Boundary value problems for general discrete systems on infinite intervals (Q1356844)

This paper deals with (parametrized) discrete systems of the form \[ x(k+1)= A(0)x(0)+\cdots+ A(k)x(k)+ b(k)+f_k(x(0),\dots, x(k),\lambda), \] where \(x(\cdot)\in\mathbb{R}^n\), \(A(0),\dots, A(k)\) are \((n,n)\) matrices and \(b(k)\) and \(f_k(\cdot)\) \(n\)-dimensional vectors. The goal of the paper is to study existence and uniqueness of the difference equation under the boundary condition that \(L(x)=\ell\) for some linear operator \(L\); it includes the class of discrete integral equations of Volterra type. Both for finite time horizon and infinite time horizon, necessary and sufficient conditions for existence and uniqueness of a finite solution \(x(\cdot)\) are established.