Stability of abstract difference equations
Let \(M\) be a closed subspace of \(\ell ^\infty (J,X)\), the space of all bounded sequences \(\{x_n\}_{n\in J}\subset X\) where \(J\in \{\mathbb Z^+,\mathbb Z\}\) and \(X\) is a complex Banach space. Consider the operator difference equation \(u(n+1)=Au(n)+\psi(n)\), where \(\psi \in \ell ^\infty (\mathbb Z,X)\) such that the restriction \(\psi |_J\) on \(J\) belongs to \(M\) and \(A\) is the generator of a \(C_0\)-semigroup of linear bounded operators \((T (t))_{t\geq 0}\) on \(X\) The author deduces that the zero equilibrium of the homogeneous equation \(u(n+1)=Au(n)\), \(n\in \mathbb Z^+\), is globally asymptotically stable if and only if every solution is bounded.
scientific article; zbMATH DE number 1992908