We define the following Calculus of Variations problem :
$\max \int_0^1[X(t)E(t)-c(E(t))]dt$ subjetct to $X(0)=1000$ $X(1)=500$
and $X'(t)=-aX(t)E(t)$ (and $a$ is a constant). with the hypothesis that $c$ is a continuous and convex function of $E$.
I tried to write it as a Calculus of Variations problems :
$\max \int_0^1[-aX'(t)-c(-a\frac{X'(t)}{X(t)})]dt$ subjetct to $X(0)=1000$ $X(1)=500$
And I'm stuck on proving that this problem has a solution (existence).
(Note that we are looking for $E(t)$ )