Fixed-point theorems for multivalued non-expansive mappings without uniform convexity

Abstract

Let $X$ be a Banach space whose characteristic of noncompact convexity is less than $1$ and satisfies the nonstrict Opial condition. Let $C$ be a bounded closed convex subset of $X$, $KC\left(C\right)$ the family of all compact convex subsets of $C$, and $T$ a nonexpansive mapping from $C$ into $KC\left(C\right)$. We prove that $T$ has a fixed point. The nonstrict Opial condition can be removed if, in addition, $T$ is a $1$-$\chi$-contractive mapping.