Research

10 pages, 326 KiB

Open AccessArticle

Precise Conditions on the Unique Solvability of the Linear Fractional Functional Differential Equations Related to the ς-Nonpositive Operators

by Natalia Dilna

Fractal Fract. 2023, 7(10), 720; https://doi.org/10.3390/fractalfract7100720 - 29 Sep 2023

Viewed by 531

Abstract

Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to

ς

-nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are [...] Read more.

Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to

ς

-nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are illustrated by an example of the pantograph-type model from electrodynamics. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

17 pages, 356 KiB

Open AccessArticle

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

by Milan Medveď, Michal Pospíšil and Eva Brestovanská

Fractal Fract. 2023, 7(8), 611; https://doi.org/10.3390/fractalfract7080611 - 08 Aug 2023

Cited by 1 | Viewed by 759

Abstract

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered

Ψ

-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered

Ψ

-Hilfer fractional integrals, are [...] Read more.

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered

Ψ

-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered

Ψ

-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered

Ψ

-Caputo fractional derivative of order

0 < α < 1

, where the right side may depend on time, the solution, or its tempered

Ψ

-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the

Ψ

-exponential stability of trivial solutions are proven for these kinds of differential equations. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

14 pages, 380 KiB

Open AccessArticle

On Pantograph Problems Involving Weighted Caputo Fractional Operators with Respect to Another Function

by Saeed M. Ali

Fractal Fract. 2023, 7(7), 559; https://doi.org/10.3390/fractalfract7070559 - 19 Jul 2023

Cited by 1 | Viewed by 924

Abstract

In this investigation, weighted

p s i

-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes [...] Read more.

In this investigation, weighted

p s i

-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where

ψ (x) = x

. With the aid of Banach’s and Krasnoselskii’s classic fixed point results, we have established a the qualitative study. Different values of

ψ (x)

and

w (x)

are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted

ψ

-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

21 pages, 886 KiB

Open AccessArticle

The Hölder Regularity for Abstract Fractional Differential Equation with Applications to Rayleigh–Stokes Problems

by Jiawei He and Guangmeng Wu

Fractal Fract. 2023, 7(7), 549; https://doi.org/10.3390/fractalfract7070549 - 16 Jul 2023

Cited by 1 | Viewed by 733

Abstract

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this [...] Read more.

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

23 pages, 385 KiB

Open AccessArticle

The Existence of Mild Solutions for Hilfer Fractional Stochastic Evolution Equations with Order μ∈(1,2)

by Qien Li and Yong Zhou

Fractal Fract. 2023, 7(7), 525; https://doi.org/10.3390/fractalfract7070525 - 02 Jul 2023

Cited by 3 | Viewed by 769

Abstract

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order

μ \in (1, 2)

and type

ν \in [0, 1]

. We prove the existence of [...] Read more.

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order

μ \in (1, 2)

and type

ν \in [0, 1]

. We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder fixed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efficacy of this method. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

32 pages, 422 KiB

Open AccessArticle

New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh

Fractal Fract. 2023, 7(5), 405; https://doi.org/10.3390/fractalfract7050405 - 16 May 2023

Cited by 3 | Viewed by 1293

Abstract

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, [...] Read more.

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the

(m, t g s)

-type s-preinvex function, Godunova–Levin

(s, m)

-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

14 pages, 1216 KiB

Open AccessArticle

Qualitatively Stable Schemes for the Black–Scholes Equation

by Mohammad Mehdizadeh Khalsaraei, Ali Shokri, Yuanheng Wang, Sohrab Bazm, Giti Navidifar and Pari Khakzad

Fractal Fract. 2023, 7(2), 154; https://doi.org/10.3390/fractalfract7020154 - 04 Feb 2023

Cited by 2 | Viewed by 947

Abstract

In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the [...] Read more.

In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the new method is positive, stable, and consistent when low volatility is assumed. The efficiency of the new method is demonstrated by a numerical example. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

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11 pages, 310 KiB

Open AccessArticle

Existence Results for a Differential Equation Involving the Right Caputo Fractional Derivative and Mixed Nonlinearities with Nonlocal Closed Boundary Conditions

by Bashir Ahmad, Manal Alnahdi and Sotiris K. Ntouyas

Fractal Fract. 2023, 7(2), 129; https://doi.org/10.3390/fractalfract7020129 - 30 Jan 2023

Cited by 4 | Viewed by 1285

Abstract

In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, we discuss the existence of solutions for a mixed nonlinear differential equation involving a right Caputo fractional derivative operator, and left and right Riemann–Liouville fractional integral [...] Read more.

