Solving the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$

Question

For $i=1,\dots,M$ vectors $x_i\in\mathbb{R}^N$ and scalars $\alpha_i$, can you find a vector $z$ satisfying the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$?

Any pointers will also be appreciated:

• does such an equation have a name?
• is it discussed systematically in some field of math or physics?
• how many such solutions can you expect to find?

Thanks!

Answer 1

Obviously, $$\lim_{\|z\|\to\infty}\sum_i \alpha_i \exp(-||x_i-z||^2)x_i=0$$ (zero vector) and for $R$ large enough $$f:B(0,R)\longrightarrow B(0,R).$$ By Brouwer, your equation will have always some solution (a fixed point of $f$).