Linearized stability analysis of discrete Volterra equations (Q596719)

Using the fundamental and resolvent matrices, the authors study the stability, the asymptotic stability and the uniform asymptotic stability for the linear discrete Volterra equation \[ x(n)= f(n)= \sum^n_{j=0} B(n,j)x(j),\quad n\geq 0,\tag{1} \] where \(B:\mathbb{Z}^+\times \mathbb{Z}^+\to \mathbb{E}^{d\times d}\) is the kernel (\(B(n,j)= 0\), \(j>n\)) and \(f: \mathbb{Z}^+\to \mathbb{E}^d\) is a given mapping. They also consider discrete Volterra equations of the form \[ x(n)= f(n)+ \sum^n_{j=0} K(n,j,x(n)),\quad (n\geq 0), \] and, under appropriate assumptions which are not discussed in the present paper, they affirm that ``it is possible to relate the stability of solutions'' of such kind of equations to the stability for equation (1).