Attraction and Stochastic Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Respect to Semimartingales

Abstract

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

高维系统周期解的存在性与唯一性

本文首先利用一种新方法讨论高维系统解的一致有界性，一致最终有界性及非常稳定性问题，然后将所获结果应用于高维系统周期解的存在性与唯一性的研究。另外，本文还提出了“阶梯形系统”的概念，并对其建立了一种利用系统部分方程周期解存在性判定整个系统周期解存在性的方法。     

ON THE STABILITY AND BOUNDEDNESS OFSOLUTIONS FOR THE RETORDED LIENARD-TYPE EQUATION

1IntroductionConsidertheretardedLi6nard--typeequationwherehisanonnegativeconstant,fitfi,WandgarecontinuousfunctionsonR.Recently,theboundedness,stability,oscillationandperiodicityofsolutionsofwhichisEq.(1)withoutdelayandW(x)=0,hasbeenwidelyinvestigated(of[1-5]).Clearly,iffi(x)=f(x),f200=0,W(x)=0,Eq.(1)isreducedtothewell-knownretardedLi6nardequationIn1963,Krasovskii[6]gaveasufficientconditionoftheuniformasymptoticstabilityofthezerosolutionforEq.(3).Hisworkcausedmuchinterest,itisseenbythewor…     

泛函微分方程解的指数渐近稳定性与有界性

本文研究了具有有限时滞中立型泛函微分方程解的有界性问题 ,得到了方程解的指数渐近稳定性蕴涵有界解的存在性的新的结果     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Global asymptotic stability and boundedness of certain multi‐delay functional differential equations of third order

In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi‐delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov‐Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright © 2014 John Wiley & Sons, Ltd.     

一类具马尔科夫链中立型随机神经网络的动力学性质（英文）

A generalized neutral stochastic functional differential equation(NSFDE) with Markovian switching is studied. We will discuss some important properties of the solutions including boundedness and exponential stability by using Lyapunov-Krasovskii functional,Matrix inequality and some analysis techniques. Finally, an numerical example for neutral stochastic neural networks with Markovian switching is given to show the effectiveness of the results in this paper.     

一类非线性时滞差分方程解的若干性质

本文研究了一类非线性时滞差分方程解的有界持久性、全局渐近稳定性以及无界解的存在性,所得结果解决及部分解决了文献「４」提出的两个ｏｐｅｎ问题,包含了文献「６」中相应的结果。     

Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses

In this paper, we investigate the dynamics of a class of the so-called semi-ratio-dependent predator-prey interaction models with functional responses based on systems of nonautonomous differential equations with time-dependent parameters. The functional responses are classified into five types and typical examples of each type are provided. Then we establish sufficient criteria for the boundedness of solutions, the permanence of system, and the existence, uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution. Some conclusive discussion is presented at the end of this paper.     

Convergence to equilibria in scalar nonquasimonotone functional differential equations

We consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by Smith and Thieme. It is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by “ignoring the delays”. The result on the boundedness of the solutions, combined with a convergence theorem due to Smith and Thieme, leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data. Under stronger conditions, guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation, a similar result is proved for the convergence of all solutions.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

Stability of nonlinear stochastic-evolution equations

Criteria for boundedness, asymptotic stability of sample paths given by solutions to nonlinear stochastic-evolution equations are presented. The analysis is based on a functional Itô formula, Liapunov and related functionals, and generalization of methods developed in finite dimensions. Applications to parabolic Itô equations are given.     

一类四阶非线性微分方程解的有界性及稳定性

运用Liapunov函数方法,研究了一类四阶非线性微分方程解的有界性及稳定性,得到了解的有界性及零解的全局渐近稳定性的充分条件。     

Global boundedness and stability of solutions for prey-taxis model with handling and searching predators

In this paper, we show the global boundedness and stability of solutions for prey-taxis model with handling and searching predators in a two-dimensional bounded domain with smooth boundary. First, entropy-like equations and boundedness criteria are derived, and it is proved that the system has a unique uniformly bounded global classical solution. In addition, we show that prey-only steady state is globally asymptotically stable if the predator is weak. The convergence rate of solutions to the steady states is derived in the paper.     

PERIODIC SOLUTIONS TO A HIGH-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION

In this paper,using the principle of a priori boundedness,we study the existence and uniqueness of periodic solutions to a high-order neutral functional differential equation.Some new results for the existence and uniqueness of the periodic solution are obtained.     

YoshiZawa型周期解定理和Massera型周期解定理研究进展简介

微分方程解的有界性和周期解的存在性是檄分方程理论研究中的两个重要课题，二者之间有着紧密的联系．在解的有界性与周期解的存在性的研究中，Yoshizawa周期解定理和Massera周期解定理是非常重要的结果，具有重要的理论意义和应用价值．本文以Yoshizawa型周期解定理和Massera型周期解定理的研究为主，简要介绍泛函微分方程周期解理论研究方面的一些新进展。     

On Stability and Boundedness for Lipschitzian Differential Inclusions: The Converse of Lyapunov's Theorems

For Lipschitzian differential inclusions, we prove that the existence of suitable Lyapunov functions is necessary for uniform stability and uniform boundedness of solutions.     

Global dynamics of a competitive system of rational difference equations

In this paper, we study the qualitative behavior of a competitive system of second‐order rational difference equations. More precisely, we investigate the boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the system. Some numerical examples are given to verify our theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.