Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces

The theory of modular spaces was initiated by Nakano 1 in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz 2 in 1959. These spaces were developed following the successful theory of Orlicz spaces, which replaces the particular, integral form of the nonlinear functional, which controls the growth of members of the space, by an abstractly given functional with some good properties. In 2007, Razani et al. 3 studied some fixed points of nonlinear and asymptotic contractions in the modular spaces. In addition, quasi-contraction mappings in modular spaces without Δ2-condition were considered by Khamsi 4 in 2008. Recently, Kuaket and Kumam 5 proved the existence of fixed points of asymptotic pointwise contractions in modular spaces. Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces. The existence of fixed points of asymptotic pointwise nonexpansive mappings was studied by Kirk and Xu 6 in 2008, that is, mappings T : C → C, such that