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Fixed point theorem in metric spaces and its application to the Collatz conjecture

Toshiharu Kawasaki

TL;DR

The work proposes a new fixed point framework for metric spaces via $(\alpha,\beta,\gamma,\delta,\varepsilon,\zeta)$-weighted generalized pseudocontractions and their $\lambda$-weighted variants, establishing conditions under which iterates converge to fixed points in complete spaces. It then specializes to the natural numbers with the standard metric and constructs a Collatz-inspired operator, analyzing its contraction properties under careful coefficient choices and parity-based cases. The main contribution is a rigorous set of convergence results (Theorems th:1–th:4) and a detailed, though partial, application to the Collatz problem, showing that under certain parameter regimes a fixed point exists and iteration converges, while highlighting the limitations posed by discreteness and case completeness. The work clarifies how fixed-point methods in generalized pseudocontractions could inform Collatz-type questions, offering a structured route to a potential proof assuming global satisfaction of the required inequalities.</nobr>

Abstract

In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.

Fixed point theorem in metric spaces and its application to the Collatz conjecture

TL;DR

The work proposes a new fixed point framework for metric spaces via -weighted generalized pseudocontractions and their -weighted variants, establishing conditions under which iterates converge to fixed points in complete spaces. It then specializes to the natural numbers with the standard metric and constructs a Collatz-inspired operator, analyzing its contraction properties under careful coefficient choices and parity-based cases. The main contribution is a rigorous set of convergence results (Theorems th:1–th:4) and a detailed, though partial, application to the Collatz problem, showing that under certain parameter regimes a fixed point exists and iteration converges, while highlighting the limitations posed by discreteness and case completeness. The work clarifies how fixed-point methods in generalized pseudocontractions could inform Collatz-type questions, offering a structured route to a potential proof assuming global satisfaction of the required inequalities.</nobr>

Abstract

In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.

Paper Structure

This paper contains 3 sections, 7 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Fixed point theorem
  3. Applying to the Collatz conjecture

Key Result

Theorem 1.1

Let $f$ be a function from $\mathbb{N}$ into $\mathbb{R}$ with $\lim_{N \to \infty}f(N) = \infty$. Then $\textup{Col}_{\textup{min}}(N) < f(N)$ for allmost all $N \in \mathbb{N}$ in the sence of logarithmic density.

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • ...and 3 more