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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2021 • Question 4

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2021 • Question 4

Let \(X\) be a Banach space and consider the non-linear mapping \(F : X \to X\) defined by

\[F(x) = y - \alpha \sin (\|x\|)x,\]

where \(y \in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C > 0\) such that for any \(|\alpha | < C, F\) is a contraction on the open ball of radius 1 centered at \(y\). Note: You may use that for any \(a, b \in \mathbb {R}\) we have \(|\sin (a) - \sin (b)| \le |a - b|\).

Raw LaTeX

Let \(X\) be a Banach space and consider the non-linear mapping \(F : X \to X\) defined by

\[F(x) = y - \alpha \sin(\|x\|)x,\]


where \(y \in X\) is fixed and \(\alpha\) is a scalar. Show that there is a constant \(C > 0\) such that
for any \(|\alpha| < C, F\) is a contraction on the open ball of radius 1 centered at \(y\).
Note: You may use that for any \(a, b \in \mathbb{R}\) we have \(|\sin(a) - \sin(b)| \le |a - b|\).

Topic Tags

Banach and Normed Spaces Contraction Mapping and Nonlinear Equations