1. Quizz 8
Let C[0, 1] be the space of real valued continuous functions on [0, 1]. Since [0, 1] is closed and bounded in R, it is sequentially compact in R. We have shown in class that C[0, 1] is a closed subspace of B[0, 1], the space of bounded real valued functions on [0, 1]. (In class, we prove a more general case.) Hence C[0, 1] with the supnorm k · k∞ is a real Banach space.
In this exercise, let us try to solve the the initial value problem dx
dt = f (t, x), x(0) = x0
via Picard iteration method/Banach contraction mapping principle.
(1) Solve for the initial value problem
x0(t) = tx(t), x(0) = 1
using following steps. In this case, f (t, x) = tx for t ∈ [0, 1] and x ∈ R (a) For any h ∈ C[0, 1], define
T (h)(t) = 1 + Z t
0
sh(s)ds, t ∈ [0, 1].
Prove that T (h) ∈ C[0, 1].
(b) Define a map T : C[0, 1] → C[0, 1] by h 7→ T (h) where T (h) is defined above.
Prove that T is a contraction mapping.
(c) Define x0(t) = 1 for all t ∈ [0, 1] and
xn(t) = T (xn−1)(t), t ∈ [0, 1].
Find xn for all n ≥ 1.
(d) Prove that (xn) is a Cauchy sequence in (C[0, 1], k · k∞).
(e) Solve for x via (xn).
(2) Solve for the initial value problem:
x0(t) = 2t(1 + x(t)), x(0) = 0.
In this case, f (t, x) = 2t(1 + x).
(a) Find b > 0 such that the map
T : C[0, b] → C[0, b], h 7→ T (h) is a contraction mapping, where
T (h)(t) = Z t
0
2s(1 + h(s))ds (b) Define x0(t) = 0 and xn(t) by
xn= T (xn−1).
Find (xn).
(c) Solve for the initial value problem by (xn).
(3) Let f : R2 → R be a continuous function. Suppose g, h : [a, b] → R are continuous.
Define k : [a, b] → R by
k(x) = f (g(x), h(x)), x ∈ [a, b].
Prove that k is continuous on [a, b].
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