Lipchitz curve selection and its application to Thamrongthanyalak's open problem

TL;DR

This paper addresses an open problem related to the definable Banach fixed point property by employing Lipschitz curve selection, and also establishes a definable version of Caristi's fixed point theorem.

Contribution

It introduces a novel application of Lipschitz curve selection to solve an open problem and extends fixed point theorems to a definable setting.

Findings

Solved Thamrongthanyalak's open problem on Banach fixed point property.

Developed a definable version of Caristi fixed point theorem.

Demonstrated the effectiveness of Lipschitz curve selection in fixed point theory.

Abstract

We solve an open problem posed in Thamrongthanyalak's paper on the definable Banach fixed point property. A Lipschitz curve selection is a key of our solution. In addition, we show a definable version of Caristi fixed point theorem.