An application of fixed point theorem

Question

I want to use the fixed point method to solve the equation to find $y$: $$ y = c_1 y^3 - c_2 y$$where $c_1, c_2$ are real valued constants. So I designed $$ y_{k+1} = c_1 y_k^3 - c_2 y_k$$ to approximate $y$. But I don't know what to do next. Also I want to know the convergence for this equation.

Answer 1

Do you want to do this on a computer or by hand? Approximating things by hand usually makes little sense, so suppose by computer.

If so, then the thing to do is first to put some value of $y_0$ (probably close to $0$, or else $y_k \to \infty$ as $k \to \infty$, but also not precisely $0$, or else $y_k = 0$ for all $k$). Keep using the recursion to generate $y_k$, until you reach a good solution (a reasonable check is $y_{k+1} \simeq y_k$), or run out of patience ($k$ is very large, say $k \gg 1000$), or you see that the sequence diverges ($y_k$ is very large, say $y_k \gg 1000$). If you found an approximate solution, then the job is done. If not, then you probably should try again with a different choice of $y_0$ (it might be reasonable to put $y_0$ random).

Of course, it is usually much simpler to solve the equation by transforming it into the form: $$ y(c_1 y^2 - (c_2+1)) = 0$$ and then computing $y = 0$ or $y = \pm \sqrt{\frac{c_2+1}{c_1}}$. In some circumstances you explicitly don't want to use square roots, however. (E.g. you want to generalise the method afterwards, or you just want to learn how the fixed point method works, or you're programming and your language does not have the square root function...). I'm assuming its one of such situations.