Grönwall bound

Definition

Let $\delta, K, \varepsilon$ be real numbers. The Grönwall bound with these parameters is the function $G_{\delta, K, \varepsilon} : \mathbb{R} \to \mathbb{R}$ given by $$G_{\delta, K, \varepsilon}(x) = \begin{cases} \delta\, e^{K x} + \dfrac{\varepsilon}{K} \left( e^{K x} - 1 \right) & \text{if } K \neq 0, \\ \delta + \varepsilon x & \text{if } K = 0. \end{cases}$$

Remarks

The function $v = G_{\delta, K, \varepsilon}$ is the unique solution of $v' = K v + \varepsilon$, $v(0) = \delta$. For $\varepsilon = 0$, it is $\delta e^{K x}$. A form of Grönwall's inequality uses it: let $f : [a, b] \to \mathbb{R}^n$ be continuous and differentiable on $(a, b)$, with $\|f(a)\| \le \delta$ and $\|f'(x)\| \le K \|f(x)\| + \varepsilon$ on $(a, b)$. Then $\|f(x)\| \le G_{\delta, K, \varepsilon}(x - a)$ on $[a, b]$. The estimate for approximate solutions is stated with this bound.

Explanations

L1 human

Suppose that a quantity starts at most at $\delta$ and grows at most at rate $K$ times itself plus $\varepsilon$. The Grönwall bound is the largest value that it can reach. It is the solution of the extreme case, where both inequalities are equalities.