AbstractLet X be a real Banach space, ω : [0, +∞) → ℝ be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫10 (ω(t))−1 dt = ∞, then for any (t0, x0) ∈ ℝ×X and any continuous map f : ℝ×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ℝ, x, y ∈ X, the Cauchy problem $\dot x$(t) = f(t, x(t)), x(t0) = x0 has a unique solution in a neighborhood of t0. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫10 (ω(t))−1 dt < ∞ then there exists a continuous map f : ℝ × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ℝ × X × X and the Cauchy problem $\dot x$(t) = f(t, x(t)), x(t0) = x0 has no solutions in any interval of the real line.