2 Ordinary differential equations
2.1 Dini derivatives
The upper right Dini derivative of \(f : \mathbb {R} \to \mathbb {R}\) at \(t\) is
When \(f\) is differentiable at \(t\) this equals \(f'(t)\).
2.2 Gronwall–Bellman inequality
Gronwall–Bellman inequality. Let \(\Lambda , \mu : [a,b] \to \mathbb {R}\) be continuous with \(\mu \ge 0\), and let \(y : [a,b] \to \mathbb {R}\) be continuous satisfying
Then
2.3 Continuous dependence on parameters
A function \(x : [t_0, t_1] \to E\) is an integral solution of \(\dot{x} = F(t,x)\) with initial value \(x_0\) when
Theorem 3.5 (Continuous dependence on parameters). If \(y\) solves \(\dot{y} = f(t,y)\) and \(z\) solves \(\dot{z} = f(t,z) + g(t,z)\) with \(\| g(t,x)\| \le \alpha \) and \(\| z_0 - y_0\| \le \alpha \), and \(\alpha (1 + 1/L)e^{L(t_1-t_0)} \le \varepsilon \), then \(\| y(t) - z(t)\| \le \varepsilon \) for all \(t \in [t_0, t_1]\).
Reduce to theorem 22 applied to \(\| y-z\| \), using the \(L\)-Lipschitz bound on \(f\) and the \(\alpha \)-bound on \(g\).
2.4 Comparison lemma
Comparison Lemma 3.4. If \(u\) solves \(\dot{u} = f(t,u)\) with \(u(t_0) = u_0\), and \(v\) is continuous with \(D^{+}v(t) \le f(t,v(t))\) and \(v(t_0) \le u_0\), then \(v(t) \le u(t)\) for all \(t \in [t_0, t_1]\).
For each \(\lambda {\gt} 0\), the perturbed solution \(z_\lambda \) of \(\dot{z} = f(t,z) + \lambda \) satisfies \(v \le z_\lambda \) by ??. By theorem 24, \(\| u - z_\lambda \| \le \varepsilon /2\). Since \(\varepsilon {\gt} 0\) is arbitrary, \(v(t) \le u(t)\).