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

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?

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I think you mean DSolve, not RSolve. –  Heike May 20 '12 at 19:04
Yes, RSolve is for recursive relations. –  Sjoerd C. de Vries May 20 '12 at 19:06
@Heike: Thanks, could you elaborate? Why is this a problem for DSolve? I am not solving for the antiderivative of g, nor do I use a derivative of it. (Though admittedly, the problem is a first-order condition of a calculus of variation problem involving G, the antiderivative.) What am I missing? –  László May 20 '12 at 19:07
This looks more like a mathematical problem, than a Mathematica problem. –  Sjoerd C. de Vries May 20 '12 at 20:19
I'm voting to close this as "Not a Real Question". You should sit down and think of your problem, reduce it to a simple case that captures the essence accurately, try out for yourself in Mathematica, and only then post your question here, and phrase it in a way that the problem and the desired result is clear. Don't take it personally — just try to collect your thoughts first and read the documentation before you try anything out (if you had done this, you would not have used RSolve when you needed DSolve). People here are willing to help, but no one likes to run in circles... –  rm -rf May 20 '12 at 20:19

## closed as off topic by rm -rf♦, Sjoerd C. de VriesMay 20 '12 at 20:20

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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 $g$.