Lecture Notes in Differential Equations - Bruce E. Shapiro

102 LESSON 12. EXISTENCE OF SOLUTIONS*
2. f(t, y) is bounded by some number M on a rectangle R, i.e,
for all t, y in R.
3. f(t, y) is differentiable on R.
|f(t, y)| < M (12.8)
4. ∂f/∂y is also bounded by M on R, i.e.,
∂f(t, y)
∣ ∂y ∣ < M (12.9)
5. Define the Picard iterates as φ 0 (t) = y 0 and
for all k = 1, 2, . . . .
∫ t
φ k (t) = y 0 + f(s, φ k−1 (s))ds
t 0
(12.10)
Definition 12.2. Lipshitz Condition. A function f(t, y) is said to satisfy
a Lipshitz Condition on y in the rectangle R or be Lipshitz in y on R if
there exists some number K > 0, that we will call the Lipshitz Constant,
such that
|f(t, y 1 ) − f(t, y 2 )| ≤ K|y 1 − y 2 | (12.11)
for all t, y 1 , y 2 in some rectangle R.
Example 12.1. Show that f(t, y) = ty 2 is Lipshitz on the square −1 <
t < 1, −1 < y < 1.
We need to find a K such that
for f(t, y) = ty 2 , i.e., we need to show
But
so we need to find a K such that
|f(t, p) − f(t, q)| ≤ K|p − q| (12.12)
|tp 2 − tq 2 | ≤ K|p − q| (12.13)
|tp 2 − tq 2 | = |t(p − q)(p + q)| (12.14)
|t(p − q)(p + q)| ≤ K|p − q| (12.15)
|t||p + q| ≤ K (12.16)