 Solution

ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2023/1/b/i/solution

For

s > t

, the coefficient bound gives

A (s) - A (t) \geq C_{1} (s - t)

. Thus for

0 \leq s \leq t

the positive kernel satisfies

K (t, s) := e^{- (A (t) - A (s))} \leq e^{- C_{1} (t - s)}, \int_{0}^{t} K (t, s) d s \leq \frac{1 - e ^{- C_{1} t}}{C _{1}} \leq \frac{1}{C _{1}} .

The starting Picard iterate is continuous and has supremum norm

M_{0} = ∣ b ∣

. If the

n

th Picard iterate is bounded and continuous, its integral defines another continuous function, and

∣ y_{n + 1} (t) ∣ \leq ∣ b ∣ e^{- A (t)} + M_{n}^{2} \int_{0}^{t} K (t, s) d s \leq ∣ b ∣ + \frac{M _{n}^{2}}{C _{1}} .

Taking the supremum norm and applying mathematical induction proves boundedness of every iterate and

M_{n + 1} \leq ∣ b ∣ + C_{1}^{- 1} M_{n}^{2} .

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