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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 6

Question

Math 6410 • ODE Fall 1994 • Question 6

Prove:

Let \(f\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(f\) is not identically zero and either nonnegative in \(E\) or non-positive in \(E\). Then \(x'=f(x)\) has no closed orbit which is entirely in \(E\).

Raw LaTeX

Prove:

Let \(f\) be of class \(C^1\) in a simply connected region \(E\) in \(\R^2\). Assume the
divergence of \(f\) is not identically zero and either nonnegative in \(E\) or
non-positive in \(E\). Then \(x'=f(x)\) has no closed orbit which is entirely in \(E\).

Topic Tags

Self-Adjoint and Positive Operators