# ODE0.1

From Example Problems

State and prove Gronwall's lemma.

Lemma: Let be continuous for and satisfy the inequalities

for some non-negative constants and . Then

Proof:

It is sufficient to consider the case . The proof for the other case is the same. Let

Then is continuous and differentiable, and . Hence the inequality takes the form

Now multiply both sides by the non-negative function , so that

Now integrate the inequality from to :

or

.

But,

or

which completes the proof.