I have the following nonlinear system $$\begin{cases} y_1 = \frac{x_1}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_2 = \frac{x_2}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_3 = \frac{x_3}{\sqrt{x_1^2+x_2^2+x_3^2}} \end{cases} $$
How can I check whether it has solution for $x_1,x_2,x_3$ and whether it is unique? How do I solve it?
What if I know that $x_1=1$ and I need to solve it for $x_2,x_3$.
This question is related to: Injectivity of transformation
At first, you should have $$ y_1^2 + y_2^2 + y_3^2 = 1. $$ Let $$ x_1 = R\sin\theta\cos\phi,\\ x_2 = R\sin\theta\sin\phi,\\ x_3 = R\cos\theta $$ (you always can find such $R$, $\theta$ and $\phi$; check this if you want). So, $$ y_1 = \sin\theta\cos\phi,\\ y_2 = \sin\theta\sin\phi,\\ y_3 = \cos\theta $$ From last you can find $\theta$. And $$ \frac{y_2}{y_1} = \tan\phi $$ Now you can determine $\phi$ and $\theta$ (with some annoying manipulations with signs). You cannot determine $R$ (because $y$'s doesn't change if you multiply all $x$'s by something).