2.2 Lipschitz conditions
Definition 2.2.1. \(f\) is Lipschitz in \(x\) on a set \(R\) with constant \(L\) if \[ \|f(t, x) - f(t, y)\| \leq L\|x - y\| \qquad \text {for all } (t, x), (t, y) \in R , \] and locally Lipschitz on \(D\) if every point of \(D\) has a neighbourhood on which \(f\) is Lipschitz in \(x\).
Lemma 2.2.2. If \(f\) and its partial derivatives \(\partial f_i/\partial x_j\) are continuous on \(D\), then \(f\) is locally Lipschitz. On a set \(R = [t_0 - a, t_0 + a] \times \bar B(x_0, b)\) contained in \(D\), one may take \(L = \max _R\|D_xf(t, x)\|\), the largest operator norm of the Jacobian matrix.
Proof. The ball \(\bar B(x_0, b)\) is convex, so for \(x, y\) in it the segment \(x + \theta (y - x)\), \(0 \leq \theta \leq 1\), stays in \(R\). By the fundamental theorem of calculus applied to \(\theta \mapsto f(t, x + \theta (y - x))\), \[ f(t, y) - f(t, x) = \int _0^1D_xf\bigl (t, x + \theta (y - x)\bigr )(y - x)\,d\theta , \] whose norm is at most \(\max _R\|D_xf\|\,\|y - x\|\). The maximum exists because \(D_xf\) is continuous on the compact set \(R\). Every point of \(D\) has such a compact neighbourhood. □
- (a)
- \(f(x) = 1 + x^2\) is Lipschitz on \(|x| \leq b\) with \(L = 2b\), since \(|x^2 - y^2| = |x + y||x - y|\); it is locally but not globally Lipschitz on \(\mathbb R\).
- (b)
- \(f(x) = \sin x\) is globally Lipschitz with \(L = 1\), and a linear \(f(t, x) = A(t)x\) is Lipschitz on \([\alpha , \beta ] \times \mathbb R^n\) with \(L = \max \|A(t)\|\).
- (c)
- \(f(x) = 3x^{2/3}\) is not Lipschitz near \(0\): \(|f(x) - f(0)|/|x - 0| = 3|x|^{-1/3} \to \infty \). Nor is \(\sqrt {|x|}\). Both are continuous, and both have non-unique solutions from \(0\).
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