Grönwall bound

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.