Knowlpedia

Proposition

Scalar comparison barriers for an ODE

Subsolutions and supersolutions bound a scalar solution when the vector field is locally Lipschitz.

Statement

Let y′=F(t,y)y'=F(t,y), with FF continuous and in its scalar state. Suppose differentiable barriers a,ba,b satisfy

a′≤F(t,a),b′≥F(t,b),a(t0)≤y(t0)≤b(t0).a'\le F(t,a),\qquad b'\ge F(t,b),\qquad a(t_0)\le y(t_0)\le b(t_0).

As long as all three functions stay in the common equation domain, a(t)≤y(t)≤b(t)a(t)\le y(t)\le b(t) for t≥t0t\ge t_0.

Why touching does not permit crossing

On a compact interval, the positive part of a−ya-y is bounded by a Lipschitz constant times its own accumulated integral. Gronwall forces it to vanish; apply the same reasoning to y−by-b. A barrier can therefore preserve positivity or separation from a singular denominator. For systems, componentwise comparison needs further order-preserving hypotheses and does not follow from this scalar statement.