Existence–uniqueness, Picard, Lipschitz

The three theorems

Theorem	Hypothesis on f(x, y)	Conclusion
Peano	continuous near $(x_{0}, y_{0})$	a local solution exists
Picard–Lindelöf	continuous + Lipschitz in y	local solution exists and is unique
Global	continuous + |f(x,y)| ≤ A + B|y|	solution extends to the whole interval

$f \in C^{1} \Rightarrow$ locally Lipschitz ⇒ uniqueness. Lipschitz is sufficient, not necessary.

The two spoilers

Non-uniqueness: y′ = $y^{1/3}$, y(0) = 0. The right side is continuous but not Lipschitz at 0 (the derivative blows up). Solutions: y ≡ 0 and y = ±$(2x/3)^{3/2}$, plus every "wait then leave" hybrid — infinitely many.

**Finite-time blow-up:$** y' = y^{2}, y(0) = 1$ has the unique solution y = 1/(1 − x), which escapes at x = 1. Uniqueness is local; superlinear growth destroys global existence.

Extension rules

• Bounded f ⇒ |y′| bounded ⇒ no blow-up ⇒ global.
• Linear growth |f| ≤ A + B|y| ⇒ global (Grönwall).
• Equilibria trap solutions: if $f(y_{0}) = 0$ then $y \equiv y_{0}$ is a solution, and by uniqueness no other solution can cross that level. This is why $y' = (1 - y^{2})^{n} \cos y$ with y(0) = 0 stays in (−1, 1) for all time.

Grönwall's inequality

$u(x) \le A + B\int{}u \Rightarrow u(x) \le A$ $e^{Bx}$. This is the engine behind continuous dependence on initial data and global existence.

Key takeaways

• Continuity ⇒ existence; Lipschitz ⇒ uniqueness; linear growth ⇒ global.
• $y^{1/3}$ kills uniqueness; $y^{2}$ kills global existence.
• Equilibria act as barriers that solutions can never cross.