Herzog and Lemmert have proven that if E is a Fréchet space and T: E → E is a continuous linear operator, then solvability (in [0,1]) of the Cauchy problem ẋ = Tx, x(0) = x₀ for any x₀ ∈ E implies solvability of the problem ẋ(t) = Tx(t) + f(t,x(t)), x(0) = x₀ for any x₀ ∈ E and any continuous map f: [0,1] × E → E with relatively compact image. We prove the same theorem for a large class of locally convex spaces including:
• DFS-spaces, i.e., strong duals of Fréchet-Schwartz spaces, in particular the spaces of Schwartz distributions 𝓢'(ℝⁿ), the spaces of distributions with compact support 𝓔'(Ω) and the spaces of germs of holomorphic functions H(K) over an arbitrary compact set K ⊂ ℂⁿ;
• complete LFS-spaces, i.e., complete inductive limits of sequences of Fréchet-Schwartz spaces, in particular the spaces 𝓓(Ω) of test functions;
• PLS-spaces, i.e., projective limits of sequences of DFS-spaces, in particular, the spaces 𝓓'(Ω) of distibutions and 𝓐(Ω) of real-analytic functions.
Here Ω is an arbitrary open domain in ℝⁿ. We construct an example showing that the analogous statement for (smoothly) time-dependent linear operators is invalid already for Fréchet spaces.