Analytical fixed point iteration method I have a system of nonlinear ordinary diffferential equations and i want to use analytical fixed point iteration method. Unfortunately i could only see computational resources but i couldnt find anything on this method. Can somebody please explain me how one obtains analytical solution to nonlinear ODE's using fixed point analytical method.

I imagine that the  iteration gives an analytic solution of nonlinear ODE only in very exceptional cases. Here is one, extremely simple example: 
$$y'=y^2, \quad y(0)=1$$
The integral form is 
$$y_{n+1}=1+\int_0^x y_n^2(t)\,dt$$ 
Starting with $y_0=1$, we get $y_1=1+x$, $y_2=1+x+x^2+x^3/3$, and so forth. All are polynomials. 
Claim: for $0\le k\le n$, the coefficient of $x^k $ in $y_n $ is equal to $1$.
Proof: by indiction on $n$. By assumption, 
$$y_{n-1}=1+x+\dots +x^{n-1} + \text{(powers above $n-1$)}$$ 
Hence, the coefficient of $x^{k-1}$ in $y_{n-1}^2$ is the same as its coefficient in 
$$(1+x+\dots+x^{k-1} )((1+x+\dots+x^{k-1})$$
which is $k$. The integration of $ kx^{k-1}$ produces $x^k$, proving the claim. 
From the above, one can conclude that the iterations (which are already known to converge) converge to $\sum_{k=0}^\infty x^k$, which is $\dfrac{1}{1-x}$, the solution of the ODE.
