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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.

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closed as unclear what you're asking by Igor Rivin, John Pardon, Andrey Rekalo, Michael Renardy, Chris Godsil Aug 1 '13 at 12:09

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I added my question. $\endgroup$ – Alan Aug 1 '13 at 15:38
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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.

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