nonuniqueness

Nonuniqueness is the property of a problem or model having more than one solution that satisfies the given conditions. It contrasts with uniqueness, where a problem has a single solution. In mathematics, nonuniqueness often arises from underdetermination, nonlinearity, symmetry, or insufficient constraints that allow multiple admissible results.

A common setting is linear systems. A x = b can have no solution, a unique solution, or infinitely many solutions depending on the rank of A and the number of variables. Even simple equations like x + y = 1 in the plane have infinitely many pairs (x, y) that fit.

Differential equations also exhibit nonuniqueness, particularly when the right-hand side fails to be Lipschitz continuous. A classic example is y' = sqrt(y) with y(0) = 0, which admits more than one solution, such as y(t) ≡ 0 and y(t) = (t/2)^2 for t ≥ 0.

In applied contexts, nonuniqueness arises in inverse problems, where different parameter values can produce indistinguishable observations, and in optimization, where objective functions with symmetry or flat regions can have multiple optima. Nonuniqueness can also reflect inherent ambiguity in models when data or constraints are incomplete.

Addressing nonuniqueness typically involves adding constraints, regularization, or applying selection criteria that favor a particular solution, or proving conditions that guarantee uniqueness under stronger assumptions. Understanding the sources of nonuniqueness helps analysts assess whether multiple solutions are meaningful, or whether additional information is needed to identify a preferred outcome.