Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.     

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

Stability analysis of one-leg methods for nonlinear functional differential and functional equations

This paper is concerned with the numerical solution of nonlinear functional differential and functional equations. The adaptation of one-leg methods is considered. It is proved that an A-stable one-leg method is globally stable and a strongly A-stable one-leg method is asymptotically stable under suitable conditions. A numerical test is given to confirm the theoretical results.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

非线性中立型泛函微分方程Runge-Kutta 法的稳定性和收敛性

本文涉及Runge-Kutta 法变步长求解非线性中立型泛函微分方程(NFDEs) 的稳定性和收敛性.为此, 基于Volterra 泛函微分方程Runge-Kutta 方法的B- 理论, 引入了中立型泛函微分方程Runge-Kutta 方法的EB (expanded B-theory)-稳定性和EB-收敛性概念. 之后获得了Runge-Kutta 方法变步长求解此类方程的EB - 稳定性和EB- 收敛性. 这些结果对中立型延迟微分方程和中立型延迟积分微分方程也是新的.     

Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in banach spaces

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.     

The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations

The asymptotic stability of theoretical and numerical solutions for neutral multidelay-differential equations (NMDEs) is dealt with. A sufficient condition on the asymptotic stability of theoretical solutions for NMDEs is obtained. On the basis of this condition, it is proved that A-stability of the multistep Runge-Kutta methods for ODEs is equivalent to NGP_k-stability of the induced methods for NMDEs. Project supported by the National Natural Science Foundation of China (Grant No. 19771034).     

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

Explicit multistep method for the numerical solution of stiff differential equations

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.     

一类奇摄动非线性Volterra型泛函微分方程边值问题

Abstract. In this paper,a kind of boundary value problems for Volterra functional differential e-quation is studied.     

Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.     

Dissipativity of Volterra functional differential equations

This paper is concerned with the dissipativity of Volterra functional differential equations in a Hilbert space. A sufficient condition for dissipativity of one class of such equations is obtained. This result is applied to delay differential equations and integro-differential equations to obtain dissipativity results that are more general and deeper than related results in the previous literature.     

Positive solutions for systems of nonlinear eigenvalue problems for functional differential equations

Values of?λ?are determined for which there exist positive solutions of the system of functional differential equations, u″?+?λa(t)f(v _t )?=?0,v″?+?λb(t)g(u _t )?=?0, for 0?t?u(s)?=?v(s)?=?φ(s), ?r?≤?s?≤?0, and the boundary conditions u(0)?=?v(0)?=?φ(0)?=?u(1)?=?v(1)?=?0. A Guo–Krasnosel'skii fixed point theorem is applied.     

Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations

In this paper, we focus on the error behavior of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (NVDIDEs) with constant delay. The convergence properties of the Runge-Kutta methods with two classes of quadrature technique, compound quadrature rule and Pouzet type quadrature technique, are investigated.      

On a class of functional and differential functional equations

One class of functional and differential functional equations with deviating argument depending on the unknown function is considered in this study. The procedure of constructing and investigating solutions to such equations is also proposed.     

Exponential stability and inequalities of solutions of abstract functional differential equations

With the Lyapunov second method, we study the abstract functional differential equation, . We obtain inequalities of solutions and exponential stability with conditions like:

(i)

,

(ii)

.

Applications in ordinary and partial functional differential equations are given.     

Periodic solutions of neutral nonlinear system of differential equations with functional delay

We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form

     

Stability and asymptotic stability of -methods for nonlinear stiff Volterra functional differential equations in Banach spaces

This paper is concerned with the stability and asymptotic stability of θ-methods for the initial value problems of nonlinear stiff Volterra functional differential equations in Banach spaces. A series of new stability and asymptotic stability results of θ-methods are obtained.