Card Recovery 5.30 Astatalk

Generalizations of the simple fixed point theorem Banach

Exercise

Let (E, d) a metric space is an application of complet.T

E in E. Show that for any x in E, {T ^ {n} x} converges

in E, to a single fixed point of T, if either of the following is true

:
<=sigma(i=0 à l'infini)IIS^{n+i}x-S^{n+i+1}xII
i) There exists an integer n> = 1 and a constant k in] 0,1 [
<=R_{n} IIz-SxII
such as

d (T ^ {n} x T ^ {n} y)

ii) There exists a sequence {k_ {n}} in IR ^ {+} such that:

d (T ^ {n} x T ^ {n} y)

where the series [k_ {n}] is convergent.

iii) E has a different distance from such as:

d (x, y)

T is continuous with respect to d and strictly contracting

over with.