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modular formulation of the local version of the set of fixed point theorem of Banach.

In modular spaces, the following result offers a generalization

the local version of the principle set

contraction in Banach spaces:

Execice (provisional text).

Let X be any set and (E, rho) a complete modular space

, where the modular rho satisfies the property of Fatou < y(p).Montrer que,pour tout t dans [0,p], la suite

Let the ball B = B (z, r) = {x: rho (XZ)

T is an application deXxB in E satisfying:

There are two constants, k in] 0,1 [and p> 1, such as:

rho (p (T (x, y)-T (x, y ')))

for all x in X and all (y, y ') in BxB.

Assume that for any x in X, rho (q (T (x, z)-z))

the conjugate of p, ie d (1 / p) + (1 / q) = 1

demonstrate that there is a unique application of X f in B such that:

f (x) = T (x, f (x)) on X. Assume further that X is a modular space

any (or topological space satisfying the first axiom of countability

) T is continuous with respect to its first argument and rho

satisfies condition delta2.

Demonstrate that f is continuous on X.
< r}.