Fixed Points of Fundamentally Non Expansive Mappings in Reflexive Banach Spaces

First, we derive several properties of a fundamentally nonexpansive self-mapping defined on a nonempty subset of a Banach space. Then, we demonstrate that if the Banach space satisfies the Opial condition, the fixed point set of such a mapping with a convex range is nonempty. In particular, we prove that when the reflexive Banach space is uniformly convex and the range of the mapping is bounded, closed, and convex, its fixed point set is also nonempty, closed, and convex.