Existence of \(\Psi\)-bounded solutions for linear matrix difference equations on \(\mathbb{Z}^{+}\) (Q2877836)

The paper studies the linear matrix difference equation NEWLINE\[NEWLINE X(n+1)=A(n)X(n)B(n)+F(n), NEWLINE\]NEWLINE where \(A(n),B(n),F(n)\) are \(m\times m\) matrix-valued functions on \(\mathbb{Z}^+\) and \(F(n)\) is \(\Psi\)-summable. Under the boundedness of \(A(n),B(n)\) on \(\mathbb{Z}^+\), the paper obtains a sufficient and necessary condition for the existence of at least one \(\Psi\)-bounded solution. They also study the asymptotic behavior of \(\Psi\)-bounded solutions \(X(n)\): \(\lim_{n\to\infty}|\Psi(n)X(n)|=0\).