Stability analysis of discrete recurrence equations of Volterra type with degenerate kernels (Q1192755)

In [Funkc. Ekvacioj 15, 101-117 (1972; Zbl 0251.45019)] \textit{J. M. Bownds} and \textit{J. M. Cushing} derived results on the uniform stability (on \(\mathbb{R}_ +)\) of solutions to the system of Volterra integral equations of the second kind \(y(t)=g(t)+\int^ t_ 0k(t,s)y(s)ds\), where the kernel \(k\) is of Pincherle-Goursat (PG) type, i.e. \(k(t,s)=\sum^ p_{i=1}A_ i(t)B_ i(s)\), with continuous \(A_ i\), \(B_ i\), or is dominated by a PG-type kernel. The present paper deals with analogous stability results for systems of (real) discrete Volterra equations of the form (1) \(y_ n=g_ n+h\sum^ n_{j=n_ 0}k_{n,j}y_ j\;(n_ 0\geq 0;\;h>0)\), where \(k_{n,j}\) is given by (or dominated by) \(k_{n,j}=\sum^ p_{i=1}A_{i,n}B_{i,j}\). The main tool in the analysis is a new representation formula in which the solution of (1) is expressed in terms of the fundamental matrix of a corresponding system of first-order difference equations.