

0 \leq x \leq 1

was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

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16 pages, 326 KiB

Open AccessArticle

Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann–Stieltjes Integro-Multipoint Boundary Conditions

by Ymnah Alruwaily, Bashir Ahmad, Sotiris K. Ntouyas and Ahmed S. M. Alzaidi

Fractal Fract. 2022, 6(2), 123; https://doi.org/10.3390/fractalfract6020123 - 20 Feb 2022

Cited by 19 | Viewed by 2327

Abstract

This paper is concerned with the existence of solutions for a fully coupled Riemann–Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example [...] Read more.

This paper is concerned with the existence of solutions for a fully coupled Riemann–Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

12 pages, 278 KiB

Open AccessArticle

Existence and Approximate Controllability of Mild Solutions for Fractional Evolution Systems of Sobolev-Type

by Yue Liang

Fractal Fract. 2022, 6(2), 56; https://doi.org/10.3390/fractalfract6020056 - 22 Jan 2022

Cited by 4 | Viewed by 2451

Abstract

This paper investigates the existence and approximate controllability of Riemann–Liouville fractional evolution systems of Sobolev-type in abstract spaces. At first, a group of sufficient conditions is established for the existence of mild solutions without the compactness of operator semigroup. Then the approximate controllability [...] Read more.

This paper investigates the existence and approximate controllability of Riemann–Liouville fractional evolution systems of Sobolev-type in abstract spaces. At first, a group of sufficient conditions is established for the existence of mild solutions without the compactness of operator semigroup. Then the approximate controllability is studied under the assumption that the corresponding linear system is approximate controllability. The proof is based on the fixed point theory and the method of operator semigroup. An example is given as an application of the obtained results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

16 pages, 337 KiB

Open AccessArticle

Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in ℝⁿ

by Yizhe Feng and Zhanbing Bai

Fractal Fract. 2022, 6(1), 25; https://doi.org/10.3390/fractalfract6010025 - 1 Jan 2022

Cited by 8 | Viewed by 1856

Abstract

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable

x \in R^{n}

and the effect of the coefficient matrices matrices B and C of boundary [...] Read more.

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable

x \in R^{n}

and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

20 pages, 389 KiB

Open AccessArticle

Positive Solutions of a Singular Fractional Boundary Value Problem with r-Laplacian Operators

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2022, 6(1), 18; https://doi.org/10.3390/fractalfract6010018 - 30 Dec 2021

Cited by 5 | Viewed by 1802

Abstract

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. [...] Read more.

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main results we apply the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

20 pages, 353 KiB

Open AccessArticle

Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary

by Areej Bin Sultan, Mohamed Jleli and Bessem Samet

Fractal Fract. 2021, 5(4), 258; https://doi.org/10.3390/fractalfract5040258 - 6 Dec 2021

Cited by 3 | Viewed by 2620

Abstract

We first consider the damped wave inequality [...] Read more.

We first consider the damped wave inequality

\frac{\partial^{2} u}{\partial t^{2}} - \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial u}{\partial t} \geq x^{σ} {| u |}^{p}, t > 0, x \in (0, L),

where

L > 0

,

σ \in R

, and

p > 1

, under the Dirichlet boundary conditions

(u (t, 0), u (t, L)) = (f (t), g (t)), t > 0 .

We establish sufficient conditions depending on

σ

, p, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated:

g \equiv 0

and

g (t) = t^{γ}

,

γ > - 1

. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality

\frac{\partial^{α} u}{\partial t^{α}} - \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{β} u}{\partial t^{β}} \geq x^{σ} {| u |}^{p}, t > 0, x \in (0, L),

where

α \in (1, 2)

,

β \in (0, 1)

, and

\frac{\partial^{τ}}{\partial t^{τ}}

is the time-Caputo fractional derivative of order

τ

,

τ \in {α, β}

. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

14 pages, 314 KiB

Open AccessArticle

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

by Shahram Rezapour, Mohammed Said Souid, Sina Etemad, Zoubida Bouazza, Sotiris K. Ntouyas, Suphawat Asawasamrit and Jessada Tariboon

Fractal Fract. 2021, 5(4), 216; https://doi.org/10.3390/fractalfract5040216 - 12 Nov 2021

Cited by 5 | Viewed by 1596

Abstract

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert [...] Read more.

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

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