What Ink Is Used For Balloon Stamp

The fixed point theorem of Banach and domain invariant (modular)

The following exercise offers a modular version (partial)

the result, well known in Banach spaces,

on the principle of contraction and the domain of invariance.

(see message from 03/29/2007)

Exercise. <= k d(x,y) sur ExE.

Let (E, rho) a modular space complet.On guess

rho (xy) is finite on ExE. T is a strongly application <= k_{n} d(x,y) sur ExE(*)

contracting E to E, ie d, there exist constants c and k such that

c> 1, k in] 0,1 [and

rho (c (Tx- Ty)) \\ leq k rho (xy) on ExE. <=d'(x,y) sur ExE,

1) shows that IT is bijective from E to E.

2) It is further assumed that the modular

rho satisfies the condition that delta2.Demontrer

T is a homeomorphism of E onto E.

Proof

1) Either the application of S E in E, defined by:

Sx = Tx + y, or y is a given element in E. Let us show

that S admits a fixed point unique.En Indeed,

rho (c (Sx-Sy)) = rho (c (Tx-Ty)) \\ leq k rho (xy)

on ExE, then S admits a unique fixed point

(see Ait-Taleb Hanebaly . references.Message 02/02/2007)) and

result, IT is invertible on E. E

2) We show that rho-IT is continuing, in effect, be

{x (n)} be a sequence converging to x. rho-

let y (n) = x (n)-Tx (n) and y = x-Tx, and let c 'combines the

c [(1 / c) + (1 / c') = 1] then:

rho (y (n)-y) = rho ((c '/ c') (x (n)-x) + (c / c) (Tx-Tx (n))

\\ leq rho (c '(x (n)-x)) + rho k (x (n)-x)
By
dlta2, rho (c '(x (n)-x)) tends to 0 when n

tends to infinity, therefore, y (n) tends to y. Let us show that

(IT) ^ { -1} is continuous, because:

let {x (n)} be a sequence converging to x. rho-

Let y (n) = (IT) ^ {-1} x (n) and y = ( IT) ^ {-1} x, so

x (n) = y (n)-Ty (n) and x = y-Ty; beyond:


rho (y (n)-y) = rho ((c '/ c') (x (n)-x) + (c / c) (Ty (n)-Ty)))

\\ leq rho ( c '(x (n)-x)) + k rho (y (n)-y)

what that entails:

rho (y (n)-y) \\ leq (1/1-k) rho (c '(x (n)-x))

By delta2, the second member of the inequality above

tends to 0 when n tends to infinity and y (n) is

rho- converging to y, so (IT) ^ {-1} is rho-continuous.