Ordinary differential equations Counterexample

Non-uniqueness for a non-Lipschitz equation

Builds on Picard–Lindelöf theorem Lipschitz condition Initial value problem Peano existence theorem

Counterexample

For a real number $x$ , let $x^{1/3}$ be its real cube root and let $x^{2/3} = \left(x^{1/3}\right)^2$ . Consider the initial value problem $x'(t) = 3\, x(t)^{2/3}$ , $x(0) = 0$ , on $\mathbb{R}$ . The constant function $x(t) = 0$ is a solution. The function $x(t) = t^3$ is also a solution, because its derivative is $3t^2 = 3 \left(t^3\right)^{2/3}$ . More generally, for each $c \ge 0$ , the function equal to $0$ for $t < c$ and to $(t - c)^3$ for $t \ge c$ is a solution. So the problem has infinitely many solutions on every interval that contains $0$ in its interior.

Remarks

The right side $f(x) = 3 x^{2/3}$ is continuous, so the Peano existence theorem applies. The function $f$ is not Lipschitz on any neighbourhood of $0$ : the quotient $f(x)/x = 3 x^{-1/3}$ is unbounded as $x \to 0$ . So the Picard–Lindelöf theorem does not apply. For an initial value $x(0) = x_0 \neq 0$ , the field is $C^1$ near $x_0$ , and the solution is unique near $t = 0$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Solution of an ordinary differential equation definition
Initial value problem definition
Peano existence theorem statement
Picard–Lindelöf hypotheses definition
Picard operator definition
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Picard–Lindelöf theorem statement
Non-uniqueness for a non-Lipschitz equation counterexample

Tag	`00K7ASZ1PR`
URI	`m2://00K7ASZ1PR`
Kind	counterexample
Writing	W0 stub
MSC	34A12
Uses	Picard–Lindelöf theorem, Lipschitz condition, Initial value problem, Peano existence theorem

Non-uniqueness for a non-Lipschitz equation

Counterexample

Remarks

Reading path

Further reading

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