Some geometric properties in modular spaces and application to fixed point theory

We prove that modular spaces Lρ have the uniform Kadec–Klee property w.r.t. the convergence ρ-a.e. when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial condition w.r.t. the convergence ρ-a.e. for both the Luxemburg norm and the Amemiya norm. Some assumptions over the modular ρ need to be assumed. The above geometric properties will enable us to obtain some fixed point results in modular spaces for different kind of mappings.  2004 Elsevier Inc. All rights reserved.