- #1

JulienB

- 408

- 12

## Homework Statement

Hi everybody! I have a math problem to solve, I'd like to check if I understand well the Banach fixed-point theorem in the case of Euclidean norm and how to deal with maximum norm.

Check if the following functions ƒ: ℝ

^{2}→ ℝ

^{2}are strictly contractive in relation to the given norms:

[tex]\mbox{a) } || \cdot ||_2 , f(x_1,x_2) = \frac{1}{2} (sin x_1, cos x_2)

\\ \mbox{b) } || \cdot ||_2 , f(x_1,x_2) = \frac{1}{2} (x_1 + x_2, x_2)

\\ \mbox{c) } || \cdot ||_{\infty} , f(x_1,x_2) = \frac{1}{2} (x_1 + x_2, x_2)

[/tex]

(note: not sure about my translation, can you say "contractive" in english?)

## Homework Equations

Banach fixed-point theorem, mean value theorem for a), definition of norms

## The Attempt at a Solution

I found it was not so easy to prove that the functions were strictly contractive or not, but here is a go:

[tex] \mbox{a) } || \frac{1}{2} \binom{sin x_1}{cos x_2} - \frac{1}{2} \binom{sin y_1}{cos y_2} ||_2 = \frac{1}{2} || \binom{sin x_1 - sin y_1}{cos x_2 - cos y_2} ||_2 \\

= \frac{1}{2} \sqrt{(sin x_1 - sin y_1)^2 + (cos x_2 - cos y_2)^2} \\

= \frac {1}{2} \sqrt{cos^2(α) (x_1 - y_1)^2 + sin^2 (β) (x_2 - y_2)^2} ≤ \frac{1}{2} \sqrt{(x_1 - y_1)^2 - (x_2 - y_2)^2} = \frac{1}{2} || \binom{x_1 - y_1}{x_2 - y_2} ||_2 \\

\mbox{(note that I made use of the mean value theorem to show the inequality)} \\

c < 1 \implies f(x_1,x_2) \mbox{ is strictly contractive.}

[/tex]

Is that correct? Am I understanding the Banach fixed-point theorem correctly? Now start the problems:

[tex] \mbox{b) } || \frac{1}{2} \binom{x_1 + x_2}{x_2} - \frac{1}{2} \binom{y_1 + y_2}{y_2} ||_2 = \frac{1}{2} \sqrt{(x_1 + x_2 - y_1 - y_2)^2 + (x_2 - y_2)^2} \\

= \frac{1}{2} \sqrt{x_1^2 + 2x_1 x_2 + x_2^2 - 2x_1y_1 + 2x_1y_2 - 2 x_2y_1 + 2x_2y_2 + y_1^2 - 2y_1y_2 + y_2^2 + (x_2 - y_2)^2} \\

= \frac{1}{2} \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_2 + y_2)^2 + 2(x_1x_2 + x_1y_2 - x_2y_1 - y_1y_2)}

[/tex]

Ehem... It seems to me that it would require a c bigger than 1 to make the inequality of the Banach theorem work, therefore the function would

**not**be strictly contractive. On another hand the ½ makes it hard to prove... Is my intuition correct? Any clue about how to prove it if so?

And for c) I don't have a clue really. According to the definition in my teacher's script:

[tex]

c \cdot || \binom{x_1 - y_1}{x_2 - y_2} ||_{\infty} = c \cdot max | \binom{x_1 - y_1}{x_2 - y_2} | \\

\mbox{and} \\

\frac{1}{2} || \binom{x_1 + x_2 - y_1 - y_2}{x_2 - y_2} ||_{\infty} = \frac{1}{2} \cdot max | \binom{x_1 + x_2 - y_1 - y_2}{x_2 - y_2} |

[/tex]

But what does that mean? It seems to me like those maximums are not existing! Is that the case? If yes, what can I conclude? If no, I must have misunderstood how to determine the maximum of a function in ℝ

^{2}...or the definition of a maximum norm altogether!

Thank you in advance for your answers.Julien.