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local version of the fixed point theorem of Banach (modular)

In modular spaces, the following result,

proposes a generalization of the local version of the principle of

contraction in Banach spaces.


Execice

Let (E, rho) a complete modular space, or modular rho

checks the Fatou property.
Let the ball B = B (y, r) = {x: rho (xy) is strictly less than r}.
T is an application of B in E sharp contraction, ie d,

there are two constants p, k, with k in] 0,1 [and p> 1

such as


rho ( p (Tx-Ty)) \\ leq k rho (xy) on ExE

Assume that rho (q (Ty-y)

ie d (1 / p) + (1 / q) = 1 . Prove that T has a fixed point.

Proof

is followed the same steps if the metric.

Let r 'in] 0, r [such that rho (q (Ty-y)) \\ leq (1 -k) r '

Consider the ball D = {x: rho (xy) \\ leq r'}. Let us show that D is stable by T;

Indeed, for x in D, we have:

rho (Tx-y) = rho ((p / p) (Tx-Ty) + (q / q) (Ty-y))

\\ leq rho (p (Tx-Ty)) + rho (q (Ty-y))

\\ leq kr '+ (1-k) r' = r '

As rho checks the Fatou property, D is rho-closed (check).

By

théorème2.1 in [Hanebaly (2005)], T has a fixed point.

< (1-k) r,ou q est le conjugué de p,