Week 6 Thursday lecture

Existence and Uniqueness for ODEs

Overview

• Study of the existence and uniqueness of solutions for ordinary differential equations (ODEs).

• Fundamental theorems help in determining whether initial value problems (IVPs) have solutions and if those solutions are unique.

Initial Value Problem

• General form: [ x'(t) = f(x(t)), \quad t \in (0, 8) \quad \text{with} \quad x(0) = x_0. ]

  • Example:
    [ x'(t) = x^3 + t^4, \quad x(0) = 1. ]

  • Solution structure can be expressed as:
    [ x(t) = e^t + c. ]

Fundamental Theorem of ODEs (Thm. 5.24)

• Assumption: Let ( f: \mathbb{R} \to \mathbb{R} ) be differentiable, and let ( \max | f'(x)| \leq M. )

• Conclusion: The IVP has a unique solution ( x: (0, 8) \to \mathbb{R}, ) where ( 8 \leq \frac{1}{M} ).

Fundamental Theorem of Calculus Application

• Using FTC, we can express a solution as follows:
  [ x(t) = x(0) + \int_0^t f(x(s)) ds. ]

• Denote: ( J(x) = x0 + \int0^t f(x(s)) ds. )

• This shows that the mapping of ( J ) is continuous over ( C([0, 8]) ).

Contraction Mapping Theorem

• Aim: To demonstrate that ( J ) acts as a contraction on space ( C([0, 3]) ) for small ( \epsilon ).

• If ( J: C([0,3]) \to C([0,3]) ) is a contraction, it guarantees a unique fixed point, which corresponds to a unique solution of the IVP.

• Specific expression derived: [ J(x) - J(y) ]

  • Leads to establishing bounds based on the maximum rates over the interval and continuous function conditions.

Establishing Contraction

• Need maximum contraction constant:
  [ M \cdot \epsilon < 1. ]

• Conclusion: Unique fixed point ( x = \sqrt{x} ) proves existence of a unique solution to the IVP.

Summary

• The contraction mapping principle facilitates finding unique solutions to IVPs, grounded in the properties of the stated function ( f ) and the continuity and differentiability of the involved mappings.

• By ensure ( M < 1 / 8 ), we confirm both the existence and uniqueness conditions required for the solution of the ODE.