TITLE: Banach contraction mapping
QUESTION [2 upvotes]: In my note, i'm studying about metric space. In this section I also study about Banach contraction mapping theorem. Can i use Banach contraction mapping to find a solution for question below ? anyone can help me. Thank you

How to find a solution of the equation $f(x) = x$ for $x = (x_1, x_2,x_3) \in \mathbb E^3$ (euclidean space) , where $f(x_1, x_2,x_3)=(\frac {x_1}{5} +\frac {x_2}{2}+\frac {x_3}{4}-1,\frac {x_1}{3} +\frac {x_2}{3}+\frac {x_3}{4}+2, \frac {x_1}{4} +\frac {x_2}{15}+\frac {x_3}{4}+3) $

REPLY [1 votes]: Yes you can.  You should review the contraction mapping theorem.  It says that if you are trying to solve $f(x)=x$ and $|f'(x)| \lt q \lt 1$ in the region of interest you can just iterate $x_{i+1}=f(x_i)$ to find a solution.  The derivative condition says that the distance from the solution is multiplied by a factor of (about) $q$ each time and since $q \lt 1, q^n \to 0$.  You need to verify that the eigenvalues of the matrix $\frac {\partial f}{\partial x_i}$ are all less than $1$.  The maximum eigenvalue becomes your $q$ in the theorem.
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