Definition
Nonuniqueness for an initial-value problem
Two distinct solutions in the same stated class satisfy the same equation, data, and prescribed force.
An initial-value problem is nonunique in a specified solution class if two distinct members of that class satisfy the same equation, initial data, boundary conditions, and prescribed force. Distinctness uses the class's equality convention: for measurable solutions it usually means failure of almost-everywhere equality, whereas continuous functions are compared pointwise.
Quantifiers
Existence of a nonunique datum does not assert nonuniqueness for every datum. Two solutions with different forces do not prove nonuniqueness for a fixed forced problem. Enlarging the solution class can destroy uniqueness even when uniqueness holds in a smaller class.