Asymptotically nonexpansive mappings in modular function spaces

Abstract

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 (Ω, µ) which is compact for the topology of local convergence in measure has a fixed point.