Some fixed point results for (c)-mappings in Banach spaces

Abstract

In this paper, we solve two fixed point problems associated with the class of (c)-mappings. The first one is devoted to obtain the existence of fixed points for such mappings defined on $$\hbox {weak}^{\star }$$ closed bounded convex subsets of duals of separable Banach spaces when the orthogonality relation $$\perp $$ is uniformly $$\hbox {weak}^{\star }$$ approximately symmetric. For the second problem, using the idea of R. Smarzewski (On firmly nonexpansive mappings, Proc. Am. Math. Soc 113(3):723–725,1991), we prove the existence of fixed points for such mappings which are defined on a finite union of weakly compact convex subsets of UCED (uniformly convex in every direction) Banach spaces.