Neutral Functional Differential Equations Research Articles

Delay‐dependent stability of highly nonlinear neutral stochastic functional differential equations

AbstractThis article focuses on the delay‐dependent stability of highly nonlinear hybrid neutral stochastic functional differential equations (NSFDEs). The delay dependent stability criteria for a class of highly nonlinear hybrid NSFDEs are derived via the Lyapunov functional. The stabilities discussed in this article include stability, asymptotically stability and exponential stability. A numerical example is given to illustrate the criteria established.

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.

Stability analysis of implicit–explicit [formula omitted]-methods for composite stiff neutral functional differential equations in Banach space

This paper is concerned with the stability and asymptotic stability of implicit–explicit (IMEX) θ-methods for composite stiff neutral functional differential equations in Banach space. The stability and asymptotic stability conditions of the methods are obtained. The given numerical experiments confirmed the accuracy of the obtained theoretical results and showed that the IMEX θ-methods have a higher computational efficiency compared to the θ-methods in the end.

Pseudo almost automorphy of stochastic neutral partial functional differential equations with Lévy noise

ABSTRACT The paper introduces and studies the concepts of (Poisson) -pseudo almost automorphy of class r for stochastic processes. Under suitable conditions on the coefficients, we establish the existence, and uniqueness of mild solutions, which are -pseudo almost automorphic of class r in distribution for stochastic neutral partial functional differential equations with Lévy noise.

Analysis of stochastic neutral fractional functional differential equations

This work deals with the large deviation principle which studies the decay of probabilities of certain kind of extremely rare events. We consider stochastic neutral fractional functional differential equation with multiplicative noise and show large deviation principle for its solution processes in a suitable Polish space. The existence and uniqueness results are presented using the Picard iterative method, which is indeed essential for further analysis. The establishment of Freidlin–Wentzell type large deviation principle is solely based on the variational representation developed by Budhiraja and Dupuis in which the weak convergence technique is used to show the sufficient condition. Examples are provided to emphasize the theory.

Boundary value problem for fractional neutral differential equations with infinite delay

In this article, we develop and extend some qualitative analyses of a class of neutral functional differential equations involving a Caputo fractional derivative over the infinite delay period. The existence and uniqueness results are proved based on an equivalent fractional integral equation with the help of Banach contraction principle and Schauder's fixed point theorem.

Global Attractiveness and Quasi-Invariant Sets of Impulsive Neutral Stochastic Functional Differential Equations Driven by Tempered Fractional Brownian Motion

Global Attractiveness and Quasi-Invariant Sets of Impulsive Neutral Stochastic Functional Differential Equations Driven by Tempered Fractional Brownian Motion

Existence of relaxed optimal control for $G$-neutral stochastic functional differential equations with uncontrolled diffusion

Existence of relaxed optimal control for $G$-neutral stochastic functional differential equations with uncontrolled diffusion

Existence of solutions for fractional impulsive neutral functional differential equations driven by fractional Brownian motion

Abstract In this paper, we consider a class of fractional impulsive neutral stochastic functional differential equations with infinite delay driven by a fractional Brownian motion in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy.

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.

A Note on Exponential Stability for Numerical Solution of Neutral Stochastic Functional Differential Equations

This paper examines the numerical solutions of the neutral stochastic functional differential equation. This study establishes the discrete stochastic Razumikhin-type theorem to investigate the exponential stability in the mean square sense of the Euler–Maruyama numerical solution to this equation. In addition, the Borel–Cantelli lemma and the stochastic analysis theory are incorporated to discuss the almost sure exponential stability for this numerical solution of such equations.

Neutral functional sequential differential equations with Caputo fractional derivative on time scales

In this paper, we establish the existence and uniqueness of a solution for a class of initial value problems for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and Krasnoselskii fixed point theorem. As applications, two examples are included to show the applicability of our results.

On the existence and mean square stability of neutral stochastic functional differential equations driven by G‐Lévy process

The goal of the present article is to investigate the concepts of existences and stability for neutral stochastic functional differential equations (NSFDEs) driven by G‐Lévy process. Under non‐Lipschitz assumption the existence and uniqueness result for solutions of NSFDEs based on G‐Lévy process has been acquired. The results have been deducted by using the Bihari inequality, H lder inequality and the Burkholder–Davis–Gundy (BDG) type inequalities of the G‐framework. In addition, the mean square stability has been studied.

Comparison theorem for neutral stochastic functional differential equations driven by [formula omitted]-Brownian motion

In this paper, we investigate sufficient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case (Bai and Jiang, 2016) and nonlinear expectation framework (Huang and Yang, 2021).

Ulam–Hyers–Rassias stability of neutral stochastic functional differential equations

In this paper, by using the Gronwall inequality, we show two new results on the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of neutral stochastic functional differential equations. Two examples illustrating our results are exhibited.

A note on controllability of impulsive neutral stochastic functional differential equations driven by a Rosenblatt process in a Hilbert space

In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process in a Hilbert space. The controllability results are obtained by the Banach fixed point theorem. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

Critical cases in neutral functional differential equations, arising from hydraulic engineering

This paper starts from several applications described by initial/boundary value problems for \(1D\) (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties.

Sequential fractional order Neutral functional Integro differential equations on time scales with Caputo fractional operator over Banach spaces

<abstract><p>In this work, we scrutinize the existence and uniqueness of the solution to the Integro differential equations for the Caputo fractional derivative on the time scale. We derive the solution of the neutral fractional differential equations along the finite delay conditions. The fixed point theory is demonstrated, and the solution depends upon the fixed point theorems: Banach contraction principle, nonlinear alternative for Leray-Schauder type, and Krasnoselskii fixed point theorem.</p></abstract>

Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients

<p style='text-indent:20px;'>In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.</p>

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