Neutral Functional Differential Equations Research Articles

New asymptotic study on the non-autonomous NFDEs involving Haddock conjecture

The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded and converges to a constant, which fully refines and generalizes the existing findings.

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.

Existence and Approximate Controllability of Random Impulsive Neutral Functional Differential Equation with Finite Delay

This study investigates second-order neutral functional differential equations with delays, prevalent in various scientific and engineering fields. These equations, characterized by their neutral nature and delays, present unique challenges within Banach spaces. The research focuses on the existence and approximate controllability of solutions, using advanced mathematical tools like cosine family theory and the Leray-Schauder theorem to establish rigorous solution conditions. These theoretical results are empirically validated through practical examples, enhancing understanding of real-life behavior and bridging theory with practice. The study’s findings advance the understanding of delayed feedback systems, facilitating effective control strategies and practical engineering solutions, thereby contributing significantly to dynamical systems and control theory.

On Uniqueness and Continuous Dependence of the Weak Solutions of Neutral Stochastic Functional Differential Equations in Infinite Dimensional Spaces

On Uniqueness and Continuous Dependence of the Weak Solutions of Neutral Stochastic Functional Differential Equations in Infinite Dimensional Spaces

Existence and uniqueness of neutral functional differential equations with sequential fractional operators.

In this research paper, we investigate the existence and uniqueness of solutions for neutral functional differential equations with sequential fractional orders, specifically involving the [Formula: see text]-Caputo operator. To obtain the desired results, we employ the Banach fixed point theorem (BFPT), a nonlinear variation of the Leray-Schauder fixed point theorem (SFPT), and the Krasnoselski fixed point theorem (KFPT). Additionally, we provide illustrative examples that demonstrate the key findings. Furthermore, we address a scenario where an initial value integral condition is considered.

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

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.