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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 5

Question

Math 6410 • ODE Fall 1993 • Question 5

Assume the conditions of d) and let \(B(t)\) be a continuous matrix function with \[ \int _0^\infty \lVert A(t)-B(t)\rVert \,dt <+\infty . \] Let \(\Psi \) be a solution of \[ \Psi '=B(t)\Psi . \] Show that \(\Psi \) must be uniformly bounded on \([0,\infty )\).

Hint: \(\Psi \) also solves \[ \Psi '=A(t)\Psi +(B(t)-A(t))\Psi . \] Use variation of constants.

Raw LaTeX

Assume the conditions of d) and let \(B(t)\) be a continuous matrix
function with
\[
\int_0^\infty \lVert A(t)-B(t)\rVert\,dt <+\infty.
\]
Let \(\Psi\) be a solution of
\[
\Psi'=B(t)\Psi.
\]
Show that \(\Psi\) must be uniformly bounded on \([0,\infty)\).

\underline{Hint:} \(\Psi\) also solves
\[
\Psi'=A(t)\Psi+(B(t)-A(t))\Psi.
\]
Use variation of constants.

Topic Tags

Calculus of Variations Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions Variation of Constants Formula