Introduction:
recall the fixed point theorem of Edelstein (1962)
(in metric):
Theorem.
Let (E, d) a metric space is an application complet.T
contraction of e in E. We assume q'il exists an x in E
as the sequence {T} x ^ {n} possesses a subsequence converging to
z.Alors, z is a unique fixed point of T.
The following result provides a modular version of this
resultat.Cette formulation was introduced in the thesis
Ait Taleb (1996). see message references.
Exercise. 
Let E (rho), a modular space, or rho verifies ownership of Fatou.K is a subset of E-compact rho (rho).
T is an application of K in K such that:
rho (Tx-Ty)
show that there exists a z in K such that inf {rho (x-Tx), x in K}
= rho (z-Tz). Deduce that Tz = z.
Proof
Let a = inf {rho (x-Tx), x in K}, so there is a minimizing sequence
{x (n)} in K such that rho (x (n)-Tx (n ))---> aK rho-being compact, we can extract
{x (n)} a subsequence, still denoted by {x (n)}, rho-
converging towards z in K; as T is contractive, therefore, Tx (n) converges to
Tz.La Fatou property, provides:
rho (z-Tz)
other part,
which implies that a = 0 and Tz = z.
< rho(x-y) pour tout (x,y) dans KxK avec x different de y
