proof of Gronwall’s lemma


The inequality

ϕ(t)K+Lt0tψ(s)ϕ(s)𝑑s (1)

is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to

ϕ(t)K+Lt0tψ(s)ϕ(s)𝑑s1

Multiply by Lψ(t) and integrate, giving

t0tLψ(s)ϕ(s)dsK+Lt0sψ(τ)ϕ(τ)𝑑τLt0tψ(s)𝑑s

Thus

ln(K+Lt0tψ(s)ϕ(s)𝑑s)-lnKLt0tψ(s)𝑑s

and finally

K+Lt0tψ(s)ϕ(s)𝑑sKexp(Lt0tψ(s)𝑑s)

Using (1) in the left hand side of this inequality gives the result.

Title proof of Gronwall’s lemma
Canonical name ProofOfGronwallsLemma
Date of creation 2013-03-22 13:22:23
Last modified on 2013-03-22 13:22:23
Owner jarino (552)
Last modified by jarino (552)
Numerical id 5
Author jarino (552)
Entry type Proof
Classification msc 26D10