1.The recall (see the message of 26/10/2007, the third demonstration of the principle of contraction) the following result:
Let (E, d) a metric space is a continuous complet.T of E into E and there exists a (any) f E in IR ^ {+} such that:
d (x, Tx) 
Then T admits a fixed point. 
This formulation generalizes the principle of contraction at the existence of fixed point. 2. In this direction, Caristi (1976) proposed the following result:
Théorème1.
<=f(x)-f(Tx) sur E
Let (E, d) a metric space complet.T is an application (any) of E into E and there exists a mapping f from E to IR-lower semicontinuous, bounded below and such that:
d (x , Tx)
Then T admits a fixed point.
For direct demonstration of this theorem using the methods of order, I was ordered by the relation: x
and we apply, for example, the general result of [Brezis-Browder].
<=f(x)-f(Tx) sur E
(see also [Dugundji-granite] [Deimling], [Goebel-Kirk]).
3.Rappelons the following variational principle due to Ekeland:
Theoreme2
<=y si, et seulement si, d(x,y)<=f(x)-f(y)
Let (E, d) a complete metric space. f: E -> IR is a lower semicontinuous function and minorée.On gives itself a strictly positif.Alors epsilon, there exists an x = x (epsilon) belonging to E such that:
for all y in E, f (y)> = f (x)-epsilon d (x, y)
4.The Theorems 1 and 2 are equivalent because: The
théorème2 drives the théorème1.
We apply the theorem with Ekelend epsilon] 0.1 [. So there is an x in E such that:
for all y in E, f (y)> = f (x) - epsilon d (x, y)
We take y = Tx, where:
d (x, T (x))
therefore d (Tx, x) = 0 and Tx = x. The
théorème1 drives the théorème2
is used the axiom of choix.Supposons the contrary, then there exists a
epsillon strictly positive such that for all x in E, there is a
<= f(x)-f(T(x))<=epsilon d(x,T(x))
g (x) of x such that
f (g ( x))
Hence g admits a point fixe.Contradiction.
5.La formulation Caristi (theoreme1) is considered an important generalization of the Banach contraction principle, only in the existence of fixed point, and still attracts the interest being reported several generalizations of this theorem in the literature (see, eg, [Downing, Kirk DW.A.]. [Bae, JS - Cho, EW - Yeom, SH (1994)]. [Bae, JS (2003)]. Also, note the result of Takahashi (1990) which provides a result of minimizing non-convex, unifying theorem and Caristi minimization principle of Ekeland.
(For a bibliography of articles cited above, see the message 02/02/2007)
