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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 2

Question

Math 6420 • PDE Fall 2016 • Question 2

Let \(X\) be a Banach space and consider the non-linear mapping \(F:X\to X\) defined by \[ F(x)=y-\alpha \lVert x\rVert x, \] where \(y\in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C>0\) such that for any \(\lvert \alpha \rvert <C\), \(F\) is a contraction on the open ball of radius \(1\) centered at \(y\).

Raw LaTeX

Let \(X\) be a Banach space and consider the \textit{non-linear} mapping \(F:X\to X\) defined by
\[
  F(x)=y-\alpha\lVert x\rVert x,
\]
where \(y\in X\) is fixed and \(\alpha\) is a scalar. Show that there is a constant \(C>0\) such
that for any \(\lvert\alpha\rvert<C\), \(F\) is a contraction on the open ball of radius \(1\)
centered at \(y\).

Topic Tags

Banach and Normed Spaces Contraction Mapping and Nonlinear Equations Linear Programming and Optimization