Fixed point theorems for asymptotically regular mappings in modular and metric spaces

Abstract

Let $$L_\rho $$ be a modular function space where $$\rho $$ is a function modular satisfying the $$\Delta _2$$-type condition and is of s-convex type for $$s\in (0,1]$$ (see Definition 2 and Remark 3.2). Under this framework, the existence of fixed points for asymptotically regular mappings defined on some classes of subsets of $$L_\rho $$ is proved. As a consequence, some previous fixed point results for convex modulars (that is, modulars of 1-convex type) are extended. In particular, some variable Lebesgue spaces are modular function spaces where the modular is of s-convex type for some $$s\in (0,1)$$ but may fail to be convex. Furthermore, our arguments are applied to wider classes of domains than those that are just compact for the topology of the convergence locally in measure. This will be illustrated with several examples.