Abstract
Let C be a nonempty, rho-bounded, rho-closed, and convex subset of a modular function space L-rho and T : C -> C be a monotone asymptotically rho-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) = alpha T-n(f(n)) + (1 - alpha)f(n),
for n is an element of N and establish the analogue to Schu's fundamental results in the setting of modular function spaces.