Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Let be a uniformly convex modular function space with a strong Opial property. Let be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of . In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point. MSC:47H09, 47H10.