Intrepreting the correction equations $x_{n+1}=x_n+\lambda f(x_n)$ and $x_{n+1}=x_n+\lambda g(x_n)f(x_n)$ in the Variational Iteration Method

Question

I am studying the Variational Iteration Method. There are some concepts related to this method that I don't understand very well. One of them is the correction function.
For example, if we want to find the root of the equation $f(x)=0$, we construct an equation $$x_{n+1}=x_n+\lambda f(x_n),$$ where $\lambda$ is a general Lagrange multiplier. They call this the correction equation and $x_n$ is an approximate root of $f(x)=0$.
I understand that this can be used to get more accurate roots by iteration but I don't understand how they came up with this equation. It has also been mentioned that we can use another type of correction function to find the root of $f(x)=0$ and it is $$x_{n+1}=x_n+\lambda g(x_n)f(x_n),$$ where $g(x_n)$ is an auxiliary function. Can anyone explain to me what these equations mean? Thank you!

Answer 1

They just mean that you are rewriting the original equation, $f(x)=0$, as $$ x = x +\lambda f(x) $$

or

$$ x = x + \lambda g(x) f(x). $$

The roots of $f$ are the fixed points of $g_1(x)=x+\lambda f(x)$ or $g_2(x)=x+\lambda g(x) f(x)$. This is convenient because it allows us to use the Fixed Point Theorem to establish existence and uniqueness of solution, as well as an approximation scheme.