非线性脉冲差分方程解的稳定性

应用不等式估值法讨论了非线性脉冲时滞差分方程解的性质,并得到它的解的一致稳定性的一些充分条件.     

On convergence of solutions of a second order nonlinear difference equations

The object of the present paper is to establish some criteria of the convergence of all bounded solutions of (1), (1').     

Stability in totally nonlinear delay difference equations

In this paper we use fixed point method to prove asymptotic stability results of the zero solution of a nonlinear delay difference equation. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Raffoul (2006) \cite{r1}, Yankson (2009) \cite{y1}, Jin and Luo (2009) \cite{jin} and Chen (2013) \cite{c}     

On convergence and stability of difference schemes for derivative nonlinear evolution equations

We investigate the initial-boundary value problem for the nonlinear equation system $$\frac{{\partial u}}{{\partial t}} = A\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u) + g(u)\frac{{\partial u}}{{\partial x}},$$ whereA is a complex diagonal matrix,f andg are complex vector-functions. The convergence and stability in theW ₂ ² norm of the proposed Crank-Nicolson type difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed.     

Permanence and oscillation of a system of two nonlinear difference equations

In this paper we study the permanence and the oscillatory behavior of the solutions of a system of two nonlinear difference equations.     

A convergence criterion for the solutions of nonlinear difference equations and dynamical systems

We show that the solutions of nonlinear higher order difference equations may have convergent subsequences, even when the solution as a whole does not converge, if the defining function sequence is suitably bounded near the origin. The delay size and pattern in the higher order equation play an essential role in determining which subsequences of its solutions converge. We then show that this method can be extended to planar systems and discuss applications to discrete dynamical systems that have been used in biological population models.     

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.     

Analysis of a set of nonlinear dynamics trajectories: Stability of difference equations

For a set of difference equations generated by discretization of the set of differential equations with Hukuhara derivative a principle of comparison with matrix Lyapunov function is specified and sufficient stability conditions of certain type are established. The analysis is carried out in terms of a matrix Lyapunov function of special structure. For an essentially nonlinear multiconnected switched difference system, conditions are obtained providing the asymptotic stability of its zero solution for any switching law. An example is presented to demonstrate efficiency of the proposed approaches.     

Nonisospectral nonlinear difference equations

A large class of evolutionary differential-difference equations of semíinfinite Toda chain type is constructed and a procedure is given for finding a solution by the method of the inverse spectral problem related to Jacobian matrices.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 555–558, April, 1990.     

A class of difference ABS-type algorithms for a nonlinear system of equations

In this paper we give a class of algorithms for solving nonlinear algebraic equations using difference approximations of derivatives. The class is a modification of the original ABS class with the advantage of requiring less function evaluations. Special cases include the methods of Brown and Brent and the discretized Newton method, which is formulated in a way requiring fewer function evaluations per iteration.     

Stability analysis of a system of second‐order difference equations

In this paper, we study the boundedness character and persistence, existence and uniqueness of the positive equilibrium, local and global behavior, and the rate of convergence of positive solutions of the following system of rational difference equations where the parameters α_i,β_i,a_i,b_i for i∈{1,2} and the initial conditions x_?1,x₀,y_?1,y₀ are positive real numbers. Some numerical example are given to verify our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

On a class of third-order nonlinear difference equations

This paper studies the boundedness character of the positive solutions of the difference equation

     

On nonlinear difference and differential equations

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results.The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.Dedicated to my teacher and friend, Professor Germund Dahlquist, on the occasion of his 60th birthday.     

Uniform convergence of a finite difference scheme for a system of coupledreaction‐diffusion equations

We study a system of coupled reaction‐diffusion equations. The equations have diffusion parameters of different magnitudes associated with them. Near each boundary, their solution exhibit two overlapping layers. A difference scheme on layer‐adapted piecewise uniform meshes is used to solve the system numerically. We show that the scheme is almost second‐order convergent, uniformly in both perturbation parameters, thus improving previous results [3].     

Stability of semi-autonomous difference equations

For a linear difference equation with constant coefficients and several bounded variable delays we obtain criteria for the uniform and uniformly exponential stability expressed in terms of parameters of the initial problem. We adduce examples that prove the exactness of the boundaries of the obtained stability domain.