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open problem in the theory of fixed point (Modular Spaces) (2) References

following problem is implicit in earlier work:

Let (E, \\ rho) a complete modular space, D is a subset of E

\\ rho-T ferme.Soit an application from D to D \\ rho-strictly <(1-k)r,montrer que T admet

contracting, ie,:

exists k in] 0,1 [such as \\ rho (Tx-Ty) \\ leq k \\ rho (xy) on DXD

Then T possesses Does a point Fixed?

Note: If < r'< r tel que d(Ty,y)\leq (1-k)r'<(1-k)r.

B \\ rho-terminal, the answer is positive and the proof is almost

immediate.Si satisfies the condition delta2 rho and rho (xy) is

finite for all x, y in D The theoreme3.1 [Hanebaly (2005)] provides a result.

. See also Ait Taleb-Hanebaly (Post-References)