Asymptotic property of nonlinear Volterra difference systems
The authors obtain a linear difference system equivalent to the Volterra difference system \[ y(n+1)=A(n)y(n)+\sum_{s=n_0}^nB(n,s)y(s)+g(n)\qquad y(n_0)=x_0 \] where \(n_0\) is a nonnegative integer, \(A(n)\) and \(B(n,s)\) are \(n\times n\) matrix functions on \(N(n_0)=\{ n_0,n_0+1,n_0+2,\dots \}\) and \(N(n_0)\times N(n_0)\) respectively, and \(g(n)\) is a vector function on \(N(n_0)\). They show that the obtained system has an asymptotic equilibrium. Furthermore, they study asymptotic equivalence between the nonlinear Volterra system \[ x(n+1)=f(n,x(n))+\sum_{s=n_0}^ng(n,s,x(s)) \] and its variational system \[ v(n+1)=\partial f/\partial x(n,0)v(n)+ \sum_{s=n_0}^n\partial g/\partial x(n,s,0)v(s) \] by using the comparison principle. Two difference systems \(x(n+1)=f(n,x(n))\) and \(y(n+1)=g(n,y(n))\) are said to be asymptotically equivalent if, for every solution \(x(n)\), there exists a solution \(y(n)\) such that \(x(n)=y(n)+o(1)\) as \(n\to \infty \), and conversely, for every solution \(y(n)\), there exists a solution \(x(n)\) such that the above asymptotic relation holds.
scientific article; zbMATH DE number 1801710