Gronwall's inequality
/ GROHN-vahl /
Gronwall's inequality is the single most useful estimate in this whole subject: a small lemma that turns a self-referential bound into an explicit one. Its job is to take a statement like 'this quantity is bounded by a constant plus an accumulation of itself' and conclude that the quantity cannot grow faster than an exponential. It is the rigorous form of the intuition that a process which feeds back on itself proportionally grows at most exponentially — never faster.
In its basic form it reads: if a nonnegative function u(x) satisfies u(x) <= a + L times the integral from x0 to x of u(t) dt, for constants a >= 0 and L >= 0, then automatically u(x) <= a times e^(L(x - x0)). Notice the shape of the input — the unknown u appears on both sides, trapped inside its own integral — and the shape of the output — a clean, explicit exponential cap with no integral left. That move, converting an implicit integral bound into an explicit exponential one, is what makes it so powerful: it is exactly the tool needed to control how fast solutions of differential equations can grow or spread.
This one inequality quietly powers three of the main theorems here. It gives uniqueness: set u to the difference of two solutions with the same start, get a = 0, and conclude u <= 0, so the solutions coincide. It gives continuous dependence: bound the gap between two solutions by e^(L|x - x0|) times the initial gap. And it underwrites continuation and a priori bounds that rule out blow-up. Whenever a proof in existence-uniqueness theory needs to say 'this cannot grow too fast,' Gronwall is almost always the hidden engine.
To prove uniqueness: if y and z both solve y' = f(x, y) with the same start and f is Lipschitz with constant L, let u = |y - z|. Then u(x) <= L times the integral from x0 to x of u(t) dt, so here a = 0. Gronwall forces u(x) <= 0 times e^(Lx) = 0, hence y = z. The two solutions cannot differ at all.
With a = 0 the exponential cap collapses to zero — that is how Gronwall delivers uniqueness in one line.
The inequality bounds growth by an exponential, not by a constant — it says 'no faster than e^(Lx),' which is the honest truth: solutions of linear-rate feedback genuinely can grow exponentially, just never faster.