ON THE STABILITY OF DIFFERENTIAL SYSTEMS WITH TIME LAG

In this paer the inequality of Lemma 1 of [1] is extended.By means of our inequality and the method of Lyapunov function we study the stability of two kinds of large seate differential systems with time lag and the stability of a higher -order differential equation with time lag.The sufficient conditions for the stability(S.),the asymptotic stability(A.S.),the uniformly asymptotic stability(U.A.S) and the exponential asymptotic stability(E.A.S.) of the zero solutions of the systerms are obtained respectively.     

On the general problem of stability for impulsive differential equations

Criteria for stability, asymptotical stability and instability of the nontrivial solutions of the impulsive system

     

On stability of some linear and nonlinear delay differential equations

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delays

     

Asymptotic expansions for time-varying scalar differential equations using the residue theorem for their discretized difference equations

A fictitious time-dependent time-varying finite sampling period is defined for each time instant at which the asymptotic expansion of the solution of a continuous-time differential equation is investigated. Such a time-dependent sampling period is defined as the quotient of each time instant and a positive integer which tends to infinity as time tends to infinity. The asymptotic expansion formulas are extendable to the case of stable Lyapunov’s equations and to the use of a constant sampling period with minor modifications in the required mathematical proofs. Additional stability results are also discussed.     

Global asymptotic stability of certain third‐order nonlinear differential equations

In this paper, we give some sufficient conditions to guarantee global asymptotic stability of the zero solution of the third‐order nonlinear differential equation: x ′ ′ ′ + g(x,x ′ ,x ′ ′ ) + f(x,x ′ )x ′ + h(x) = 0. Two examples are also given to illustrate our results. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic differential delay equations with jumps,under nonlinear growth condition

In this paper, we investigate the existence and uniqueness of solutions to stochastic differential delay equations under a local Lipschitz condition but without linear growth condition on its coefficients. Moreover, we prove convergence in probability of the Euler–Maruyama approximation as well as of the stochastic theta method approximation to the exact solution.     

On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and S?omiński.     

On positivity and stability of linear time-varying Volterra equations

Linear time-varying Volterra integro-differential equations of non-convolution type are considered. Positivity is characterized and a sufficient condition for exponential asymptotic stability is presented.

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The second author thanks the Alexander von Humboldt Foundation for their support.     

On a general result for backward stochastic differential equations

The existence of the solution of a general infinite dimensional backward stochastic differential equation is discussed. In our setting, we generalize many works concerning the existence problem (by a new approach).     

On positive solutions of fractional differential equations with change of sign

We study the existence and uniqueness of positive solutions of fractional differential equations with change of sign where 1 < α ≤ 2, is continuous and does not vanish identically on any subinterval of [0,1].     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

On the asymptotic stability in functional differential equations

Consider a system of functional differential equations where is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional and negative definite functional . In applications one can construct a positive definite functional , whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional in functional differential equations is autonomous ( does not depend on ), and N. N. Krasovskii created such criterion for the case where the functional is periodic in . For the general case of the non-autonomous functional V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when is almost periodic in . This case is a particular case of the class of non-autonomous functionals.

     

Averaging of time-varying differential equations revisited

Employing comparison of integrals on a fast time scale, we offer a new criterion and simple proofs of the averaging principle for time-varying ordinary differential equations. The method allows straightforward extensions and generalizations. Comparisons with available criteria and estimates, along with examples and applications, are offered.     

Fixed points and stability of neutral stochastic delay differential equations

In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized.     

On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

This paper is addressed to some questions concerning the exponential stability and its robustness measure for linear time-varying differential-algebraic systems of index 1. First, the Bohl exponent theory that is well known for ordinary differential equations is extended to differential-algebraic equations. Then, it is investigated that how the Bohl exponent and the stability radii with respect to dynamic perturbations for a differential-algebraic system depend on the system data. The paper can be considered as a continued and complementary part to a recent paper on stability radii for time-varying differential-algebraic equations [N.H. Du, V.H. Linh, Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbations, J. Differential Equations 230 (2006) 579-599].     

On the theory of stability of matrix differential equations

We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.     

On exponential stability of nonlinear differential systems with time-varying delay

General nonlinear differential systems with time-varying delays are considered. Several explicit criteria for exponential stability are presented. An example is given to illustrate the obtained results. To the best of our knowledge, the results of this note are new.     

Stability criteria for differential equations with variable time delays

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results.     

On the numerical solution of second order ordinary differential equations in the high-frequency regime

We describe an algorithm for the numerical solution of second order linear ordinary differential equations in the high-frequency regime. It is based on the recent observation that solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate this and other properties of the method with several numerical experiments.     

On stability of linear periodic differential difference equations

The paper considers the equation

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.