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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2022 • Question 1

Question

Math 6410 • ODE Fall 2022 • Question 1

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\) and \(L\in \mathbb {R}\) be such that

\[ |f(t,x)-f(t,y)|\le L|x-y| \quad \text {for all } x,y\in \mathbb {R}^n \text { and all } t\in \mathbb {R}. \]

Show that there is \(T>0\) such that the Picard iterates converge to a continuously differentiable solution of the initial value problem

\[ \dot {x}=f(t,x) \]
\[ x(0)=x_0 \]

on \([0,T]\). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

Raw LaTeX

Let \(f:[0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d\) be a continuous
function and \(x_0\in\mathbb{R}^d\) and \(L\in\mathbb{R}\) be such that

\[
|f(t,x)-f(t,y)|\le L|x-y| \quad\text{for all } x,y\in\mathbb{R}^n
\text{ and all } t\in\mathbb{R}.
\]

Show that there is \(T>0\) such that the Picard iterates converge to a
continuously differentiable solution of the initial value problem

\[
\dot{x}=f(t,x)
\]

\[
x(0)=x_0
\]


on \([0,T]\). [Don't just quote a result. State the theorem and give as complete and detailed
a proof as you can.]

Topic Tags

Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions