Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Implicit Function Theorems and Critical Point Theory in Fréchet Spaces

Kaveh Eftekharinasab

Abstract

We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's $ C_c^1 $-mappings between arbitrary Fréchet spaces. Subsequently, within this framework, we apply these theorems to establish the global existence and uniqueness of solutions to initial value problems that involve the loss of one derivative. Moreover, we prove a Lagrange multiplier theorem by employing indirect applications of the global implicit function theorems through submersions and transversality.

Global Implicit Function Theorems and Critical Point Theory in Fréchet Spaces

Abstract

We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's -mappings between arbitrary Fréchet spaces. Subsequently, within this framework, we apply these theorems to establish the global existence and uniqueness of solutions to initial value problems that involve the loss of one derivative. Moreover, we prove a Lagrange multiplier theorem by employing indirect applications of the global implicit function theorems through submersions and transversality.
Paper Structure (5 sections, 16 theorems, 103 equations)

This paper contains 5 sections, 16 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Differentiability
  3. Global Implicit Function Theorems
  4. An Initial Value Problem
  5. Lagrange Multipliers

Key Result

Lemma 2.1

Let $\varphi, \psi \in \mathop{\mathrm{Lip_{loc}}}\nolimits (\mathbbm {F},\mathbb{R})$, and let $x \in \mathbbm {F}$.

Theorems & Definitions (30)

  • Lemma 2.1: k1, Lemma 1.1
  • Definition 3.1: k1, Definition 2.1, Chang PS-Condition
  • Theorem 3.1: k1, Theorem 3.2, Mountain Pass Theorem
  • Lemma 3.1: k1, Lemma 4.1
  • Theorem 3.2: k1, Lemma 1.4, Chain Rule
  • Theorem 3.3: Global Implicit Function Theorem I
  • proof
  • Definition 3.2: k2, Definition 3.2, PS-Condition
  • Theorem 3.4: k2, Corollary 4.7
  • Theorem 3.5: k3, Theorem 2.3
  • ...and 20 more