Neutral Functional Differential Equations Research Articles

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

A New Method for Solving a Neutral Functional-Differential Equation with Proportional Delays

This study presents and implements a new hybrid technique that combines the Sawi transform (ST) and Homotopy perturbation method (HPM) to solve neutral functional-differential equations with proportional delays. Some of the important properties of the method are established and validated. We start the method by first applying ST to obtain the recurrence relation. We, next, implement HPM to find convergent series solutions of the recurrence relation. The series is free of assumptions and restrictions, highlighting its adaptability and robustness. Moreover, the convergence of the method is established through convincing proof. To demonstrate its effectiveness and applicability, we provide five examples. The method yields accurate approximate solutions, or in some cases exact solutions, with a few number of iterations, reinforcing its reliability and validity. Moreover, the performance of the method is compared with some available methods and demonstrates its superiority and efficiency.

Tackling critical cases of the difference operator stability occurring in applications described by 1D distributed parameters

AbstractThe article starts from the challenge of the critical cases in difference operator stability for neutral functional differential equations (NFDE). Such cases occur in the NFDE associated to hyperbolic partial differential equations (PDE) dynamics in mechanical and hydraulic engineering. For some of such applications it resulted that the aforementioned critical (nonasymptotic) stability is connected to the character and level of the energy losses. It is shown that suitable choice of the losses to be taken into account can remove the critical stability properties and give the difference operator the asymptotic stability thus ensuring asymptotic stability for the system's dynamics and also other asymptotic properties.

On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition

Abstract The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.

Stability results of neutral fractional functional differential equations with multi Caputo fractional derivatives

Stability results of neutral fractional functional differential equations with multi Caputo fractional derivatives

Existence and Regularity of solutions in alpha norm for some second order partial neutral functional differential equations in Banach Spaces

The purpose of this paper is to study the existence and regularity of solutions in the $\alpha$-norm for some second order partial neutral functional differential equations with finite delay in Banach spaces. To do this, we use the cosine family theory and Schauder's fixed point Theorem to establish the local existence of solutions and we give some sufficients conditions to enssure the regularity of solutions. Finally, we give an example to illustrate the theoretical results.

A System of Coupled Impulsive Neutral Functional Differential Equations: New Existence Results Driven by Fractional Brownian Motion and the Wiener Process

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper).

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.

Novel Criteria for Exponential Stability of Neutral Functional Differential Equations

We address the challenging problem of exponential stability of general neutral functional differential equations. By a novel approach, we present some new explicit criteria for the exponential stability of neutral functional differential equations. Some remarks and illustrative examples are given.

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.

Study of Non-Linear Impulsive Neutral Fuzzy Delay Differential Equations with Non-Local Conditions

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

Enhanced affine formation maneuver control using historical acceleration command

SummaryHow a network of multiple vehicles gets to a goal formation from an initial formation is as important as reaching the goal formation. Despite that stress matrix has been well applied to construct distributed affine formation control, most works have focused on the stabilization or stability issues and paid little attention to the performance regulation or the control parameterization issues. In this paper, we aim to design an error‐model based distributed control for the affine formation maneuver of second‐order multi‐agent systems (MASs), and to demonstrate the advantages of the control approach in performance tuning. The basic idea behind the control design is to force the actual error‐trajectory dynamics of each controlled follower to approximate that of an ideal second‐order error model. The designed controller possesses a notable feature that the historical acceleration command (HAC) is used to enhance system performance during the formation maneuvers. By a theory tool of neutral functional differential equations, we develop a delay‐independent condition on the design parameters, under which the system stability is robust to the actively introduced HAC. Furthermore, we derive an inequality describe the relationship between the ultimate bounds of tracking errors and the delay parameter. This paper presents a clear mechanism to tune the comprehensive performance of affine formation control. Simulation results show the effectiveness of the stability condition and the performance‐tuning mechanism, as well as the possible fragility of the system stability to the delay when the condition is not satisfied.

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.

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

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

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.

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Controllability of second order neutral impulsive fuzzy functional differential equations with Non-Local conditions

Controllability of second order neutral impulsive fuzzy functional differential equations with Non-Local conditions