Hilfer Fractional Differential Equations Research Articles

Solvability of nonlocal fractional vector functional boundary value problems with p‐Laplacian on a half‐line at resonance

In this paper, the solvability of a system of fully nonlinear Hilfer fractional differential equations on the half‐line is studied. Firstly, considering that the half‐line as the domain has the problem of insufficient compactness, we defined the space of weighting functions and normed the convergence at infinity. Secondly, by constructing appropriate operators in these Banach spaces and using the extension for the continuous theorem, some sufficient conditions for the solvability of the problem are obtained. Finally, an example is given to illustrate the main results.

On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals

In the present research, we investigate the existence and uniqueness of solutions for a system of (k; χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann–Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. Examples are also constructed to illustrate the obtained results.

Approximate controllability for Hilfer fractional stochastic differential systems of order 1

In this paper, we analyse the approximate controllability of Hilfer fractional stochastic differential systems of order 1 < μ < 2 in Hilbert spaces. The primary findings are carried out by using fractional calculus, stochastic analysis theory, cosine family, and Schauder's fixed point theorem. In particular, we draw a new set of sufficient conditions for the approximate controllability of Hilfer fractional stochastic differential equations of order 1 < μ < 2 under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the derived theory.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples.

The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions

This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results.

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.

Controllability of Hilfer Fractional Semilinear Integro-Differential Equation of Order α ∊ (0, 1), β ∊ [0, 1

In this paper, Hilfer fractional differential equation is studied. Firstly, we used Laplace transform and semigroup theory to find the mild solution of the system. Then exact controllability of proposed system is established using the Arzela-Ascoli theorem and Schauder fixed point theorem. To illustrate the developed theory we provide an example at the end.

Null controllability of Hilfer fractional stochastic differential equations with nonlocal conditions

&lt;p style='text-indent:20px;'&gt;This paper assesses a class of nonlocal Hilfer fractional stochastic differential equations (NHFSDEs) governed by fractional Brownian motion (fBm). New sufficient condition of exact null controllability for this stochastic setting has been investigated using fractional calculus, fixed point theory and a theory of resolvent operator. The derived result is new because it generalizes several of previously published results. However, the obtained results are proven by an illustration using stochastic partial differential equations to demonstrate the present application characteristic of null controllability. A filter example is demonstrated to make use of resolvent operator in a practical way.&lt;/p&gt;

Approximate controllability of neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.

Study on the controllability of Hilfer fractional differential system with and without impulsive conditions via infinite delay

In this manuscript, we investigate the controllability of two different kinds of Hilfer fractional differential equations with an almost sectorial operator and infinite delay. First, we demonstrate the exact controllability of the Hilfer fractional system using the measure of noncompactness. Next, we develop the results for the controllability of the system under impulsive conditions. Finally, to show how the key findings may be utilised, applications are presented.

Trajectory controllability of neutral stochastic integrodifferential equations with mixed fractional Brownian motion

This work focuses on the existence and trajectory (T-)controllability of mixed fractional Brownian motion (fBm) with the Hurst index and neutral stochastic integrodifferential equations (NSIDEs) with deviating argument and fBm. Stochastic integrodifferential equations (SIDEs) are solved in Hilbert space using stochastic analysis, the resolvent operator, and Krasnoselskii's fixed point theorem (KFPT). Furthermore, providing adequate assumptions, the T-controllability of the considered system is organised by using extended Gronwall's inequality. We demonstrate the theoretical insights and numerical simulations are included which is unique and makes this work more interesting. The obtained results generalise existing results from [Chalishajar, D. N., George, R. K., & Nandakumaran, A. K. (2010). Trajectory controllability of nonlinear integro-differential system. Journal of Franklin Institute, 347(7), 1065–1075.; Durga, N., Muthukumar, P., & Malik, M. (2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 1–27.; Muslim, M., & George, R. K. (2019). Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27, 529–537.; Dhayal, R., Malik, M., & Abbas, S. (2021). Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680.].

Existence of Mild Solution of the Hilfer Fractional Differential Equations with Infinite Delay on an Infinite Interval

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli–Arzelà theorem and Mönch’s fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results.

Hilfer fractional delay differential equations: Existence and uniqueness computational results and pointwise approximation utilizing the Shifted-Legendre Galerkin algorithm

In this research, we investigate Hilfer-type FDDEs of both linear and nonlinear nature, with orders α and β. We studied some Riemann-Liouville, Caputo-Liouville, and Hilfer fractional derivatives properties; and then used them to convert our fractional delay problem into an equivalent fractional Volterra delay problem. After that, for the resulting fractional Volterra delay problem, using the CMT, existence-uniqueness is demonstrated. For numerical solutions, we studied the orthogonal shifted Legendre polynomials and used them as the basis for the Galerkin algorithm to find an approximation solution to the corresponding FDDE. By employing this algorithm, the provided FDDE is converted into a series of algebraic systems. By solving this system, the approximated solutions for the equation are obtained. The main algorithm advantage is that the number of needed iterations is low compared with the soft gained results. Some linear and nonlinear numerical applications together with several figures and tables are utilized to prove the performance and accuracy of the algorithm. The outcomes results are considered according to the given analysis and the numerical method was offered in the last section with several hints to guide future work.

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.

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.

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.

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.

Maximum Principle for Nonlinear Fractional Differential Equations with the Hilfer Derivative

In this paper, two significant inequalities for the Hilfer fractional derivative of a function in the space ACγ([0,b],Rn), 0≤γ≤1 are obtained. We first verified the extremum principle for the Hilfer fractional derivative. In addition, we estimated the Hilfer derivative of a function at its extreme points. Furthermore, we derived and proved a maximum principle for a nonlinear Hilfer fractional differential equation. Finally, we analyzed the solutions of a nonlinear Hilfer fractional differential equation. Our results generalize and extend some obtained theorems on this topic.

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.

New best proximity point (pair) theorems via MNC and application to the existence of optimum solutions for a system of $$\psi $$-Hilfer fractional differential equations

New best proximity point (pair) theorems via MNC and application to the existence of optimum solutions for a system of $$\psi $$-Hilfer fractional differential equations