Hi all,

I'd be very happy if you could help me solve a problem in my research.

I need to prove the following:

$\displaystyle H'(r) = -y(r) - k H(r) $

k is a constant.

y is strictly increasing, but not continuous.

Let $\displaystyle (a,b]\subset R $.

$\displaystyle (H_x, y_x) $ denotes solution x.

$\displaystyle H_1(a)<H_0(a)<0 $.

$\displaystyle H_0(s)<0, H_1(s)<0 $ for all $\displaystyle s\in(a,b] $.

$\displaystyle y_1(s)>y_0(s) $ for all $\displaystyle s \in (a,b] $.

Show:

$\displaystyle H_1(r)<H_0(r) $ for all $\displaystyle r \in (a,b] $.