How Can I Prove Gronwall's Inequality?

[onie mti]

i am asked to prove phi(t) ≤ phi(s)e^(c(|t-s|)) for t in I. I have to proof this

how can I start supposing that t>s and that d/dt(e^(-ct) phi(t))

 

[gtoguy97]

<=0 for t in ILet's start by supposing that t>s and that d/dt(e^(-ct) phi(t))<=0 for t in I. We can then use the Mean Value Theorem to conclude that there exists a c_1 in (s,t) such that: phi(t) = phi(s) + d/dt(e^(-ct) phi(c_1))(t-s). Using the fact that d/dt(e^(-ct) phi(t))<=0 for t in I and rearranging terms, we get: phi(t) <= phi(s) + d/dt(e^(-ct) phi(c_1))(t-s), which can be rewritten as: phi(t) <= phi(s)e^(c(|t-s|)). Therefore, we have proved that phi(t) ≤ phi(s)e^(c(|t-s|)) for t in I.