Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 5.7

Dependence on Initial Conditions and Parameters

Mathematics — Calculus · implicit-function-theorem-integral-equation

Problem

Let

A

be an

n \times n

matrix over

\mathbb{R}

all of whose eigenvalues satisfy

\mathrm{Re}\, \lambda < 0

. Suppose that

f \in C^1(\mathbb{R}^n, \mathbb{R}^n)

with

f(0) = 0

and

Df(0) = 0

. Use the Implicit Function Theorem 5.7 to prove that there exists an

r > 0

such that the integral equation

y(t) = \exp At\, y_0 + \int_0^t \exp A(t-s)\, f(y(s))\, ds

has a solution

y \in C^0_b([0,\infty), \mathbb{R}^n)

, for all

y_0 \in \mathbb{R}^n

with

\|y_0\| < r

. (Compare with Theorem 3.11.)