Nonlinear 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.

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.

Variational formulation for nonlinear impulsive fractional differential equations with (p, q)‐Laplacian operator

In this paper, we focus on a class of generalized (p, q)‐Laplacian‐type impulsive fractional differential equation for 1 < p≤q < ∞, with a nonlinearity f containing fractional derivatives and simultaneously and being imposed on mild assumptions contrasting with previous works. The underlying idea is based on the Mountain pass theorem and the iterative technique to obtain the existence of nontrivial solutions for our equation. Our work extends and supplements some existing results.

RETRACTED: Existence of solutions for fractional order impulsive differential equations with integral boundary conditions

In this paper, we prove the expression and the existence of a class of nonlinear impulsive fractional order differential equations with integral boundary conditions. The unique solution of the differential equations by Green’s function is given. By using Schauder fixed point theorem and Leray-Schauder fixed point theorem, several sufficient conditions for the existence and uniqueness results are established.

Convergence Analysis of Implicit Euler Method for a Class of Nonlinear Impulsive Fractional Differential Equations

For a class of nonlinear impulsive fractional differential equations, we first transform them into equivalent integral equations, and then the implicit Euler method is adapted for solving the problem. The convergence analysis of the method shows that the method is convergent of the first order. The numerical results verify the correctness of the theoretical results.

Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

The main purpose of this paper is to study the existence theorem for a common solution to a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations. In this respect, we study the fuzzy version of some essential common fixed-point results from metric spaces in the newly introduced notion of complex valued fuzzy metric spaces. Also, we provide an illustrative example to demonstrate the validity of our derived results.

Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations

The aim of this paper is to use the combination of Reproducing kernel Hilbert space method (RKHSM) and fractional differential transform method (FDTM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solutions of this class of equations are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.

Multiple positive solutions for a coupled system of nonlinear impulsive fractional differential equations with parameters

We consider the multiplicity of positive solutions (PSs) for a coupled system involving nonlinear impulsive fractional differential equations with parameters. By employing the classical Guo–Krasnosel’skii fixed point theorem, some sufficient criteria for the existence of multiple PSs in terms of different values of parameters are derived. As an application, an example is given to illustrate the theoretical results.

ON THE NONLINEAR IMPULSIVE Ψ–HILFER FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider the nonlinear Ψ-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of solutions. The acquired results are extended to the nonlocal Ψ-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we obtained. Further, examples are provided in support of the results we got.

Ulam\u2013Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions

In this article, we discuss the sufficient conditions for the existence, uniqueness and stability of solutions to a class of nonlinear impulsive boundary value problem of fractional order differential equations. Using classical fixed point theorems, we develop the required conditions. Further, using the techniques of nonlinear functional analysis, we investigate Ulam–Hyers stability results to the proposed problem. For applications of our derived results, we present two numerical examples.

Numerical methods for solving initial value problems of some kinds of nonlinear impulsive fractional differential equations

Numerical methods for solving initial value problems of some kinds of nonlinear impulsive fractional differential equations

Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients

In this paper, the nonlinear multiple base points boundary value problems of the impulsive fractional differential equations are studied. By using the fixed point theorem and the Mittag-Leffler functions, the sufficient conditions for the existence of the solutions to the given problems are formulated. An example is presented to illustrate the result.

Existence of solutions for the nonlinear multiple base points impulsive fractional differential equations with the three-point boundary conditions

In this paper, we study a class of nonlinear multiple base points impulsive fractional differential equations involving the three-point boundary conditions. A new result on the existence of a solution is established by using fixed point theorems. An example is presented to illustrate the result.

Solvability of Boundary Value Problems for Impulsive Fractional Differential Equations Via Critical Point Theory

In this paper, we consider boundary value problems for impulsive fractional differential equations containing left and right Riemann–Liouville fractional integral operators. Variational structure for these problems are established in a proper fractional derivative space, which can be regarded as a novelty item. Some sufficient conditions for the existence of solutions to this boundary value problem for nonlinear impulsive fractional differential equations are established by applying critical point theorems and some skills of inequalities. Finally, two examples are presented to show the feasibility and effectiveness of our results.

Boundary value problems of multi‐term fractional differential equations with impulse effects

We point out some mistakes in a known paper. Some existence results for solutions of two classes of boundary value problems for nonlinear impulsive fractional differential equations are established. Our analysis relies on the well‐known Schauder fixed point theorem. Examples are given to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.

A Survey of Lyapunov Functions, Stability and Impulsive Caputo Fractional Differential Equations

Abstract We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives

Abstract In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory. Examples are given to illustrate main results. This paper is motivated by [Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19], [Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J, Appl, Math, Comput. 39(2012) 421-443] and [Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl. 2013, 2013: 80].

Nonlinear Four-point Impulsive Fractional Differential Equations with p-Laplacian Operator

In this paper, we investigate the existence of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order α ∈ (2,3]. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. Some illustrative examples are also discussed.

Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions

By constructing Green’s function, we give the natural formulae of solutions forthe following nonlinear impulsive fractional differential equation with generalizedperiodic boundary value conditions: where is a real number, is the standard Caputo differentiation. We present theproperties of Green’s function. Some sufficient conditions for the existence ofsingle or multiple positive solutions of the above nonlinear fractional differentialequation are established. Our analysis relies on a nonlinear alternative of theSchauder and Guo-Krasnosel’skii fixed point theorem on cones. As applications,some interesting examples are provided to illustrate the main results. MSC: 34B10, 34B15, 34B37.

Existence of solution to a class of boundary value problem for impulsive fractional differential equations

By means of the Green function, the boundary value problem of a fractional differential equation can be reduced to the equivalent integral equation. Recently, this method has been used successfully to discuss the existence of the solution to the boundary value problem of a nonlinear fractional differential equation. By applying the nonlinear alternative of the Leray-Schauder type and the Krasnoselskii fixed point theorem, we investigate the boundary value problem of a nonlinear impulsive fractional differential equation, and we obtain two existence results for the solution.MSC:26A33, 34B15.