Proving that one solution always lies above the other

Hi all,

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

I need to prove the following:

k is a constant.

y is strictly increasing, but not continuous.

Let .

denotes solution x.

.

for all .

for all .

Show:

for all .

Re: Proving that one solution always lies above the other

What does "solution x" mean? What is "x" here?

Re: Proving that one solution always lies above the other

x is just an indicator. A solution is a pair of functions y and H.

Ultimately, I need to prove it also for:

,

where k is a strictly increasing positive function.

Thanks a lot!

Re: Proving that one solution always lies above the other

The solutions of a differential equation obviously must be "differentiable" so must be continuous. So if for some and for some , there must exist between and such that .

But by the "existence and uniqueness" theorem, since there can be only **one** solution to this first order differential equation having a given value at .