how to solve an implicit integral equation? (iterate to a functional fixed point?) [closed]

Question

I reduced a (special case) of my problem to the following code. Even though in this special case all related functions are analytical, DSolve is not the tool for this, though I am indeed looking for a continuous function as a solution.

I am OK with looping over guesses for the function, though as a newbie, I can only give you some pseudocode. Your help would be greatly appreciated.

1. Guess a g1, say a constant function.
2. Take a list of points, and solve the given equation for g2[z] for each z in the list, with g2[z] being the value between z and next value in the list, while g1[z] above.
3. Interpolate over the g2[z]-s to get a new g1, and iterate until g2[z] is close to g1[z] for each z.

Or is this a stupid algorithm?

H = ParetoDistribution[1.18709*10^6, 0.938482]
Hstar = H
k = 10/3
T[z_] = 2/3*z
DSolve[T'[z]/(1 - T'[z]) == 
  k/(z PDF[Hstar, z]) Integrate[(1 - g[zz]) Exp[zz - z] PDF[H, 
      zz], {zz, z, \[Infinity]}], g, z]

Or if I shall proceed another way, what pattern should I exploit? What pattern could Mathematica exploit? And in what construct?

Answer 1

I think what you're trying to do is to solve $f(z)=h(z)\int _z^\infty dz'\,\left(1-g(z')\right)\rho(z')$ for $g$, with $f$, $h$ and $\rho$ known functions. This is an integral equation.

However, dividing both sides by $h(z)$ (I'm assuming $h\neq0$) and differentiating with respect to $z$ we get $f'(z)/h(z)-f(z)h'(z)/h(z)^2=\rho(z)(g(z)-1)$, which is an algebraic equation for $g$.

So no computation needed.