Math205c Spring Term 2003

Assignment 18

1. Find an example of a connected set which is not pathwise connected.

2. Assume that x_{0} ∈ ℝ^{n} and that f ∈ C^{1−}(ℝ^{n}, ℝ^{n}) is such that

|f(x) − f(y)| ≤ L|x − y| , x, y ∈ ℝ^{n}

for some L < 1. Show that the operator

T : C([0, 1], ℝ^{n}) → C([0, 1], ℝ^{n}) , u → T u

defined through

(T u)(t) = x_{0} + ∫_{0}^{t} f(u(τ)) dτ , t ∈ [0, 1]

has a fixed point in C([0, 1], ℝ^{n}) which solves

\begin{cases}u′ = f(u) \\ u(0) = x_{0} \end{cases} .

3. Let f ∈ C(D, ℝ^{m}) for some D ⊂^{o} ℝ^{n} and assume it is differentiable at x ∈ D. Show that there exists a constant M > 0 such that

|f(x) − f(y)| ≤ M |x − y|

for y in a neighborhood of x.

4. Let f ∈ C^{1}(D, ℝ) for some D ⊂^{o} ℝ^{n}. Fix x ∈ D and assume that ∇f(x) ≠ 0. Show that ∇f(x) points in the direction of maximal growth of f.

5. You ask a question.

The Homework is due Friday, April 18.