On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system $y'=A(t)y$ where $A(t)$= \begin{matrix} -1 -9 \cos^2 6t + 12 \sin 6t \cos 6t & 12 \cos^2 6t + 9 \sin 6t \cos 6t \\ -12 \sin^2 6t + 9 \sin 6t \cos 6t & -1 - 9 \sin^2 6t - 12 \sin 6t \cos 6t \end{matrix}

Although the eigenvalues of $A(t)$ are $\lambda_1=-1$ and $\lambda_2=-10$ (constants), the zero solution is unstable.

On page 159 they present the following statement which I would appreciate if someone can address me a proof.

"It can be shown, however, that the strict negativity of all the eigenvalues of $A(t)$ for $t\geq T >0$ plus, for example, the conditions (i) the eigenvalues of $A(\infty)=\lim_{t\to +\infty}^{ } A(t)$ all have real parts negative and (ii) the elements of $A(t)$ are continuous and have only a finite number of maxima and minima on the interval $T\leq t < \infty$, lead to the asymptotic stability of the zero solution of $y'=A(t) y$."

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a $2 \times 2$ counterexample by Vinograd to the system $y'=A(t)y$, where $$ A(t) = \left( \begin{matrix} -1 -9 \cos^2 6t + 12 \sin 6t \cos 6t & 12 \cos^2 6t + 9 \sin 6t \cos 6t \\ -12 \sin^2 6t + 9 \sin 6t \cos 6t & -1 - 9 \sin^2 6t - 12 \sin 6t \cos 6t \end{matrix} \right) $$ Although the eigenvalues of $A(t)$ are $\lambda_1=-1$ and $\lambda_2=-10$ (constants), the zero solution is unstable.

On page 159 they present the following statement which I would appreciate if someone can address me a proof.

"It can be shown, however, that the strict negativity of all the eigenvalues of $A(t)$ for $t\geq T >0$ plus, for example, the conditions (i) the eigenvalues of $A(\infty)=\lim_{t\to +\infty}^{ } A(t)$ all have real parts negative and (ii) the elements of $A(t)$ are continuous and have only a finite number of maxima and minima on the interval $T\leq t < \infty$, lead to the asymptotic stability of the zero solution of $y'=A(t) y$."

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system y'=A(t)y where A(t)= \begin{matrix} -1 -9 cos^2 6t + 12 sin 6t cos 6t & 12 cos^2 6t + 9 sin 6t cos 6t \\ -12 sin^2 6t + 9 sin 6t cos 6t & -1 - 9 sin^2 6t - 12 sin 6t cos 6t \end{matrix}

Although the eigenvalues of $A(t)$ are $\lambda_1=-1$ and $\lambda_2=-10$ (constants), the zero solution is unstable.

On page 159 they present the following statement which I would appreciate if someone can address me a proof.

"It can be shown, however, that the strict negativity of all the eigenvalues of $A(t)$ for $t\geq T >0$ plus, for example, the conditions (i) the eigenvalues of $A(\infty)=\lim_{t\to +\infty}^{ } A(t)$ all have real parts negative and (ii) the elements of $A(t)$ are continuous and have only a finite number of maxima and minima on the interval $T\leq t < \infty$, lead to the asymptotic stability of the zero solution of $y'=A(t) y$."

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system $y'=A(t)y$ where $A(t)$= \begin{matrix} -1 -9 \cos^2 6t + 12 \sin 6t \cos 6t & 12 \cos^2 6t + 9 \sin 6t \cos 6t \\ -12 \sin^2 6t + 9 \sin 6t \cos 6t & -1 - 9 \sin^2 6t - 12 \sin 6t \cos 6t \end{matrix}

Although the eigenvalues of $A(t)$ are $\lambda_1=-1$ and $\lambda_2=-10$ (constants), the zero solution is unstable.

On page 159 they present the following statement which I would appreciate if someone can address me a proof.

"It can be shown, however, that the strict negativity of all the eigenvalues of $A(t)$ for $t\geq T >0$ plus, for example, the conditions (i) the eigenvalues of $A(\infty)=\lim_{t\to +\infty}^{ } A(t)$ all have real parts negative and (ii) the elements of $A(t)$ are continuous and have only a finite number of maxima and minima on the interval $T\leq t < \infty$, lead to the asymptotic stability of the zero solution of $y'=A(t) y$."