Advancements in fixed point theory in modular function spaces

The purpose of this paper is to give an outline of the recent results in fixed point theory for asymptotic pointwise contractive and nonexpansive mappings, and semigroups of such mappings, defined on some subsets of modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces such as Lebesgue, Orlicz, Musielak–Orlicz, Lorentz, Orlicz–Lorentz, Calderon–Lozanovskii spaces and many others. In the context of the fixed point theory, we will discuss foundations of the geometry of modular function spaces, and other important techniques like extensions of the Opial property and normal structure to modular spaces. We will present a series of existence theorems of fixed points for nonlinear mappings, and of common fixed points for semigroups of mappings. We will also discuss the iterative algorithms for the construction of the fixed points of the asymptotic pointwise nonexpansive mappings and the convergence of such algorithms.[Figure not available: see fulltext.]. © 2012, The Author(s).