# Continuity of DE solution in the _density functions_?

1. Feb 1, 2009

### Anja

Hi there,
I'm an economics grad student and looking for a pointer to a theorem/paper that solves the problem below.

Here goes:

I have the system
$$\dot{B(i)} =- \int_0^J \alpha(i,t)(\pi(i,t)-B(i)-G(t))m(t) d t$$
$$\dot{G(j)} =- \int_0^I \alpha(t,j)(\pi(t,j)-B(t)-G(j)w(t) d t$$
with fixed boundary conditions on B(I) and G(J) (both strictly positive).

$$\pi$$ is twice cont. diff, m and w are from a uniformly equicontinuous set of functions on a closed interval, and $$\alpha$$ is the indicator function for the $$(\pi(i,j)-B(i)-G(j))$$ term being nonnegative. I can impose conditions on $$\pi$$ so that this expression is zero at only two points i or j for any given j or i (the functions are "strictly single-peaked" in each dimension).

I would need the solution (B*, G*) to this system to be continuous w.r.t. (m,w) (all with the sup norm). This is part of proving existence of an equilibrium (fixed point) for a larger system, in case you are wondering... if I forgot anything, please let me know. Any help is greatly appreciated. Honorable mention in my thesis/paper if successful!