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On Zeros of Homotopic Mappings, Fixed Point Problems, and Inverse Functions

Oleg Zubelevich

Abstract

This article examines a family of smooth mappings between Banach spaces and establishes conditions for the existence of their zeros. Applications to fixed-point problems and the Implicit Function Theorem are also discussed.

On Zeros of Homotopic Mappings, Fixed Point Problems, and Inverse Functions

Abstract

This article examines a family of smooth mappings between Banach spaces and establishes conditions for the existence of their zeros. Applications to fixed-point problems and the Implicit Function Theorem are also discussed.
Paper Structure (11 sections, 7 theorems, 55 equations)

This paper contains 11 sections, 7 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. The Main Theorems
  3. The case of general domain
  4. The case of coercive $F$ and an unbounded domain
  5. Applications
  6. A Fixed Point Problem
  7. An Implicit Function Theorem
  8. Proof of the Theorems
  9. Proof of Theorem \ref{['q12e']}
  10. Proof of Theorem \ref{['xfb6t']}
  11. Appendix: Extensibility of Solutions to ODE

Key Result

Theorem 1

Under hypotheses 00, h4, and h, suppose that $F(0, x_0)=0$ for some $x_0\in D$. Then, there are two possibilities:

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: re
  • Lemma 1
  • Theorem 6
  • Remark 1