Neutral Stochastic Functional Differential Equations Research Articles

Stability and boundedness criteria for certain second-order nonlinear neutral stochastic functional differential equations

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

Stochastic averaging principle for neutral stochastic functional differential equations driven by G-Lévy process

In this paper, we aim to establish an averaging principle for neutral stochastic functional differential equations driven by G-Lévy processes with non-Lipschitz coefficients. Utilizing the theory of sub-linear expectations, we show that the solutions of the considered stochastic systems can be approximated by the solutions of averaged neutral stochastic functional differential equations driven by G-Lévy processes in the mean square convergence and convergence in the sense of capacity. Finally, we give an example to support our main results.

Existence of a Class of Doubly Perturbed Stochastic Functional Differential Equations with Poisson Jumps

In this paper, we use successive approximations and Picard iterative method to establish the existence and uniqueness of mild solution for a class of doubly perturbed impulsive neutral stochastic functional differential equations with Poisson jumps in Hilbert spaces. An example of a doubly perturbed stochastic differential equation with delays is given to illustrate our main results.

Stability of switched neutral stochastic functional systems with different structures under high nonlinearity

This paper studies the stability of the hybrid neutral stochastic functional differential equations (NSFDEs) with different structures under highly nonlinear conditions. NSFDEs are of wide suitability for simulating random processes with memory, such as tumor evolution mechanism in life science field. The new stochastic system studied in this paper has completely different system structures in different switching subspaces, and the coefficients are highly nonlinear. Moreover, the neutral term is subject to the Markovian switching. This work fills the gap of the stability analysis for differently structured highly nonlinear neutral stochastic functional systems. By using the Lyapunov functional method and the generalized Khasminskii-type conditions, we first establish the existence and uniqueness theorem as well as the asymptotical bounded property of the unique global solution. Then the important stable properties are obtained including H∞ stability, asymptotic stability and exponential stability in Lp. Finally a numerical example is given to illustrate the effectiveness of the results.

Stabilization by discrete‐time feedback control for highly nonlinear hybrid neutral stochastic functional differential equations with infinite delay

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example.

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Existence and Stability of Ulam–Hyers for Neutral Stochastic Functional Differential Equations

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of dxa(s)+g(s,xa(s-ω(s)))=Ixa(s)+f(s,xa(s-ϱ(s)))ds+ς(s)dϖH(s),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{d}}\\left[ {x}_{a}(s) + {\\mathfrak {g}}(s, {x}_{a}(s - \\omega (s)))\\right] =\\left[ {\\mathfrak {I}}{x}_a(s) + {\\mathfrak {f}}(s, {x}_a(s -\\varrho (s)))\\right] {\ extrm{d}}s + \\varsigma (s){\ extrm{d}}\\varpi ^{{\\mathbb {H}}}(s),$$\\end{document}0≤s≤T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le s\\le {\\mathcal {T}}$$\\end{document}, xa(s)=ζ(s),-ρ≤s≤0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${x}_a(s) = \\zeta (s),\\ -\\rho \\le s\\le 0. $$\\end{document} The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples.

Transportation inequalities for mean-field neutral stochastic functional differential equation driven by a fractional Brownian motion

In this paper we demonstrate the uniqueness and existence of a mild solution for a mean-field neutral stochastic differential equation that involves finite delay. The equation is driven by a fractional Brownian motion with Hurst parameter H>1/2 in a Hilbert space. Additionally, we establish the transportation inequalities for the law of the mild solution, with respect to the uniform distance.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations with Markovian Switching

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations with Markovian Switching

Stepanov-like doubly weighted pseudo almost automorphic mild solutions for fractional stochastic neutral functional differential equations

This paper first investigates the equivalence of the space and translation invariance of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes for nonequivalent weight functions; secondly, based on semigroup theory, fractional calculations, and the Krasnoselskii fixed-point theorem, we obtain the existence and uniqueness of Stepanov-like doubly weighted pseudo almost automorphic mild solutions for a class of nonlinear fractional stochastic neutral functional differential equations under non-Lipschitz conditions. These results enrich the complex dynamics of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes.

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

The generalized Khasminskii-type conditions in establishing existence, uniqueness and moment estimates of solution to neutral stochastic functional differential equations

The main objective of this paper involving neutral stochastic functional differential equations (NSFDEs), with finite or infinite history dependence, is to prove the existence and uniqueness of their global solutions by imposing conditions that hold for highly non-linear NSFDEs? coefficients. For that purpose, Yamada-Watanabe condition or the local Lipschitz condition for the drift and diffusion coefficients are imposed, together with contractivity condition for the neutral term. Also, instead of the linear growth condition for the drift and diffusion coefficients of the equations, generalized Khasminskii-type conditions are applied. The proof of the existence and uniqueness of the solution also leads us to estimates of the moments to the solution. Consequently, we discuss some asymptotic properties of the solution in terms of the generalized Lyapunov exponent. Additionally, we consider a class of neutral stochastic differential equations with state-dependent delay, as a special case of NSFDEs. The theoretical results are illustrated with two examples.

Discrete-time feedback stabilization for neutral stochastic functional differential equations driven by G-Lévy process

This paper mainly discusses the exponential stability of neutral stochastic functional differential equations with G-Lévy jump. For a given mean square exponential unstable neutral stochastic differential equations with G-Lévy jump, a discrete-time feedback control in the drift part is designed to realize feedback stability and the H∞ stable càdlàg solution of corresponding systems is obtained. The upper bound of state observation duration of the corresponding systems is derived by extending Mao′s method. Further up, the exponential stabilization conditions are established and some exponential stabilities of the solutions are proved by using the G-Lyapunov functional method. Moreover, an example is presented to verify the obtained results.

Asymptotic behaviour analysis of hybrid neutral stochastic functional differential equations driven by Lévy noise

This paper focuses on the existence and uniqueness, the pth moment and almost sure stability with a general decay rate and the practical stability with general decay rate of a class of hybrid neutral stochastic functional differential equations driven by Lévy noise. Our crucial techniques include Lyapunov functions and the nonnegative semi-martingale convergence theorem. The conditions on the diffusion operator and neutral term are weaker than those in the related existing works. Examples are given to show the effectiveness of the obtained results.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations

This paper focuses on the problem of the pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations (INSFDEs). Based on the Lyapunov function and average dwell time (ADT), two sufficient criteria for the exponential stability of INSFDEs are derived, which manifest that the result obtained in this paper is more convenient to be used than those Razumikhin conditions in former literature. Finally, two numerical examples and simulations are given to verify the validity of our result.

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.

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

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.