In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, we discuss the existence of solutions for a mixed nonlinear differential equation involving a right Caputo fractional derivative operator, and left and right Riemann–Liouville fractional integral operators of different orders. We apply a decent and fruitful approach of fixed point theory to establish the desired results. Examples are given for illustration of the main results. The paper concludes with some interesting observations. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

17 pages, 343 KiB

Open AccessArticle

Nonlocal Problems for Hilfer Fractional q-Difference Equations

by Chunping Tian, Haibo Gu and Zunkai Yang

Fractal Fract. 2023, 7(2), 101; https://doi.org/10.3390/fractalfract7020101 - 17 Jan 2023

Viewed by 908

Abstract

In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an [...] Read more.

In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an ordered Banach space is discussed by monotone iterative method. In that following, we consider the Ulam stability results for equations. Finally, two examples are given to illustrate the effectiveness of theory results. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

10 pages, 314 KiB

Open AccessArticle

Unique Solvability of the Initial-Value Problem for Fractional Functional Differential Equations—Pantograph-Type Model

by Natalia Dilna

Fractal Fract. 2023, 7(1), 65; https://doi.org/10.3390/fractalfract7010065 - 05 Jan 2023

Cited by 1 | Viewed by 868

Abstract

Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the [...] Read more.

Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the unique solvability conditions for the initial-value problem for linear functional differential equations of the fractional order. The advantage is the lack of the calculation of fractional derivatives, which is a complicated task. The unique solution is represented by the Neumann series. In addition, as examples, the model with a discrete memory effect and a pantograph-type model from electrodynamics are studied. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

20 pages, 380 KiB

Open AccessArticle

A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress

by Andaluzia Cristina Matei

Fractal Fract. 2022, 6(11), 651; https://doi.org/10.3390/fractalfract6110651 - 04 Nov 2022

Cited by 3 | Viewed by 947

Abstract

In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational [...] Read more.

In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement field, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

11 pages, 308 KiB

Open AccessArticle

Qualitative Properties of Positive Solutions of a Kind for Fractional Pantograph Problems using Technique Fixed Point Theory

by Hamid Boulares, Abbes Benchaabane, Nuttapol Pakkaranang, Ramsha Shafqat and Bancha Panyanak

Fractal Fract. 2022, 6(10), 593; https://doi.org/10.3390/fractalfract6100593 - 14 Oct 2022

Cited by 12 | Viewed by 1310

Abstract

The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation [...] Read more.

The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

12 pages, 285 KiB

Open AccessArticle

Results for Fuzzy Mappings and Stability of Fuzzy Sets with Applications

by Aqeel Shahzad, Abdullah Shoaib, Nabil Mlaiki and Suhad Subhi Aiadi

Fractal Fract. 2022, 6(10), 556; https://doi.org/10.3390/fractalfract6100556 - 30 Sep 2022

Cited by 2 | Viewed by 932

Abstract

The purpose of this paper is to develop some fuzzy fixed point results for the sequence of locally fuzzy mappings satisfying rational type almost contractions in complete dislocated metric spaces. We apply our results to obtain new results for set-valued and single-valued mappings. [...] Read more.

The purpose of this paper is to develop some fuzzy fixed point results for the sequence of locally fuzzy mappings satisfying rational type almost contractions in complete dislocated metric spaces. We apply our results to obtain new results for set-valued and single-valued mappings. We also study the stability of fuzzy fixed point

γ

-level sets. An example is presented in favor of these results. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

21 pages, 550 KiB

Open AccessArticle

Hermite Fitted Block Integrator for Solving Second-Order Anisotropic Elliptic Type PDEs

by Emmanuel Oluseye Adeyefa, Ezekiel Olaoluwa Omole, Ali Shokri and Shao-Wen Yao

Fractal Fract. 2022, 6(9), 497; https://doi.org/10.3390/fractalfract6090497 - 05 Sep 2022

Cited by 3 | Viewed by 1251

Abstract

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite [...] Read more.

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

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11 pages, 294 KiB

Open AccessArticle

Solution of the Ill-Posed Cauchy Problem for Systems of Elliptic Type of the First Order

by Davron Aslonqulovich Juraev, Ali Shokri and Daniela Marian

Fractal Fract. 2022, 6(7), 358; https://doi.org/10.3390/fractalfract6070358 - 26 Jun 2022

Cited by 4 | Viewed by 996

Abstract

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space

R^{m}

. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability [...] Read more.

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space

R^{m}

. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

16 pages, 330 KiB

Open AccessArticle

Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers

by Haiyong Xu, Bashir Ahmad, Guotao Wang and Lihong Zhang

Fractal Fract. 2022, 6(7), 352; https://doi.org/10.3390/fractalfract6070352 - 23 Jun 2022

Cited by 1 | Viewed by 1090

Abstract

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: [...] Read more.

