April 2007

Sunday, April 29, 2007

The Wild Thornberrys Games Wildlife

between the fixed point theorem of Banach and Kannan (modular)

The following result provides a modular version of a synthesis between

fixed point theorems of Banach and

Kannan (see 9/10/2006 post).

Theorem.

Let rho} {E_ complete modular space, or mdulaire

rho satisfies the condition delta2. D is a subset

rho-rho E_ {farm}. T is an application (not necessarily

rho-continuous) from D to D satisfying:

There are two constants c> 1 and k in ( 0.1) as

rho (c (Tx-Ty)) <=liminf rho(x(n)-Tx(n))=a.

Then T admits a fixed point. <= rho(Tz-T^{2}z)< rho(z-Tz)<=a;

Monday, April 9, 2007

Ipod Shareaza Connect

Minimization of convex functions. The

can give two answers, well known

on the minimization of convex functions: The first

. We recall the following formulation

(perhaps more practical):

Theorem.

Let (E, II) a reflexive Banach space.

A is a subset of non-empty, convex and firm E;

f: A ----]- \\ infty, \\ infty] is a convex function

, semi-continuous - inferieuremet, non-

-identity equal to \\ infty and such that:

lim f (x) = \\ infty <= k max(rho(x-y),rho(x-Tx),rho(y-Ty)) sur DxD (*)
IXI -> \\ infty
As a simple example f (x) = Ix-aI. (A in E)

For proof of this result, see

H. Brezis (References 02/02/2007).

deuxieme.Le The following result generalizes a

resulat well known in the real right

:

Theorem.

Let E be a Banach space is a convex quelconque.f

IR.On in E implies that there exists a z in E such that

d_ {+} f (z) = 0.

(d_ {+} f (z). X = lim [f (z + hx)-f (z)] / h,
h -> 0 ^ {+}

with z, x E h IR)

Then f has a minimum at z

for information on d_ {+} f (z), see

RHMartin (References 02/02/2007).

Then, we easily show the inequality

f (x)> = f (z) + d_ {+} f (z). (Xz)

Finally, note that if f is strictly convex, then , f admits a unique minimum

.