2.4 Gronwall’s inequality and continuous dependence
Theorem 2.4.1 (Gronwall). Let \(u\) be continuous and non-negative on \([t_0, t_1]\), and suppose that for constants \(C \geq 0\) and \(K \geq 0\), \[ u(t) \leq C + K\int _{t_0}^tu(s)\,ds \qquad (t_0 \leq t \leq t_1) . \] Then \(u(t) \leq Ce^{K(t - t_0)}\). More generally, if \(u(t) \leq c(t) + \int _{t_0}^tk(s)u(s)\,ds\) with \(k \geq 0\) continuous and \(c\) continuous and non-decreasing, then \(u(t) \leq c(t)\exp \int _{t_0}^tk(s)\,ds\).
Proof. We prove the general form. Let \(v(t) = \int _{t_0}^tk(s)u(s)\,ds\), so \(v' = ku \leq k(c + v)\). Fix \(\tau \in [t_0, t_1]\); for \(t \leq \tau \), \(c(t) \leq c(\tau )\), so \(v' - kv \leq kc(\tau )\). Multiplying by the integrating factor \(e^{-\int _{t_0}^tk}\) gives \(\bigl (v e^{-\int _{t_0}^tk}\bigr )' \leq kc(\tau )e^{-\int _{t_0}^tk}\), and integrating from \(t_0\) to \(\tau \), with \(v(t_0) = 0\), \[ v(\tau )e^{-\int _{t_0}^\tau k} \leq c(\tau )\bigl (1 - e^{-\int _{t_0}^\tau k}\bigr ), \qquad \text {that is}\qquad v(\tau ) \leq c(\tau )\bigl (e^{\int _{t_0}^\tau k} - 1\bigr ) . \] Then \(u(\tau ) \leq c(\tau ) + v(\tau ) \leq c(\tau )e^{\int _{t_0}^\tau k}\). The first statement is the case \(c \equiv C\), \(k \equiv K\). □
The inequality converts an integral inequality into an exponential bound, and it is the main tool for comparing solutions.
Theorem 2.4.2 (Continuous dependence). Let \(f\) and \(g\) be continuous on a region containing the graphs of \(x(t)\) and \(y(t)\), solutions of \(x' = f(t, x)\), \(x(t_0) = x_0\) and \(y' = g(t, y)\), \(y(t_0) = y_0\) on an interval \(J \ni t_0\). If \(f\) is Lipschitz in \(x\) with constant \(L\) and \(\|f(t, z) - g(t, z)\| \leq \varepsilon \) there, then for \(t \in J\) \[ \|x(t) - y(t)\| \leq \|x_0 - y_0\|e^{L|t - t_0|} + \frac \varepsilon L\bigl (e^{L|t - t_0|} - 1\bigr ) . \]
Proof. Take \(t \geq t_0\). Subtracting the integral equations, \[ x(t) - y(t) = x_0 - y_0 + \int _{t_0}^t\bigl [f(s, x) - f(s, y)\bigr ]ds + \int _{t_0}^t\bigl [f(s, y) - g(s, y)\bigr ]ds , \] so \(u = \|x - y\|\) satisfies \(u(t) \leq \|x_0 - y_0\| + \varepsilon (t - t_0) + L\int _{t_0}^tu\). Put \(w = u + \varepsilon /L\): then \(w(t) \leq \|x_0 - y_0\| + \varepsilon /L + L\int _{t_0}^tw\), because \(L\int _{t_0}^t\varepsilon /L = \varepsilon (t - t_0)\). Gronwall gives \(w(t) \leq (\|x_0 - y_0\| + \varepsilon /L)e^{L(t - t_0)}\), which rearranges to the claim. □
With \(f = g\) the theorem says that solutions depend continuously, indeed Lipschitz-continuously, on their initial values, with a constant that may grow like \(e^{L|t - t_0|}\). With \(x_0 = y_0\) it says that a small error \(\varepsilon \) in the equation itself, a modelling error or a rounding error in a computation, produces an error at time \(t\) of at most \(\varepsilon (e^{Lt} - 1)/L\). Together these make the initial value problem well posed in the sense of Hadamard: a solution exists, is unique, and depends continuously on the data. The exponential factor is not an artefact: for \(x' = Lx\) the bound is attained, and it is the root of the sensitivity of chaotic systems studied at the end of the course.
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