Fixed points of monotone asymptotically nonexpansive mappings in modular function spaces

Let C be a nonempty, ρ-bounded, ρ-closed, and convex subset of a modular function space L_ρ and T: C → C be a monotone asymptotically ρ-nonexpansive mapping. In this paper, we investigate the existence of fixed points of T. In particular, we establish a modular monotone analogue to the original Goebel and Kirk's fixed point theorem for asymptotically nonexpansive mappings. We will also investigate the behavior of the modified Mann iteration process defined by f_n+1 = α Tⁿ(f_n) + (1 - α)f_n for n ∈ N and establish the analogue to Schu's fundamental results in the setting of modular function spaces.