Hilfer Fractional Research Articles

Trajectory Controllability of the Systems Governed by Hilfer Fractional Systems

In this manuscript, we consider a nonlinear system governed by Hilfer fractional integro-differential equations in a Banach space. Using the concept of operator semigroup and Gronwall’s inequality, we have established the trajectory controllability of the integro-differential equation with local and non-local conditions. Finally, we have given an example to illustrate the application of the derived results

Analytical Study of Two Nonlinear Coupled Hybrid Systems Involving Generalized Hilfer Fractional Operators

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving ϕ-Hilfer fractional derivatives. The existence results are established in the weighted space of functions using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra and Dhage’s helpful generalization of Krasnoselskii fixed- point theorem. Finally, simulated examples are provided to demonstrate the obtained results.

Approximate Controllability of a Class of Semilinear Hilfer Fractional Differential Equations

Approximate Controllability of a Class of Semilinear Hilfer Fractional Differential Equations

A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions

This paper is a survey of the recent results of the author for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in (1,2] supplemented with different kinds of nonlocal boundary conditions.

Sensitivity Analysis of Optimal Control Problems Governed by Nonlinear Hilfer Fractional Evolution Inclusions

This article studies sensitivity properties of optimal control problems that are governed by nonlinear Hilfer fractional evolution inclusions (NHFEIs) in Hilbert spaces, where the initial state $$\xi $$ is not the classical Cauchy, but is the Riemann–Liouville integral. First, we obtain the nonemptiness and the compactness properties of mild solution sets $$\mathbb {S}(\xi )$$ for NHFEIs, and also establish an extension Filippov’s theorem. Then we obtain the continuity and upper semicontinuity of optimal control problems connected with NHFEIs depending on a initial state $$\xi $$ as well as a parameter $$\lambda $$ . Finally, An illustrating example is given.

Existence Results for a Class of Coupled Hilfer Fractional Pantograph Differential Equations with Nonlocal Integral Boundary Value Conditions

This paper deals with the existence and uniqueness of solutions for a new class of coupled systems of Hilfer fractional pantograph differential equations with nonlocal integral boundary conditions. First of all, we are going to give some definitions that are necessary for the understanding of the manuscript; second of all, we are going to prove our main results using the fixed point theorems, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem; in the end, we are giving two examples to illustrate our results.

Boundary Value Problems for Hilfer Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions

In this paper, we study boundary value problems for differential inclusions, involving Hilfer fractional derivatives and nonlocal integral boundary conditions. New existence results are obtained by using standard fixed point theorems for multivalued analysis. Examples illustrating our results are also presented.

Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

On Coupled Systems for Hilfer Fractional Differential Equations with Nonlocal Integral Boundary Conditions

In this paper, we study a coupled system involving Hilfer fractional derivatives with nonlocal integral boundary conditions. Existence and uniqueness results are obtained by applying Leray-Schauder alternative, Krasnoselskii’s fixed point theorem, and Banach’s contraction mapping principle. Examples illustrating our results are also presented.

Bounded Weak Solutions for Hilfer Fractional Differential Equations on the Half Line

Bounded Weak Solutions for Hilfer Fractional Differential Equations on the Half Line

Constructive Existence of Solutions of Multi-point Boundary Value Problem for Hilfer Fractional Differential Equation at Resonance

In this paper, we study the constructive existence of solutions of multi-point boundary value problem for Hilfer fractional differential equation at resonance. The constructive existence of solutions is very important in practice, because it gives approximation method to solve equation as well as the existence. First, we obtain reversibility of some fractional integral operator which is useful for description and get analytic expressions of the inverse operator with Mittag–Leffler function. Next, we have the solution expressions of initial value problem and boundary-value problem for linear Hilfer fractional differential equations. And we get the sufficient conditions for the existence and nonexistence of solutions of multi-point value problem for nonlinear Hilfer fractional differential equation at resonance and establish the approximation method to solve it by the method of upper and lower solutions. Finally, we illustrate our results with an example.

Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

Existence and Attractivity Results for Hilfer Fractional Differential Equations

We present some results on the existence of attracting solutions of some fractional differential equations of the Hilfer type. The results on the existence of solutions are applied to the Schauder fixed-point theorem. It is proved that all solutions are uniformly locally attracting.

On the Novel Ulam–Hyers Stability for a Class of Nonlinear $$\psi $$-Hilfer Fractional Differential Equation with Time-Varying Delays

In this paper, we present some alternative results concerning the uniqueness and Ulam–Hyers stability of solutions for a kind of $$\psi $$ -Hilfer fractional differential equations with time-varying delays. Under some updated criteria along with the generalized Gronwall inequality, the new constructive results have been established in the literature. The derived analysis has the ability to generalize and improve some other results from the literature. As an application, two typical examples are delineated to demonstrate the effectiveness of our theoretical results.

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Existence and Stability Results for Boundary Value Problem for Differential Equation with ψ -Hilfer Fractional Derivative

Existence and Stability Results for Boundary Value Problem for Differential Equation with ψ -Hilfer Fractional Derivative

Controllability of Hilfer Fractional Differential Systems with Non-Dense Domain

In this paper, some exact controllability results are investigated for an integral solution of a non-densely defined abstract fractional differential system involving Hilfer fractional derivative. We tend to implement semigroup theory, fractional calculus, and measure of noncompactness to obtain the main results by fixed point technique. Finally, an application is given to illustrate the obtained results.

Results on Controllability of Nonlinear Hilfer Fractional Stochastic System

Abstract The main objective of this paper is to present sufficient conditions for controllability of Hilfer fractional nonlinear stochastic systems in finite dimensional space. The main results are obtained by using the Nussbaum fixed point theorem, stochastic analysis approach and generalized fractional calculus (Hilfer fractional derivative) which is universality of Riemann-Liouville and Caputo fractional derivative. Finally, a numerical example is provided to show the effectiveness of the obtained theoretical result. The obtained result is more generalized one than the existing results on fractional stochastic system in finite dimensional space.

Random Coupled Hilfer and Hadamard Fractional Differential Systems in Generalized Banach Spaces

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.