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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 3

Question

Math 6410 • ODE Fall 1996 • Question 3

Let \[ E_0=E\cap \{u:u(0)=u(\pi )=0\}, \] and restrict \(f\) to \(E_0\). Show that, except for a countable set of values of \(\lambda \) (describe this set !), the following holds: There exists \(\delta =\delta (\lambda )\) such that for all \(h\in X\), \(\norm {h}_X<\delta \), there exists a unique \(u\in E_0\) such that \[ f(u)=h, \tag {2.2} \] i.e. the problem \[ \begin {cases} u''+\lambda u+u^3=h,\\ u(0)=0=u(\pi ) \end {cases} \tag {2.3} \] has a unique solution.

Raw LaTeX

Let
\[
E_0=E\cap\{u:u(0)=u(\pi)=0\},
\]
and restrict \(f\) to \(E_0\). Show that, except for a countable set of values of
\(\lambda\) (describe this set !), the following holds: There exists
\(\delta=\delta(\lambda)\) such that for all \(h\in X\), \(\norm{h}_X<\delta\), there exists
a unique \(u\in E_0\) such that
\[
f(u)=h,
\tag{2.2}
\]
i.e. the problem
\[
\begin{cases}
u''+\lambda u+u^3=h,\\
u(0)=0=u(\pi)
\end{cases}
\tag{2.3}
\]
has a unique solution.

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Fundamental Solutions Poincare Maps and Periodic Solutions