On stability of the equation \(Bu'(t)=Au(t)\) (Q5944188)

The author relies on Laplace transform method to obtain information about the solutions of the equation \(Bu'(t)= Au(t)\) on the positive semi-axis. It is assumed that \(A\) and \(B\) are closed linear operators on a (complex) Banach space. In particular, conditions are provided for the asymptotic almost periodicity of a bounded uniformly continuous solution \(u(t)\) on \(\mathbb{R}_+\). Exponential stability is also discussed in the paper. Spectral properties of the pencil \(\lambda B-A\) play an important role in the investigation.