Hilfer Fractional Evolution Research Articles

Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation

Abstract In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Attractive solutions for Hilfer fractional neutral stochastic integro-differential equations with almost sectorial operators

<abstract><p>This paper studies the integro-differential equations of Hilfer fractional (HF) neutral stochastic evolution on an infinite interval with almost sectorial operators and their attractive solutions. We use semigroup theory, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. We establish the existence and attractivity theorems for mild solutions by considering the fact that the almost sectorial operator is both compact and noncompact. Example that highlight the key findings are also provided.</p></abstract>

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

The Averaging Principle for Hilfer Fractional Stochastic Evolution Equations with Lévy Noise

This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results.

Hilfer fractional neutral stochastic Sobolev-type evolution hemivariational inequality: Existence and controllability☆

This paper discusses the approximate controllability of hemivariational inequalities of the Sobolev-type Hilfer fractional neutral stochastic evolution system. Firstly, by using fixed point theorems, fractional calculus, Hilfer fractional derivatives, multivalued maps and stochastic analysis, for stochastic fractional evolution systems, we demonstrate mild solutions. Whereupon we introduce necessary conditions that ensure the approximate controllability of stochastic fractional systems. As a final step, we define our primary outcomes using an example.

An analysis concerning to the existence of mild solution for Hilfer fractional neutral evolution system on infinite interval

In this article, we focus on the new class of neutral Hilfer fractional differential equations on infinite interval in Banach space. The results are obtained with the help of the theory of fractional calculus, mild solutions, the fixed point theorem, semigroup theory, and infinite interval. First, we introduce a mild solution for neutral Hilfer fractional differential equations. Then, we prove the existence of a mild solution for the neutral Hilfer fractional differential equations. Finally, an example is provided to illustrate the obtained results.

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

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ∈(1,2) and type ν∈[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.

$$\alpha $$-Whittaker controllability of $$\vartheta $$-Hilfer fractional stochastic evolution equations driven by fractional Brownian motion

$$\alpha $$-Whittaker controllability of $$\vartheta $$-Hilfer fractional stochastic evolution equations driven by fractional Brownian motion

Controllability of Hilfer fractional Langevin evolution equations

The existence of fractional evolution equations has attracted a growing interest in recent years. The mild solution of fractional evolution equations constructed by a probability density function was first introduced by El-Borai. Inspired by El-Borai, Zhou and Jiao gave a definition of mild solution for fractional evolution equations with Caputo fractional derivative. Exact controllability is one of the fundamental issues in control theory: under some admissible control input, a system can be steered from an arbitrary given initial state to an arbitrary desired final state. In this article, using the (α, β) resolvent operator and three different fixed point theorems, we discuss the control problem for a class of Hilfer fractional Langevin evolution equations. The exact controllability of Hilfer fractional Langevin systems is established. An example is also discussed to illustrate the results.

An analysis of approximate controllability for Hilfer fractional delay differential equations of Sobolev type without uniqueness

This study focused on the approximate controllability results for the Hilfer fractional delay evolution equations of the Sobolev type without uniqueness. Initially, the Lipschitz condition is derived from the hypothesis, which is represented by a measure of noncompactness, in particular, nonlinearity. We also examined the continuity of the solution map of the Sobolev type of Hilfer fractional delay evolution equation and the topological structure of the solution set. Furthermore, we prove the approximate controllability of the fractional evolution equation of the Sobolev type with delay. Finally, we provided an example to illustrate the theoretical results.

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a $$C_0$$ -semigroup $$W(\cdot )$$ generated by the operator $$-A$$ and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding $$C_0$$ -semigroup $$W(\cdot )$$ and non-instantaneous impulsive function $$\gamma _k$$ are compact, $$W(\cdot )$$ is not compact and $$\gamma _k$$ is compact, $$W(\cdot )$$ and $$\gamma _k$$ are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given.

Decay mild solutions of Hilfer fractional differential variational–hemivariational inequalities

The primary objective of this paper is to explore a decay mild solution governed by a class of dynamical systems, called Hilfer fractional differential variational–hemivariational inequality (HFDVHVI, for short), which is composed of a Hilfer fractional evolution differential inclusion and a variational–hemivariational inequality involving two history-dependent operators in the framework of spaces. Our first aim is to investigate the solvability of the mild solutions to (HFDVHVI) by means of fixed point principle. The second step of the paper is to study the existence of decay mild solutions to (HFDVHVI) via giving expression for the Mittag-Leffler function and the Wright function.

A Study on the Approximate Controllability of Hilfer Fractional Evolution Systems

In this article, the existence and uniqueness of mild solutions are investigated for Hilfer fractional evolution systems. Particularly, the approximate controllability is also investigated under some essential conditions by applying the sequence method. An example, as an application, is provided to demonstrate the obtained results.

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.

A discussion concerning approximate controllability results for Hilfer fractional evolution equations with time delay

A discussion concerning approximate controllability results for Hilfer fractional evolution equations with time delay

Hilfer fractional stochastic evolution equations on infinite interval

Abstract This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our results.

Approximate controllability for Hilfer fractional stochastic evolution inclusion with nonlocal conditions

In this paper, the approximate controllability of Hilfer fractional stochastic evolution inclusion with nonlocal conditions is investigated. By using fractional calculus, semigroups theory, stochastic analysis and the fixed point theorem for multi-valued maps, a new sufficient condition for the existence of mild solution of the Hilfer fractional stochastic system in the space of weighted continuous functions is derived. Furthermore, a novel sufficient condition for the approximate controllability of the Hilfer fractional stochastic evolution inclusion with nonlocal conditions is established. Finally, an example is presented to illustrate the main results.