An ergodic theorem in the qualitative behavior of (non)linear semiflows (Q450581)

A case of von Neumann's mean ergodic theorem states that, if \({T(t)}_t\geq 0\) is a strongly continuous one-parameter group of unitary operators on a Hilbert space, then the operator \(\frac{1}{h} \int_0^h T(t)\, dt\) converges in the strong operator topology as \(h\rightarrow\infty\).NEWLINENEWLINE This result has applications to autonomous systems of differential equations in infinite-dimensional Banach spaces. The authors show here the equivalence between the exponential decay of solutions of a variational equation and some condition that the uniformly weighted means of the solutions converge to zero.