Mathematics 118A, Winter 1998

Homework Assignment 3, due Friday, January 30

1. Let X be a Banach space with norm ∥·∥. Recall that the norm of a linear transformation T: X → X is defined by

∥T ∥_{L} = sup_{\begin{matrix}x∈X \\ x≠0 \end{matrix}} \frac{∥T x∥}{∥x∥}.

Prove that ∥A^{k}∥ ≤ ∥A∥^{k} for every k ∈ ℕ.

2. Let f(t) = 5(1 − e^{t}). Use the contraction mapping theorem to show that the equation f(t) = t has exactly one solution, which is located in the interval [4, 5].

3. (a) Acquaint yourself with Mathematica, for more details see

        http://www.math.uci.edu/~moliver/math118/math118.html

(b) Revisit Problem 3 on Homework Assignment 1. Let Mathematica solve the equation, and generate the graphs of a few solutions.

(c) Revisit Problems 3 and 4a on Homework Assignment 2. Repeat the calculations on Mathematica, and graph the vector fields (for different values of λ) as well as some phase curves. Can Mathematica find exact solutions to these equations?