How Do I Stop My Baby From Choking On Flehm

Local version of parameterized fixed point theorem Banach

Exercise

Let X be any set and E a metric space

complet.B is an open ball in E with center z and radius <= k rho(y-y')

rT is an application of XXB in E satisfying: There

k in] 0,1 [such that d (T (x, y), T (x, y ')) \\ leq kd (y, y') < r(1-k), où q est

for all x in X and all (y, y ') in BxB. Assume that for all x

in X, d (T (x, z), z)

Prove that there exists a unique f X

B such that f (x) = T (x, f (x)) on X. If moreover X is a topological space

and T is continuous with respect to its first argument,

show that f is continuous on X.

Proof: From

after the local version (see results below), for

each x in X, the mapping T (x,.) B in E has a fixed point

single Note f (x), ie d T (x, f (x)) = f (x), which

can define a map f from X to B. We prove that f

is unique; Indeed: Let g

another application of X in B such that T (x, g (x)) = g (x)

then for all x in X:
< r(1-k).
d (f (x), g (x)) = d (T (x, f (x)), T (x, g (x))) \\ leq kd (f (x), g (x))

This entails that f = g on X.

for the continuity of f, x, x 'in X, then we have:

d (f (x), f (x')) = d (T (x, f (x)), T (x, f (x ')))

\\ leq d (T (x, f (x)), T (x, f (x')))+ d (T (x, f (x') ), T (x, f (x ')))

\\ leq kd (f (x), f (x')) + d (T (x, f (x ')), T (x', f (x ')))

From: d (f (x), f (x')) \\ leq (1/1-k) d (T (x, f (x ')), T (x '(x')) or

On the continuity of f at x '

Note.

For a direct demonstration in Banach spaces,

of this result, see [Dieudonne]. Note that this result is utlise

for a direct proof of the theorem of implicit functions.

see [Dieudonne] - [Deimling].