Impulsive Fractional Differential Equations Research Articles

Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Finite-Time Stability of Impulsive Fractional Differential Equations with Pure Delays

This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results.

Well posedness of second-order non-instantaneous impulsive fractional neutral stochastic differential equations

In this manuscript, the authors established existence, uniqueness and stability results for the second-order non-instantaneous impulsive fractional neutral stochastic differential equations (NIIFNSDEs). At first, existence and uniqueness results are obtained by using Caputo fractional derivative (fractional calculus), stochastic technique and fixed point approach with appropriate hypotheses on non-linear continuous functions. Secondly, we discuss the Ulam-Hyers Rassias stability and henceforth, we study Ulam-Hyers stability for NIIFNSDEs. Finally, an example is provided to validate the theoretical findings.

A novel numerical approach for time-varying impulsive fractional differential equations using theory of functional connections and neural network

In this paper, we propose a physics-informed neural network-based scheme to solve time-varying impulsive fractional differential equations without any labeled data. At first, the existence and uniqueness of the solution of the impulsive fractional system are proved theoretically using the Bourdin state transition matrix-based solution representation and the Banach fixed point theorem. The proposed method uses the physics-informed neural network, extreme learning machines, and the theory of functional connections. A neural network is employed as the free function to construct the constrained expression. Based on the constrained expression, the linear system is trained using the extreme learning machine algorithm, and the nonlinear system is trained using the iterative least squares method, as the only trainable parameters are the output weights. The convergence and error analysis for the algorithm is derived. The neural network training is graphically demonstrated for the nonlinear system using the loss obtained in each iteration. For the linear case, the results are validated with the original solution, while for the nonlinear case, it is verified with the original solution and other classical methods. The primary use of the theory of functional connections is it helps the solution satisfy the physical constraint.

An analysis on approximate controllability results for impulsive fractional differential equations of order 1 < r < 2 with infinite delay using sequence method

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order with infinite delay. Sectorial operator of type , the nonlinear alternative of the Leray–Schauder fixed point theorem, sequence method, and impulsive systems have been used to establish these results. First, we investigate the existence of mild solutions for impulsive fractional differential equations of order . We also establish the approximate controllability results for the nonlocal fractional delay differential equations. An example is also given to illustrate our main results.

Solving Coupled Impulsive Fractional Differential Equations With Caputo-Hadamard Derivatives in Phase Spaces

In this manuscript, we incorporate Caputo-Hadamard derivatives in impulsive fractional differential equations to obtain a new class of impulsive fractional form. Further, the existence of solutions to the proposed problem has been inferred under a state-dependent delay and suitable hypotheses in phase spaces. Finally, the considered problem has been supported by an illustrative application.

(ω,ρ)-BVP Solutions of Impulsive Differential Equations of Fractional Order on Banach Spaces

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer’s fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results.

Fractional neutral functional differential equations driven by the Rosenblatt process with an infinite delay

Abstract This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result.

On the (k,Ψ)‐Hilfer nonlinear impulsive fractional differential equations

In this paper, the ‐Hilfer fractional derivative, the most generalized fractional derivative operator, is used to analyze nonlinear impulsive fractional differential equations. The initial goal of this work is to provide a suitable representation formula for the solutions of the ‐Hilfer nonlinear impulsive fractional differential equations. The second goal is to use the representation formula we came up with to show that there is a solution to the ‐Hilfer nonlinear impulsive fractional differential equations. Analysis of the work done is based on the calculus of the ‐Hilfer fractional derivative, the fixed point theorems, and the measure of noncompactness. We illustrate the outcomes of our work through examples.

Some Results and Analysis of Nonlocal Special Random Impulsive Fractional Differential Equations

Some Results and Analysis of Nonlocal Special Random Impulsive Fractional Differential Equations

Some sufficient conditions of existence and trajectory controllability for impulsive and initial value fractional order functional differential equations

For a class of impulsive fractional differential equation with fractional order r∈(1,2) by means of solution operator Sr(t), we develop the mild solution and deal with the existence results of mild solution for the considered equation with state-dependent delay through three fixed point theorems, namely Banach, Nonlinear Leray–Schauder Alternative and Krasnoselskii’s fixed point theorem. Next, we define the mild solution via an operator function Wr,β(τ) for abstract version of the considered equation. Further, we study the trajectory controllability (which is a stronger notion than other controllability) of the equation. In the last, we present some examples to illustrate the established results through partial fractional differential equations.

