Temperature Circuit Diagram

Comparison of fixed point theorems of Brouwer and Banach in the real line

The following exercise shows that the principle of Banach

contraction on the real line is

a consequence of the fixed point theorem of Brouwer

at the existence of a fixed point and non-uniqueness.

Exercise.

i) Let T be a continuous mapping of [a, b] in [a, b];

[a, b] denotes a closed interval in IR terminal.

demonstrate that T admits a point fixe.Montrer by

an example, that the point fixed is not necessarily unique.

(fixed point theorem of Brouwer IR)

ii) Let T be a strictly contracting in IR

IR.Démontrer, with the help of i), that T admits a unique fixed point

.

(Hint: Show that for any x (0), there exists an r> 0 such

the interval [x (0)-r, x (0) + r] is stable by T)

Note . The year is extended, without difficulty,

to Banach spaces of finite or infinite dimension,

using the theorem of Brouwer's fixed point theorem or the

Darbo fixed point.