GLOBAL EXPONENTIAL STABILITY TO A CLASS OF DIFFERENTIAL SYSTEM WITH DELAY

The global exponential stability of the zero solution to a class of differential system with delay is considered.By constructing a suitable type of Lyapunov functional and using some analytical techniques,we derive some criteria to check exponential stability of this system.The results establish a relation between the delay time and the parameters of the system.Two examples are also given to illustrate the validity of the results.     

EXPONENTIAL STABILITY TO A NEUTRAL DIFFERENTIAL EQUATION OF FIRST ORDER WITH DELAY

In this paper, we study the exponential stability of the zero solution to a neutral diferential equation. By applying the Lyapunov-Krasovskiì functional approach, we prove a result on the stability of the zero solution. The result we obtained extends and generalizes the existing ones in the previous literature. Comparing with the previous results, our result is new and complements some known results.     

Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

The present work aims to determine the solution of trigonometric functional equation f with involution from group to field by using the properties of involution function,and the solution and Ulam-Hyers stability of the trigonometric functional equation are also discussed.Furthermore,this method generalizes the main theorem and gives the supplement in some reference.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

A Fixed Point Approach to the Fuzzy Stability of a Mixed Typ e Functional Equation

Through the paper, a general solution of a mixed type functional equation in fuzzy Banach space is obtained and by using the fixed point method a generalized Hyers-Ulam-Rassias stability of the mixed type functional equation in fuzzy Banach space is proved.     

STABILITY OF THEORETICAL SOLUTION AND NUMERICAL SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH PIECEWISE DELAYS

This paper is concerned with the stability of theoretical solution and numerical solutionof a class of nonlinear differential equations with piecewise delays.At first,a sufficientcondition for the stability of theoretical solution of these problems is given,then numericalstability and asymptotical stability are discussed for a class of multistep methods whenapplied to these problems.     

STABILITY OF SOLUTIONS TO CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS

By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to a certain fourth-order non-linear differential equation with delay are obtained.     

SAMPLED-DATA STATE ESTIMATION FOR NEURAL NETWORKS WITH ADDITIVE TIME–VARYING DELAYS

In this paper, we consider the problem of delay-dependent stability for state estimation of neural networks with two additive time–varying delay components via sampleddata control. By constructing a suitable Lyapunov–Krasovskii functional with triple and four integral terms and by using Jensen's inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities(LMIs) to ensure the asymptotic stability of the equilibrium point of the considered neural networks. Instead of the continuous measurement,the sampled measurement is used to estimate the neuron states, and a sampled-data estimator is constructed. Due to the delay-dependent method, a significant source of conservativeness that could be further reduced lies in the calculation of the time-derivative of the Lyapunov functional. The relationship between the time-varying delay and its upper bound is taken into account when estimating the upper bound of the derivative of Lyapunov functional. As a result, some less conservative stability criteria are established for systems with two successive delay components. Finally, numerical example is given to show the superiority of proposed method.     

ON LIAPUNOV FUNCTIONAL IN THEORY OF STABILITY OF SYSTEMS WITH TIME-LAG

It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.     

Numerical estimation of the piecewise constant Robin coefficient by identifying its discontinuous points

We propose a new numerical method for estimating the piecewise constant Robin coefficient in two-dimensional elliptic equation from boundary measurements. The Robin inverse problem is recast into a minimization of an output least-square formulation. A technique based on determining the discontinuous points of the unknown coefficient is suggested, and we investigate the differentiability of the solution and the objective functional with respect to the discontinuous points. Then we apply the Gauss-Newton method for reconstructing the shape of the unknown Robin coefficient. Numerical examples illustrate its efficiency and stability.     

Adjoint Method for an Inverse Problem of CCPF Model

The problem for determining the exchange rate function of 2D CCPF model by measurements on the partial boundary is considered and solved as one PDE-constraint optimization problem. The optimal variant is the minimum of a cost functional that quantifies the difference between the measurements and the exact solutions. Gradientbased algorithm is used to solve this optimization problem. At each step, the derivative of the cost functional with respect to the exchange rate function is calculated and only one forward solution and one adjoint solution are needed. One method based on the adjoint equation is developed and implemented. Numerical examples show the efficiency of the adjoint method.     

多元二次映射的广义稳定性

In this paper we unify the system of functional equations defining multi-quadratic mappings to a single equation, find out the general solution of it and prove its generalized Hyers-Ulam stability.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Periodic Solutions of Schrdinger Flow from S~3 to S~2

This paper deals with the periodic solutions of Schrodinger flow from S3 to S2. It is shown that the periodic solution is related to the variation of some functional and there exist S1-invariant critical points to this functional. The proof makes use of the Principle of Symmetric Criticality of Palais.     

ASYMPTOTIC STABILITY OF RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS BY LYAPUNOV’ DIRECT METHOD

This paper is devoted to studying the asymptotic stability of retarded nonlinear functional differential equations by the method of Lyapunov functionals. Under the as-sumption that there exists a positive definite time-invariant Lyapunov functional with negative semi-definite derivative, we focus on the extra conditions to guarantee the asymptotic stability, and present a new criterion, which is less conservative than the classical one. Finally, an example is given to illustrate the effectiveness of the res...     

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.     

Hamilton-Jacobi方程的单调有限元格式

A monotone finite element scheme is obtained by applying the finite element method to the viscosity equation of the Hamilton-Jacobi equation on unstructured meshes. Under some constraints, we show that this scheme is monotone and its numerical solution converges to the viscosity solution of the Hamilton-Jacobi equa-tion. Numerical examples test the stability and the convergence of this scheme.     

Asymptotic stability of periodic solution for compressible viscous van der Waals fluids

This paper is concerned with the asymptotic stability of the periodic solution to a one-dimensional model system for the compressible viscous van der Waals fluid in Eulerian coordinates. If the initial density and initial momentum are suitably close to the average density and average momentum, then the solution is proved to tend toward a stationary solution as t -→∞.     

混合型分段连续微分方程的数值稳定性

This paper deals with the stability analysis of the Euler-Maclaurin method for differential equations with piecewise constant arguments of mixed type. The expression of analytical solution is derived and the stability regions of the analytical solution are given. The necessary and sufficient conditions under which the numerical solution is asymptotically stable are discussed. The conditions under which the analytical stability region is contained in the numerical stability region are obtained and some numerical examples are given.