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

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 weak⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {weak}^{\star }$$\end{document} closed bounded convex subsets of duals of separable Banach spaces when the orthogonality relation ⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\perp $$\end{document} is uniformly weak⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {weak}^{\star }$$\end{document} 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.