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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 4

Question

Math 6410 • ODE Fall 1993 • Question 4

Assume that \(\overline {X}(t)\) is a matrix solution of (2) with \(\overline {X}(0)\) nonsingular and \(\overline {X}(t)\) uniformly bounded on \([0,\infty )\). Also assume \[ \liminf _{t\to \infty }\operatorname {Re}\int _{0}^{t}\operatorname {trace} A(s)\,ds >-\infty . \] Prove that \(\overline {X}^{-1}(t)\) is uniformly bounded on \([0,\infty )\) and no solution \(\varphi (t)\) of (2), not identically zero, can satisfy \(\varphi (t)\longrightarrow 0\) as \(t\longrightarrow \infty \).

Raw LaTeX

Assume that \(\overline{X}(t)\) is a matrix solution of (2) with
\(\overline{X}(0)\) nonsingular and \(\overline{X}(t)\) uniformly bounded on
\([0,\infty)\). Also assume
\[
\liminf_{t\to\infty}\operatorname{Re}\int_{0}^{t}\operatorname{trace} A(s)\,ds
>-\infty.
\]
Prove that \(\overline{X}^{-1}(t)\) is uniformly bounded on \([0,\infty)\) and no
solution \(\varphi(t)\) of (2), not identically zero, can satisfy
\(\varphi(t)\longrightarrow0\) as \(t\longrightarrow\infty\).

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Fundamental Solutions Poincare Maps and Periodic Solutions