Existence and multiplicity of solutions for (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type fractional differential equations with impulses

In this paper, we investigate the existence and multiplicity of solutions for a class of ‐Laplacian Kirchhoff‐type impulsive fractional differential equations. First, under a weaker condition than the Ambrosetti–Rabinowitz condition and the Miyagaki–Souto condition, the existence of an unbounded sequence of nontrivial solutions follows from the fountain theorem. Then, some new criteria are given to guarantee that the fractional differential equation has at least two nontrivial solutions using the Nehari manifold method combined with the fibering map.

Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory

This article aims to investigate the existence and uniqueness of solutions to impulsive fractional differential equations. Firstly, we show a formula for solutions to an impulsive fractional problem involving a generalization of the classical Caputo derivative with a nonlocal condition in a Banach space. Secondly, some sufficient conditions for the existence of solutions are established by applying fixed point methods. Furthermore, we reconsider some of operators for deriving the existence and uniqueness of the solution to impulsive fractional problem using topological degree techniques. The results support this in one example.

Ulam–Hyers stability for second-order non-instantaneous impulsive fractional neutral stochastic differential equations

In this paper, sufficient conditions are established for the Ulam–Hyers stability of second-order non-instantaneous impulsive fractional neutral stochastic differential equations (NIIFNSDEs) with supremum norm in the pth means square sense. The existence of solution of NIIFNSDEs is derived by using the cosine family of linear operator, Itô’s formula, and Mönch fixed point theorem in infinite-dimensional space. Finally, an example is demonstrated to illustrate the obtained theoretical results.

Existence and Finite-Time Stability Results for Impulsive Caputo-Type Fractional Stochastic Differential Equations with Time Delays

ABSTRACT This paper mainly discusses the existence and finite-time stability of solutions for impulsive fractional stochastic differential equations (IFSDEs). By applying the Picard-Lindelöf iteration method of successive approximation scheme, we establish the existence results of solutions. Subsequently, the uniqueness of solution is derived by the method of contradiction. In addition, we investigate the finite-time stability by means of the generalized Grönwall-Bellman inequality. As an application, examples are provided to expound our theoretical conclusions.

Cauchy problem for impulsive fractional differential equations with almost sectorial operators

We study a class of non-instantaneous impulsive fractional differential equations in a Banach space. We consider the Cauchy problem with Caputo derivative having variable lower limits and almost sectorial operators. We establish the existence of mild solutions in the space of piecewise continuous functions. We employ the properties of semigroup associated with the almost sectorial operator, theory of measure of non-compactness and fixed point theorems.

Controllability results to non-instantaneous impulsive with infinite delay for generalized fractional differential equations

This paper discusses controllability results for active types with infinite-time delay of non-instantaneous impulsive fractional differential equations. The model is constructed based on the generalized Caputo (Caputo-Katugampola) fractional derivative and the control function with non-local Katugampola fractional integral as a boundary condition. Our principal results are established by giving some sufficient hypotheses, utilizing well-known fractional calculus truths and using Krasnoselskii’s fixed point theorem. The infinite time delay has been treated with the abstract phase space techniques and fulfilling the ensuing axioms due to Hale and Kato. It turns out that under some sufficient conditions, the problem has at least one controllable solution. An implementation of our theoretical results is demonstrated by a numerical example.

Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions

In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results.

The existence of positive solutions for φ-Hilfer fractional differential equation with random impulses and boundary value conditions

In this paper, we consider a class of impulsive fractional differential equations involving φ-Hilfer fractional derivatives. Since the φ-Hilfer fractional derivative depends on the kernel φ, for a suitable choice of the kernel φ, our results in this paper can be applied to most of the fractional derivatives. First, we give a correspondence in this article between our problem and a Volterra-type integral equation. Then, sufficient conditions are given to ensure the existence of positive solutions and by applying the cone stretch and cone compression immobilization theorem and the Green function to study the existence of positive solutions for the problem. Finally, an example is given to illustrate the viability of the main theories.