×

Global stability of a nonlinear Volterra difference system. (English) Zbl 0868.39003

Summary: We study the global asymptotic stability of the zero solution of the nonlinear Volterra difference equation \[ x(n+1)= \sum^n_{j=0} K(n,j,x(j)), \qquad n=0,1,\dots \] where \(x(0)= x_0\in \mathbb{R}^p\) is a given initial condition and where for all \(j,n= 0,1,\dots\), the function \(K(n,j,x)\) maps \(\mathbb{R}^p\) into \(\mathbb{R}^p\) and is continuous in \(x\in \mathbb{R}^p\). We consider also the linear equation \[ y(n+1)= \sum^n_{j=0} A(n,j)y(j) \] where \(y(0)= y_0\in\mathbb{R}^p\) is a given initial condition and where \(A(n,j)\) are given \(p\times p\) matrices.

MSC:

39A11 Stability of difference equations (MSC2000)
PDFBibTeX XMLCite