1. Existence and Uniqueness

Question 1

Definition 1.3
A solution to an IVP.

Question 2

Definition 1.6
Locally Lipschitz.

Question 3

Theorem 1.7 (Picard’s Theorem for Locally Lipschitz)

Proof.

Question 4

Proposition 1.8
Suppose f \in C^1(R^n, R^n), then…

Proof.

Question 5

Definition 1.9
Globally Lipschitz.

Question 6

Theorem 1.10 (Picard’s Theorem for Globally Lipschitz)

Proof.

Question 7

Integral form of solution to IVP.

Question 8

Lemma 1.11
Solutions to IVPs are equivalent to satisfying the integral equation (full statement).

Proof.

Question 9

Picard Iteration Method

Question 10

Corollary 1.15 (Solutions Cannot Cross)

Proof.

Question 11

Corollary 1.17 (Uniqueness on Large Intervals)

Proof.

Question 12

Definition 1.18
Solution to IVP on interval.

Question 13

Lemma 1.20 (Gluing Lemma)

Question 14

Theorem 1.22 (Globally Lipschitz Implies Global Existence)

Proof.

Question 15

Definition 1.23
The Maximal Interval of Existence.

Question 16

Theorem 1.26
Maximal Interval of Existence is what?

Proof

Question 17

Definition 1.27
Neighbourhood and Epsilon-neighbourhood.

Question 18

Proposition 1.29
If K \subset R^n is non-empty and compact, and V is a neighbourhood of K, then…

Question 19

Lemma 1.30 (Uniform Existence)

Proof.

Question 20

Theorem 1.31 (Finite Time Blowup)

Proof.

Question 21

Lemma 1.33 (Gromwall’s Inequality)

Proof.

Question 22

Theorem 1.34 (Continuous Dependence)

Proof.