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Fixed point of nonexpansive

1) Exercise.

Let (E, d) a compact space MetriQ.

T is a continuous

(or more generally closed or closed graph)

of E in E

i) Show by an example that T does not necessarily

a fixed point.

(see an example in message 12/18/2007)

ii) Prove that T has a fixed point if

T admits a sequence of fixed points

approaches, ie d, there exists a Following

{x_ {n}} in E such that d (x_ {n} {n} Tx_)

tends to 0 when n tends to infinity

2.Exercice

A. Let (E , D in D.

i) Show, for example, that T does not necessarily

a fixed point.

(see an example in message 12/18/2007)

ii) Prove that T admits approaches fixed points.

B. Let [H ,(.,.), II] a Hilbert space.

D is a subset of H, non-empty, convex, and terminal

ferme.T Contracting is an application of D in D.

{u (n)} is a sequence of fixed points approaches T.

u is the weak limit of a subsequence of {u (n)}

justify the existence of u.

iii) Let v be any element of H and P is the operator

projection on D. Justifying the existence of P.

It is assumed that P is a nonexpansive

of H in D.

Show that

(u (n)-v-Tu (n) + TP (v), given (n))

any v in H.

conclude that:

(TP (v)-v, vii)

any v in H.

iv) Consider any element w in H

and define v by: v =

u + tw with t> 0
<=0,
Show that (TP (u)-u, w)

deduce that u is a fixed point of T.

v) Show that Fix (T) = {u D: Tu = u}
<=0,
is a convex set, terminal and firm.

NOTE.The fixed point theorem presented in year2

above, is due to Browder.

It is demonstrated that this result is valid

in Banach spaces uniformly convex
<=0.
(Browder and Gohde) and more generally

in Banach spaces reflexive

normal structure (Kirk). All four items

were published independently,

during the same year 1965;

was the starting point for research

important quantitatves and qualitative

on Applications and Contracting

point theory that the approach fixe.Soulignons

Kirk has raised more interest in this research.

For an overview on this work, we can

see, for example, readings

Kirk (1981), Goebel-Kirk (1990) and Kirk (1996).

The last paper is devoted to problems

open.

Finally, note that the proof of the theorem of Browder and

Browder-Gohde, in Hilbert spaces

and uniformly convex spaces,

is presented in a fairly simple

with the notion of asymptotic center.


(see 12/18/2007 post for examples

this message sujet.voir-references to 02/02/2007

articles cited above)