Hilfer Fractional Research Articles

Dynamics and Stability of $\Xi$-Hilfer Fractional Fuzzy Differential Equations with Impulses

This paper deals with the existence, uniqueness, and Ulam-stability outcomes for $\Xi$-Hilfer fractional fuzzy differential equations with impulse. Further, by using the techniques of nonlinear functional analysis, we study the Ulam-Hyers-Rassias stability.

Hilfer fractional neutral stochastic integro‐differential evolution hemivariational inequalities and optimal controls

The focus of this study is a control system with a nonlocal state that is driven by a neutral stochastic integro‐differential evolution hemivariational inequality known as Hilfer fractional (HF). In order to demonstrate effective requirements for the existence of mild solutions to the relevant control system, we first make use of the features of the extended Clarke subdifferential and a fixed point theorem for multivalued functions. We next prove that there are optimal state‐control sets that are induced by HF neutral stochastic integro‐differential evolution hemivariational inequalities with a nonlocal condition by using limited Lagrange optimal structures. The distinctiveness of the control system's solutions is not taken into consideration while developing optimal control outcomes. At last, an example is employed to go over the major findings.

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results.

Controllability Results of Hilfer Fractional Derivative Through Integral Contractors

Controllability Results of Hilfer Fractional Derivative Through Integral Contractors

The Regional Enlarged Observability for Hilfer Fractional Differential Equations

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ν01 on an internal subregion ω from the whole domain Ω with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results.

Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

This paper aims to study the existence and uniqueness of the solution for nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains a , ∞ , a ≥ 0 , in an applicable Banach space by utilizing the Banach contraction principle. Furthermore, we discuss various types of stability such as Ulam–Hyers–Rassias (UHR), Ulam–Hyers (UH), and semi-Ulam–Hyers–Rassias (sUHR) for nonlocal boundary value problem. Absolutely, our results cover that several outcomes have existed in the literature.

Existence of Sobolev-Type Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities and Optimal Controls

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control system, we first use the characteristics of the modified Clarke sub-differential and a fixed point approach for multivalued functions. Then, we show that there are optimal state-control sets that are driven by Sobolev-type HF neutral stochastic evolution HVIs utilizing constrained Lagrange optimal systems. The optimal control (OC) results are created without taking the uniqueness of the control system solutions into account. Finally, the main finding is shown by an example.

On Implicit k-Generalized $$\psi $$-Hilfer Fractional Differential Coupled Systems with Periodic Conditions

On Implicit k-Generalized $$\psi $$-Hilfer Fractional Differential Coupled Systems with Periodic Conditions

Trajectory Controllability of Hilfer Fractional Neutral Stochastic Differential Equations with Deviated Argument Using Rosenblatt Process and Poisson Jumps

Trajectory Controllability of Hilfer Fractional Neutral Stochastic Differential Equations with Deviated Argument Using Rosenblatt Process and Poisson Jumps

Exact Controllability of Hilfer Fractional Differential System with Non-instantaneous Impluleses and State Dependent Delay

Exact Controllability of Hilfer Fractional Differential System with Non-instantaneous Impluleses and State Dependent Delay

Analysis on Controllability Results for Impulsive Neutral Hilfer Fractional Differential Equations with Nonlocal Conditions

In this paper, we investigate the controllability of the system with non-local conditions. The existence of a mild solution is established. We obtain the results by using resolvent operators functions, the Hausdorff measure of non-compactness, and Monch’s fixed point theorem. We also present an example, in order to elucidate one of the results discussed.

Approximate controllability of Hilfer fractional neutral stochastic systems of the Sobolev type by using almost sectorial operators

<abstract><p>The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.</p></abstract>

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.

Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness

Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness

Analysis and Applications of Sequential Hybrid $$\psi $$-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra

Analysis and Applications of Sequential Hybrid $$\psi $$-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example.

Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

Existence and Controllability Results for Hilfer Fractional Evolution Equations via Integral Contractors

We investigate the existence and controllability of integral solutions to non-densely defined Hilfer fractional evolution equations. Our results are established by implementing the concept of a bounded integral contractor and by using a sequencing technique. In contrast to the papers available in the literature, in order to establish our exact controllability results, we do not need to define the induced inverse of the controllability operator and we do not need the non-linear function to satisfy a Lipschitz condition. We also establish trajectory controllability (which is stronger than other controllability concepts) results for Hilfer fractional evolution equations. Finally, we present and discuss an example which illustrates our results.

Investigation of a Nonlinear Coupled (k, ψ)–Hilfer Fractional Differential System with Coupled (k, ψ)–Riemann–Liouville Fractional Integral Boundary Conditions

This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,ψ)–Hilfer fractional differential equations subject to coupled (k,ψ)–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results.

Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann–Liouville and Hadamard-Type Iterated Integral Boundary Conditions

We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence of solutions to the given problem are proven. The third result provides a sufficient criterion for the existence of a unique solution to the problem at hand. Numerical examples are constructed to demonstrate the application of the results obtained. Two graphs show asymmetric solutions when a Hilfer parameter is varied. The work presented in this paper is novel and significantly enriches the literature on the topic.