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Boundedness and stability in nonlinear delay difference equations employing fixed point theory. (English) Zbl 1103.39009

This paper is devoted to derive some results on the qualitative properties of linear and nonlinear difference equations such as stability, asymptotic stability and equi-boundedness, by using the contraction mapping principle. First, the authors study a linear difference equation \[ x(t+1)=a(t)x(t)+b(t)x(t-g(t))+c(t)\Delta x(t-g(t)), \] where \(a,b,c:\mathbb{Z}\to \mathbb{R}\), \(g:\mathbb{Z}\to\mathbb{Z}^+\), and \(\Delta x(t)=x(t+1)-x(t)\). Then, a nonlinear equation is considered by replacing the term \(b(t)x(t-g(t))\) by \(q(x(t))x(t-g(t))\), where \(q:\mathbb{R}^2\to\mathbb{R}\) satisifies a local Lipschitz condition.
Reviewer: Eduardo Liz (Vigo)

MSC:

39A11 Stability of difference equations (MSC2000)
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