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23 pages, 480 KB

Open AccessArticle

Impulsive Tempered Ψ-Fractional Differential Equations with Boundary and Integral Conditions

by Chayapat Sudprasert, Suphawat Asawasamrit, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2026, 10(2), 113; https://doi.org/10.3390/fractalfract10020113 - 5 Feb 2026

Viewed by 882

This paper studies mixed impulsive boundary value problems involving tempered

Ψ

-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The [...] Read more.

This paper studies mixed impulsive boundary value problems involving tempered

Ψ

-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered

Ψ

-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application, Second Edition)

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21 pages, 404 KB

Open AccessArticle

Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications

by Qingzhe Wu, Muming Zhang, Jing Shao, Muhammad Samraiz, Humaira Javaid and Saima Naheed

Fractal Fract. 2025, 9(8), 519; https://doi.org/10.3390/fractalfract9080519 - 8 Aug 2025

Cited by 1 | Viewed by 1155

Abstract

In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in

L_{p}

norms for

p > 1

by using Hölder’s inequality. Special cases are discussed in the [...] Read more.

In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in

L_{p}

norms for

p > 1

by using Hölder’s inequality. Special cases are discussed in the form of remarks by highlighting their relationships with the existing literature. The derived results are also verified through illustrative examples, including tables and graphs. Moreover, applications of the obtained inequalities are discussed in the context of arithmetic and geometric means. Full article

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25 pages, 325 KB

Open AccessReview

Advances in Fractional Lyapunov-Type Inequalities: A Comprehensive Review

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Foundations 2025, 5(2), 18; https://doi.org/10.3390/foundations5020018 - 27 May 2025

Cited by 3 | Viewed by 1518

Abstract

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard,

ψ

-Riemann–Liouville, [...] Read more.

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard,

ψ

-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

(This article belongs to the Section Mathematical Sciences)

19 pages, 318 KB

Open AccessArticle

A Study of Fuzzy Fixed Points and Their Application to Fuzzy Fractional Differential Equations

by Nawal Alharbi and Nawab Hussain

Fractal Fract. 2025, 9(5), 270; https://doi.org/10.3390/fractalfract9050270 - 22 Apr 2025

Cited by 1 | Viewed by 1760

Abstract

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework

θ

-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued [...] Read more.

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework

θ

-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued mappings, thereby broadening the scope of fuzzy analysis. Furthermerefore, we have for aore, we apply one of our key results to derive conditions, ensuring the existence and uniqueness of a solution to Hadamard

Ψ

-Caputo tempered fuzzy fractional differential equations, particularly in the context of the SIR dynamics model. These theoretical advancements are expected to open new avenues for research in fuzzy fixed-point theory and its applications to hybrid models within

θ

-fuzzy metric spaces. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractals)

16 pages, 326 KB

Open AccessArticle

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

by Michal Pospíšil and Lucia Pospíšilová Škripková

Axioms 2024, 13(10), 680; https://doi.org/10.3390/axioms13100680 - 1 Oct 2024

Cited by 3 | Viewed by 1626

Abstract

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include [...] Read more.

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

23 pages, 344 KB

Open AccessArticle

On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

by Mieczysław Cichoń, Hussein A. H. Salem and Wafa Shammakh

Mathematics 2024, 12(17), 2631; https://doi.org/10.3390/math12172631 - 24 Aug 2024

Cited by 5 | Viewed by 1903

Abstract

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional [...] Read more.

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation

\frac{d_{ψ}^{β, μ}}{d t^{β}} (\frac{d_{ψ}^{α, μ}}{d t^{α}} + λ) x (t) = f (t, x (t)), t \in [a, b], λ \in R

, for

f \in C ([a, b] \times R)

and some critical orders

β, α \in (0, 1)

, combined with appropriate initial or boundary conditions, and with general classes of

ψ

-tempered Hilfer problems with

ψ

-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

16 pages, 323 KB

Open AccessArticle

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

by Milan Medved’, Michal Pospíšil and Eva Brestovanská

Axioms 2024, 13(5), 301; https://doi.org/10.3390/axioms13050301 - 1 May 2024

Cited by 6 | Viewed by 2360

Abstract

In this paper, the tempered

Ψ

–Riemann–Liouville fractional derivative and the tempered

Ψ

–Caputo fractional derivative of order

n - 1 < α < n \in N

are introduced for

C^{n - 1}

–functions. A nonlinear version of the second Henry–Gronwall inequality [...] Read more.

In this paper, the tempered

Ψ

–Riemann–Liouville fractional derivative and the tempered

Ψ

–Caputo fractional derivative of order

n - 1 < α < n \in N

are introduced for

C^{n - 1}

–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered

Ψ

–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered

Ψ

–Caputo fractional differential equations are proved. Illustrative examples are given. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

15 pages, 1025 KB

Open AccessArticle

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

by Donal O’Regan, Snezhana Hristova and Ravi P. Agarwal

Fractal Fract. 2024, 8(1), 11; https://doi.org/10.3390/fractalfract8010011 - 22 Dec 2023

Cited by 7 | Viewed by 2524

Abstract

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study [...] Read more.

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order

Ψ

-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the

Ψ

-tempered Caputo fractional derivative. Full article

(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)

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17 pages, 356 KB

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 - 8 Aug 2023

Cited by 7 | Viewed by 2032

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)

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