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Quasi-intermediate value theorem and outflanking arc theorem for plane maps

Jiehua Mai, Enhui Shi, Kesong Yan, Fanping Zeng

Abstract

For a disk $D$ in the plane $\mathbb R^2$ and a plane map $f$, we give several conditions on the restriction of $f$ to the boundary $\partial D$ of $D$ which imply the existence of a fixed point of $f$ in some specified domain in $D$. These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of the main results, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if $f$ is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point.

Quasi-intermediate value theorem and outflanking arc theorem for plane maps

Abstract

For a disk in the plane and a plane map , we give several conditions on the restriction of to the boundary of which imply the existence of a fixed point of in some specified domain in . These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of the main results, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point.
Paper Structure (12 sections, 15 theorems, 140 equations)

This paper contains 12 sections, 15 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notions and notations
  4. Complementary domains of a Peano continuum in the plane
  5. Extensibility of homeomorphisms between $\sigma$-graphs
  6. Topological conjugacy
  7. Local injectivity and injectivity
  8. Orientations of manifolds
  9. Directed arcs, directed angles, and directed circles
  10. Sides of directed circles and directed arcs
  11. Quasi-Intermediate Value Theorem
  12. Outflanking arc theorem

Key Result

Theorem 1.1

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be an orientation preserving homeomorphism. If $f$ has a periodic point, then $f$ has a fixed point.

Theorems & Definitions (27)

  • Theorem 1.1: Brouwer's Lemma
  • Example 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3: The Schoenfiles Theorem
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 17 more