Fixed point results for generalized nonexpansive and Suzuki mappings with application in $L^{1}(\Omega, \Sigma, \mu)$

Abstract

It is natural to ask whether the weak fixed point property for nonexpansive mappings in Banach spaces is inherited by other generalized nonexpansive mappings without using weak normal structure or close-to normal structure (also called quasi-normal structure) (see C.S. Wong, Close-to-normal structure and its applications, J. Func. Anal. 16 (1974), no. 4, 353-358). In this paper, we give an affirmative answer to this question for Suzuki mappings and other generalized nonexpansive mappings in the setting of various Banach spaces. In addition, we prove the existence of common fixed points for commuting affine $(c)$-mappings and Suzuki mappings acting on convex bounded $L^{0}$-closed subsets in the Banach space $L^{1}(\Omega, \Sigma, \mu)$.