Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Almost sure polynomial asymptotic stability of stochastic difference equations

In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is perturbed by a random term with a fading state-independent intensity. In particular, we show that if the unbounded noise has tails that fade more quickly than polynomially, then the state-independent perturbation dies away at a sufficiently fast polynomial rate in time, and if the autonomous difference equation has a polynomial nonlinearity at the origin, then the almost sure polynomial rate of decay of solutions can be determined exactly. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

On global asymptotic stability of nonlinear higher-order difference equations

In this paper, we generalize the main theorem of Liz and Ferreiro [E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659] and some other global stability results for nonautonomous higher-order difference equations to the case when contraction-type steps are incorporated together with the steps when the difference sequence can increase.     

Global asymptotic stability of a second-order system of difference equations

In this paper a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations $$x_{n + 1} = \frac{{x_n y_{n - 1}^b + 1}} {{x_n + y_{n - 1}^b }}, y_{n + 1} = \frac{{y_n x_{n - 1}^b + 1}} {{y_n + x_{n - 1}^b }}n = 0,1,2 \ldots$$ where the parameter b ∈ [0, ∞) and the initial values (x _k , y _k ) ∈ (0, ∞) (for k = ?1, 0).     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

New explicit global asymptotic stability criteria for higher order difference equations

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.     

The asymptotic estimates of solutions of difference equations

The asymptotic estimate of the solution y(t) of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y(t) and y₀(t) are given, where y₀(t) is the corresponding solution of linear equations with constant coefficients. A brief and elementary proof of the results is presented. The method of proof carries over to differential equations in which some similar results are stated.     

On explicit conditions for the asymptotic stability of linear higher order difference equations

We derive some explicit sufficient conditions for the asymptotic stability of the zero solution in a general linear higher order difference equation, and compare our estimations with other related results in the literature. Our main result also applies to some nonlinear perturbations satisfying a kind of sublinearity condition.     

An asymptotic result and a stability criterion for linear nonautonomous delay difference equations

Linear delay difference equations with variable coefficients and constant delays are considered. By the use of an appropriate solution of the so called generalized characteristic equation, an asymptotic result is obtained and a stability criterion is established.     

Stability and asymptotic behavior of difference equations

The concept of h-stability is studied and compared with the classical stabilities. Basically, the h-stability is applied to obtain a uniform treatment for the concept of stability in difference equations.