# An ultraproduct method via left reversible semigroups to study Bruck's generalized conjecture

## TL;DR

This paper introduces an ultraproduct-inspired method to analyze fixed points of left reversible semigroups acting on Banach spaces, leading to a new fixed point theorem for nearly uniformly convex spaces.

## Contribution

It develops a novel ultraproduct-like approach for fixed point analysis and proves a Bruck-type theorem for nearly uniformly convex Banach spaces.

## Key findings

- Nearly uniformly convex Banach spaces have the weak fixed point property for left reversible semigroups.
- The method extends fixed point results to broader classes of Banach spaces.
- Provides a new tool for studying fixed points in semigroup actions.

## Abstract

We use a method similar to ultraproducts to study the common fixed point of a left reversible semitopological semigroup acting on a Banach space. As an application, we prove a Bruck-type theorem for nearly uniformly convex Banach spaces to the effect that such spaces have weak fixed point property for left reversible semigroups.