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

Take σσ to be a continuous semiflow on the locally compact metric space ΘΘ, and let {A(θ)}θ∈Θ{A(θ)}θ∈Θ be a family of (possibly unbounded) densely defined closed operators on the Banach space XX.We show that the solutions of the variational equation u̇=A(σ(θ,t))u,θ∈Θ,t∈R+ have a uniform exponential decay if and only if there exist p∈[1,∞)p∈[1,∞) and a sequence of positive real numbers (γn)n∈N(γn)n∈N with γn→∞γn→∞ satisfying limn→∞γnn∫0n‖Φ(θ,τ)x‖pdτ=0uniformly w.r.t. θ∈Θ, for all x∈X, where (θ,t)→Φ(θ,t)(θ,t)→Φ(θ,t) is a strongly continuous cocycle solving (in the classical sense) the above variational equation.Thus, we generalize some known results obtained by Datko and Pazy.