Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

一类泛函微分方程的稳定性定理及其应用

本文采用一种新方法来研究 RFDE 稳定性问题,其特点是不必构造 Liapunov 泛函,用起来比较简单,应用得到的稳定性定理,本文还研究了许多领域中有重要意义的Volterra 积分微分方程的周期解的唯一性和稳定性问题.     

Asymptotic behavior of alternative Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-2005 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1998 S.-M. Jung and in 2002-2005 the authors of this paper investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve our bounds and thus our results obtained, in 2003 for Jensen type mappings and establish new theorems about the Ulam stability of additive mappings of the second form on restricted domains. Besides we introduce alternative Jensen type functional equations and investigate pertinent stability results for these alternative equations. Finally, we apply our recent research results to the asymptotic behavior of functional equations of these alternative types. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.     

Stability of impulsive functional differential equations

In this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied. By using Lyapunov functions and Razumikhin techniques, some criteria of stability, asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided. Some examples are also presented to illustrate the efficiency of the results obtained.     

Strict stability of impulsive functional differential equations

Strict stability is the kind of stability that can give us some information about the rate of decay of the solutions. There are some results about strict stability of differential equations. In the present paper, we shall extend the strict stability to impulsive functional differential equations. By using Lyapunov functions and Razumikhin technique, we shall get some criteria for the strict stability of impulsive functional differential equations, and we can see that impulses do contribute to the system's strict stability behavior.     

Spectral Problems Arising in the Theory of Differential Equations with Delay

In this article we present a review of results on asymptotic behavior and stability of strong solutions for functional differential equations (FDE). We also formulate several results about spectral properties (completeness and basisness) of exponential solutions of the above-mentioned equations. It is relevant to emphasize that our approach for the research of FDE is based on the spectral analysis of operator pencils that are symbols (characteristic quasi-polynomials) with operator coefficients. The article is divided into two parts. The first part is devoted to the research on FDE in a Hilbert space; the second part is devoted to the research on FDE in a finite-dimensional space.     

Stability of functional equations in single variable

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers-Ulam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.     

Stability for retarded functional differential equations

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 107–126, January, 2008.     

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.