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

TL;DR

The paper advances fixed-point theory in definably complete, locally o-minimal structures by proving a Lipschitz definable curve selection lemma and a definable Caristi fixed point theorem. It uses a tangent-space projection framework, definable compactness, and Jacobian estimates to construct Lipschitz curves and derive fixed-point conclusions, yielding a negative answer to whether strong BFPP forces o-minimality. The results show that a non-o-minimal yet definably complete locally o-minimal expansion can enjoy the strong BFPP and establish a definable Caristi fixed-point equivalence tied to closedness. Collectively, the work extends Caristi-Banach-type fixed-point theory to definable settings and clarifies the relationship between BFPP, open cores, and o-minimality.

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.

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: Definable Caristi fixed point theorem
• Lemma 2.1
• proof
• Lemma 2.2: Lipchitz definable curve selection
• proof
• proof : Proof of Theorem \ref{['thm:local_case']}
• Lemma 3.1
• proof