×

On global asymptotic stability of nonlinear stochastic difference equations with delays. (English) Zbl 1113.39005

In a previous work of Y. Hamaya and S. Sato [Int. J. Pure Appl. Math. 25, No. 4, 487–495 (2005; Zbl 1100.39007)], the global attractivity of the equilibrium point of the difference equation with several delays
\[ x_{n+1}=a\tanh (x_{n}-\sum_{i=1}^{m}b_{i}x_{n-k_{i}}),~~\;n\in \mathbb{N}_0, \]
where \(a>0,~\;b_{i}\geq 0,~\) and \(k_{i}\geq 0,~1\leq i\leq m\) has been investigated. In this paper, the authors contemplate the equation
\[ x_{n+1}=(1-h)x_{n}+ahF(x_{n}-bx_{n-k}),~\;n\in \mathbb{N}_0,\tag{\(*\)} \]
with arbitrary initial conditions \(x_{0},x_{-1},\dots,x_{-k}\) and \(a,b\in \mathbb{R},~0<h\leq 1,\) and \(F:\mathbb{R}\to \mathbb{R}\) is a nonrandom continuous function. They obtain the following:
Theorem 1. Assume that \(| F(u)| \leq | u| ~\;\forall u\in R~,~\) and \(| a| (1+| b| )<1.~\) Then \(\lim_{n\to \infty}x_n=0,\) where \((x_n)\) is is a solution to equation (\(*\)).
Also, the following equation is considered
\[ X_{n+1}=(1-h)X_{n}+ahF(X_{n}-\sum_{\ell =1}^{k}b_{\ell }X_{n-\ell })+\sqrt{h}g(n,X_{n},X_{n-1},\dots ,X_{n-k})\xi _{n+1}~, \tag{\(**\)} \]
\(n\in \mathbb{N}_{0}~\) with arbitrary nonrandom initial values \(X_{0},X_{-1},\dots X_{-k}\in \mathbb{R}~,h\in (0,1],\) the functions \(F:\mathbb{R}\to \mathbb{R}~\) and \(g:\mathbb{R}^{k+2}\to \mathbb{R}~\) are continuous, and \(\;\xi _{n}~\) are independent random variables such that \(E\xi _{n}=0,~E\xi _{n}^{2}=1.\)
The following theorem is obtained: Theorem 2. Suppose that \(| F(u)| \leq | u| ~\;\forall u\in R~,\) the function \(g~\) is non-random and there exist \(c_{\ell }\geq 0,~~\gamma _{n}\in R,~\ell =0,1,\dots ,k,n\in N,\) such that for all \(u_{\ell }\in R,\ell =0,1,\dots k,\) satisfies \[ | g(n,u_{0},u_{1},\dots ,u_{k})| ^{2} \leq \sum_{\ell =0}^{k}c_{\ell }| u_{\ell }| ^2+\gamma_n^2,\;g(n,0,\dots ,0)=0, \]
\[ \sum_{i=0}^{\infty }\gamma _i^2 <\infty ,~\text{ and } a^2(1+\sum_{\ell =1}^{k}| b_{\ell }| )^{2}+\sum_{j=0}^{k}c_{j}~<1~. \] Then \(\lim_{n\to \infty }~X_{n}=0~\) almost sure, where \((X_n)\) is a solution to equation (\(**\)).

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
60H25 Random operators and equations (aspects of stochastic analysis)
37H10 Generation, random and stochastic difference and differential equations

Citations:

Zbl 1100.39007
PDFBibTeX XMLCite
Full Text: Link