Fractional Neutral Functional Differential Equations Research Articles

Existence results for fractional neutral functional differential equations with infinite delay and nonlocal boundary conditions

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate 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.

Ulam–Hyers stability for an impulsive Caputo–Hadamard fractional neutral stochastic differential equations with infinite delay

This paper addresses existence and Ulam–Hyers stability (UHS) problems for an impulsive Caputo–Hadamard fractional neutral functional stochastic differential equation with infinite delay (FNFSDEwID). We first prove the existence and uniqueness of the solution using Banach fixed point theorem and standard stochastic analysis techniques. We then tackle the UHS under a Lipschitz condition on a bounded and closed interval. We end up with an illustrative example that corroborates our theoretical findings.

Continuity and approximation properties of solutions to fractional neutral stochastic functional differential equations with non-Lipschitz coefficients

ABSTRACT This paper aims to investigate a fractional neutral stochastic functional differential equation (FNSFDE) with non-Lipschitz coefficients. Under the assumptions, we first establish the continuity of the solution in the fractional order of the equation. Furthermore, an Euler-Maruyama (EM) approximation is constructed and then we obtain the strong convergence of the numerical scheme. Specially, if the non-Lipschitz conditions are replaced with the Lipschitz conditions, we shall get a definite convergence rate, which is related to the fractional order of the equation. Finally, we consider the averaging principle for the fractional neutral stochastic equation, which provides us with an easy way to study the properties of the equation.

Complete controllability of nonlinear fractional neutral functional differential equations

This paper is concerned with the complete controllability of a nonlinear fractional neutral functional differential equation. Some sufficient conditions are established for the complete controllability of the nonlinear fractional system. The conditions are established based on the fractional power of operators and the fixed-point theorem under the assumption that the associated linear system is completely controllable. Finally, an example is presented to illustrate our main result.

Retarded stochastic fractional neutral functional differential equations driven by Rosenblatt process with unbounded delay

Retarded stochastic fractional neutral functional differential equations driven by Rosenblatt process with unbounded delay

Controllability of fractional neutral functional differential equations with infinite delay driven by fractional Brownian motion

Abstract In this work, we establish a controllability result for a class of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion. To attain our objective we adapt the argument of Lakhel & McKibben (2018, Stochastics 90, no. 3, 313–329) where the existence of mild solutions to such stochastic equations was studied. An example is provided to show the applicability of the theoretical result.

Initial value problem for a fractional neutral differential equation with infinite delay

We consider the initial value problem for a class of nonlinear fractional neutral functional differential equations with infinite delay involving the standard fractional derivative in the sense of Caputo. By using a variety of tools of fractional calculus including the Banach contraction principle and the Schaefer fixed point theorem, the existence, uniqueness and continuous dependence results are obtained in the space of continuous functions.

Asymptotic behavior of solutions to time fractional neutral functional differential equations

In this paper, we derive a new fractional Halanay-like inequality, which is used to characterize the long-term behavior of time fractional neutral functional differential equations (F-NFDEs) of Hale type with order α∈(0,1). The contractivity and dissipativity of F-NFDEs are established under almost the same assumptions as those for classical integer-order NFDEs. In contrast to the exponential decay rate for NFDEs, the F-NFDEs are proved to have a polynomial decay rate. The numerical scheme based on the L1 method together with linear interpolation is constructed and applied in several examples to illustrate the theoretical results and to reveal the quite different long-term decay rate in the solutions between F-NFDEs and NFDEs.

Existence of nonoscillatory solutions for fractional neutral functional differential equation

Existence of nonoscillatory solutions for fractional neutral functional differential equation

The existence and Hyers\u2013Ulam stability of solution for an impulsive Riemann\u2013Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1

This paper deals with the existence of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1<beta<2 and its Hyers–Ulam stability. We prove the mild solutions for the equation using basic theorems of fractional differential equation. The existence result of the equation is obtained by Mönch’s fixed point theorem. Finally, we prove the Hyers–Ulam stability of the solution.

Weighted Fractional Neutral Functional Differential Equations

Weighted Fractional Neutral Functional Differential Equations

Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

In this paper, we consider a class of fractional neutral stochastic functional differential equations with infinite delay driven by a cylindrical fractional Brownian motion (fBm) in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delay

This paper addresses a class of fractional stochastic impulsive neutral functional differential equations with infinite delay which arise from many practical applications such as viscoelasticity and electrochemistry. Using fractional calculations, fixed point theorems and the stochastic analysis technique, sufficient conditions are derived to ensure the existence of solutions. An example is provided to prove the main result.

Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

Existence and stability of Hadamard p-type fractional functional differential equations

In this paper, we investigate initial value problems for Hadamard p-type fractional order functional and neutral functional differential equations. Sufficient conditions which guarantee the existence and Hyers-Ulam stability are new and obtained by some classical fixed point theory. Examples are also given to illustrate our theoretical results.

Existence and stability for Hadamard p-type fractional functional differential equations

In this paper, we study the existence and Hyers–Ulam stability of solutions for a Hadamard p-type fractional order functional and neutral functional differential equations involving initial value problems. Sufficient conditions which guarantee the existence and Hyers–Ulam stability are new and obtained.

Existence of Mild Solutions for Fractional Neutral Functional Differential Equations

Existence of Mild Solutions for Fractional Neutral Functional Differential Equations

Existence Results for Fractional Neutral Functional Differential Equations with Random Impulses

In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results.

Existence and continuous dependence of mild solutions for fractional neutral abstract evolution equations

This paper is mainly concerned with the existence, uniqueness and continuous dependence of mild solutions for fractional neutral functional differential equation with nonlocal initial conditions and infinite delay. The results are obtained by means of the classical fixed point theorems combined with theory of resolvent operators for integral equations.