On Osgood theorem in Banach spaces

✍ Scribed by Stanislav Shkarin

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 165 KB
• Volume: 257
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310080

✦ Synopsis

Abstract

Let 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 ∫^1^~0~ (ω(t))^−1^ dt = ∞, then for any (t~0~, x~0~) ∈ 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(t~0~) = x~0~ has a unique solution in a neighborhood of t~0~. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫^1^~0~ (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(t~0~) = x~0~ has no solutions in any interval of the real line.