Asymptotic equivalence of Volterra difference systems
Let \(N(n_0)= \{n_0, n_0+1, \dots\}\), where \(n_0\) is a natural number or zero, \(\mathbb{R}^k\) the \(k\)-dimensional real Euclidean space, and \(M^k\) the space of all \(k\times k\) matrices. The author considers the following systems of difference equations \[ \begin{aligned} x(n+1) &= [A+ B (n) ]x(n)+ \sum^n_{r=0} [K(n- r)+ Q(n, r)] x(r),\\ x(n+1) &= Ax (n)+ \sum^n_{r=0} K(n- r) x(r)+ f(n)+ F(n, x(n)), \end{aligned} \tag{1} \] thought as perturbations of \[ \begin{aligned} y(n+ 1)&= Ay(n)+ \sum^n_{r=0} K(n- r) y(r),\\ y(n+ 1)&= Ay(n)+ \sum^n_{r=0} K(n- r) y(r)+ f(n), \end{aligned} \tag \(1'\) \] where \(x\), \(y\), \(f\) are \(k\)-dimensional vectors, \(A\) is a constant matrix, \(B,K: N(n_0)\mapsto M^k\), \(Q: N(n_0) \times N(n_0) \mapsto M^k\), \(F: N(n_0) \times U\mapsto \mathbb{R}^k\) is for any \(n\in N(n_0)\) continuous as a function of \(x\in U\) (\(U\) a region in \(\mathbb{R}^k\)). The purpose of this paper is to give some results on the asymptotic relationship between the solutions of (1) and \((1')\), using a resolvent matrix \(R(n, m)\) defined in the paper.
scientific article; zbMATH DE number 889851