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers:

{(- Δ)}^{\frac{α}{2}} u (x) + u^{- γ} (x) + v^{- q} (x) = 0, x \in R^{N},

{(- Δ)}^{\frac{β}{2}} v (x) + v^{- σ} (x) + u^{- p} (x) = 0, x \in R^{N},

u (x) ≳ {| x |}^{a}, v (x) ≳ {| x |}^{b} as | x | \to \infty,

where

α, β \in (0, 2)

, and

a, b > 0

are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

28 pages, 413 KiB

Open AccessArticle

Existence and Stability Results for a Tripled System of the Caputo Type with Multi-Point and Integral Boundary Conditions

by Murugesan Manigandan, Muthaiah Subramanian, Thangaraj Nandha Gopal and Bundit Unyong

Fractal Fract. 2022, 6(6), 285; https://doi.org/10.3390/fractalfract6060285 - 26 May 2022

Cited by 8 | Viewed by 1479

Abstract

In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and [...] Read more.

In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and the alternative of Leray–Schauder. The stability of the Hyer–Ulam solutions are investigated. A few examples are provided to identify the major results. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

17 pages, 356 KiB

Open AccessArticle

Nonlocal Coupled System for (k,φ)-Hilfer Fractional Differential Equations

by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2022, 6(5), 234; https://doi.org/10.3390/fractalfract6050234 - 23 Apr 2022

Cited by 8 | Viewed by 1514

Abstract

In this paper, we study a coupled system consisting of

(k, φ)

-Hilfer fractional differential equations of the order

(1, 2],

supplemented with nonlocal coupled multi-point boundary conditions. The existence and uniqueness of the results are [...] Read more.

In this paper, we study a coupled system consisting of

(k, φ)

-Hilfer fractional differential equations of the order

(1, 2],

supplemented with nonlocal coupled multi-point boundary conditions. The existence and uniqueness of the results are established via Banach’s contraction mapping principle, the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed-point theorem. Numerical examples are constructed to illustrate the obtained results. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

13 pages, 301 KiB

Open AccessArticle

Some Fixed-Disc Results in Double Controlled Quasi-Metric Type Spaces

by Nabil Mlaiki, Nihal Taş, Salma Haque and Doaa Rizk

Fractal Fract. 2022, 6(2), 107; https://doi.org/10.3390/fractalfract6020107 - 12 Feb 2022

Cited by 1 | Viewed by 1430

Abstract

In this paper, we introduce new types of general contractions for self mapping on double controlled quasi-metric type spaces, where we prove the existence and uniqueness of fixed disc and circle for such mappings. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

16 pages, 296 KiB

Open AccessArticle

Common Fixed Point Theorems for Two Mappings in Complete b-Metric Spaces

by Lili Chen, Xin Xia, Yanfeng Zhao and Xin Liu

Fractal Fract. 2022, 6(2), 103; https://doi.org/10.3390/fractalfract6020103 - 11 Feb 2022

Cited by 1 | Viewed by 1940

Abstract

Our paper is devoted to the issue of the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and

θ

, respectively. Moreover, two specific examples to indicate the validity [...] Read more.

Our paper is devoted to the issue of the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and

θ

, respectively. Moreover, two specific examples to indicate the validity of our results are also given. Eventually, the generalized forms of Jungck fixed point theorem in the above spaces is investigated. Different from related literature, the conditions that the function F needs to satisfy are weakened, and F only needs to be non-decreasing in this paper. To some extent, our conclusions and methods improve the results of previous literature. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

Review

Jump to: Research, Other

35 pages, 424 KiB

Open AccessReview

A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Fractal Fract. 2022, 6(5), 273; https://doi.org/10.3390/fractalfract6050273 - 18 May 2022

Cited by 7 | Viewed by 1382

Abstract

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville,

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, [...] Read more.

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville,

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

Other

Jump to: Research, Review

10 pages, 315 KiB

Open AccessBrief Report

Fractional p-Laplacian Equations with Sandwich Pairs

by Jose Vanterler da C. Sousa

Fractal Fract. 2023, 7(6), 419; https://doi.org/10.3390/fractalfract7060419 - 23 May 2023

Cited by 1 | Viewed by 895

Abstract

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in

ψ

-fractional space [...] Read more.

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in

ψ

-fractional space

H_{p}^{α, β; ψ} (Ω)

. The results obtained in this paper are the first to make use of the theory of

ψ

-Hilfer fractional operators with p-Laplacian. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

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