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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 4

Question

Math 6410 • ODE Fall 1996 • Question 4

Define the mapping \[ y\in \mathbb {R}^N\mapsto F_t(y)\in \mathbb {R}^N \tag {2.5} \] by \[ F_t(y)=x(t,y). \tag {2.6} \] where \(x(t,y)\) solves \[ \begin {aligned} x'&=f(t,x),\\ x(0,y)&=y. \end {aligned} \tag {2.7} \] Give reasons that \(F_t\) is well-defined for each small \(t\) and well-defined for all \(t\in \mathbb {R}\), whenever \(\lvert y\rvert \le R\).

Raw LaTeX

Define the mapping
\[
y\in\R^N\mapsto F_t(y)\in\R^N
\tag{2.5}
\]
by
\[
F_t(y)=x(t,y).
\tag{2.6}
\]
where \(x(t,y)\) solves
\[
\begin{aligned}
x'&=f(t,x),\\
x(0,y)&=y.
\end{aligned}
\tag{2.7}
\]
Give reasons that \(F_t\) is well-defined for each small \(t\) and well-defined for all
\(t\in\R\), whenever \(\lvert y\rvert\le R\).

Topic Tags

Topological Degree and Fixed Point Methods