Asymptotically Nonexpansive Mappings in Modular Function Spaces

In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying a 2-type condition, C a convex, ρ-bounded, ρ-a.e. compact subset of Lρ, and T C → C a ρ-asymptotically nonexpansive mapping, then T has a fixed point.In particular, any asymptotically nonexpansive self-map defined on a convex subset of L1 μ which is compact for the topology of local convergence in measure has a fixed point.