Proving convergence of an integral-differential equation [closed]

Question

I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating.

My question is where can I find methods of showing convergence of this nonlinear differntial-integral eqaution?

I mean I got to a nonlinear relation between the function and its derivative, and obviously I can iterate y into the derivative, and then I want to show that this iteration method converges or not, do you have good refernces on this issue?

Thanks in advance, Alan.

OK, you ask me to clarify, then I will.

I am looking at the next ODE:

$$ y'' + (y')^2 + y^3 =0$$

I didn't yet imposed initial conditions.

Now I multiplied this ODE by $y'$ and integrated it to get:

$$1/2 (y')^2 + 1/4 y^4 + \int (y')^3 dx = 0$$

So I get the next equation:

$$ y = (-2(y')^2 -4 \int (y')^3 dx)^{1/4} $$

which I want to see if I can find if the iteration solution can solve this ODE.

Answer 1

I think the most likely way to prove convergence of iterates $T^j(f)$ with a nonlinear operator $T$ is to show that $T$ is a contraction mapping on a suitable complete metric space. This is what happens with the classical Picard method, which you can find in any good rigourous differential equations text.