On the Suzuki nonexpansive-type mappings

Abstract

‎It is shown that if $C$ is a nonempty convex and weakly compact subset of a‎ ‎Banach space $X$ with $M(X)>1$ and $T:C\rightarrow C$ satisfies condition $%‎ ‎(C)$ or is continuous and satisfies condition $(C_{\lambda })$ for some $%‎ ‎\lambda \in (0,1),$ then $T$ has a fixed point‎. ‎In particular‎, ‎our theorem‎ ‎holds for uniformly nonsquare Banach spaces‎. ‎A similar statement is proved‎ ‎for nearly uniformly noncreasy spaces‎.