Uniqueness of solutions

Uniqueness of solutions is the property that a specified equation or problem has at most one solution. It does not require that a solution exists.

Uniqueness proof: An argument showing that a mathematical problem has at most one solution. This is the direct proof task: assume two solutions and show they are equal.
Existence of solutions: The property that a specified equation or problem has at least one solution. Existence and uniqueness are separate claims; uniqueness alone permits no solution at all.
Picard–Lindelöf theorem: A theorem guaranteeing local existence and uniqueness for ordinary differential equations with locally Lipschitz right-hand sides. It gives a standard condition under which an initial value problem has a unique local solution.
Nonuniqueness in differential equations: The occurrence of multiple solutions to a differential equation with the same prescribed data. It shows how uniqueness can fail when regularity conditions on the equation are weakened.
Lipschitz continuity: A condition bounding how rapidly a function’s values can change with its input. Lipschitz bounds prevent distinct solution trajectories from separating under suitable conditions.
Well-posed problem: A problem whose solution exists, is unique, and depends continuously on its data. Uniqueness is one of the three classical requirements for a well-posed problem.
Cauchy–Lipschitz theorem: Another name for the Picard–Lindelöf theorem on local existence and uniqueness of ordinary differential equation solutions. The alternate name appears in treatments of the same uniqueness result.
Peano existence theorem: A theorem guaranteeing local solutions to certain ordinary differential equations with continuous right-hand sides. Continuity can ensure existence without ensuring that the solution is unique.
Contraction mapping theorem: A theorem stating that a contraction on a complete metric space has exactly one fixed point. A contraction forces any two fixed points to coincide, giving uniqueness alongside existence.
Initial value problem: A differential equation problem specifying a solution’s value at a chosen initial point. Its prescribed initial data make uniqueness questions concrete for differential equations.

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