Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 5.6

Dependence on Initial Conditions and Parameters

Mathematics — Calculus · lipschitz-c1-derivative-bound

Problem

Let XX be a Banach space. Suppose that f∈C1(X,X)f \in C^1(X, X) is Lipschitz continuous, i.e. there exists 0<α0 < \alpha such that ∥f(x)−f(y)∥≤α∥x−y∥,\|f(x) - f(y)\| \leq \alpha \|x - y\|, for all x,y∈Xx, y \in X. Prove that ∥Df(x)∥≤α\|Df(x)\| \leq \alpha, for all x∈Xx \in X.

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