Common Fixed Points in Modular Spaces

In this paper,there are new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved, Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.

Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

It is shown that ifXis a Banach space andCis a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets{Ci:1≤i≤n }ofX, and eachCihas the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping ofChas a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.