danielpancake / existence-uniqueness-for-cauchy-problem-theory

Covers basics of metric spaces, normed spaces, pointwise/uniform convergence, the Banach Fixed Point Theorem and its application in the Picard-Lindelöf Theorem for existence and uniqueness of solutions to IVPs for ODEs

Existence and Uniqueness of the Solution to Cauchy Problem

This presentation covers some key concepts in metric spaces, including Banach's fixed-point theorem, notions of convergence, normed spaces, spaces of continuous functions, and the Picard–Lindelöf theorem.

Overview

• Metric spaces
• Banach's fixed-point Theorem
  • Proof for uniqueness of a fixed-point
  • Proof for existence of a fixed-point
• Pointwise and Uniform convergence
• Normed spaces
• Space of continuous functions
• Picard–Lindelöf Theorem
• References