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Find a solution of the equation $f(x) = x$ for $x = (x_1, x_2,x_3) \in \mathbb E^3$ (euclidean space) , where $f(x_1, x_2,x_3)= (4x_1+10x_2+5x_3-20, 4x_1+4x_2+3x_3+24, 15x_1+4x_2+15x_3+180)$

How to find a solution for the equation using Banach contraction mapping? the hint from book, it can be solve using matrix form to find the fixed point. Anyone can help me to solve this question ?

user273952
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2 Answers2

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Let $x = (1,1,1), y = (0,0,0)$, then we have that $d(x,y) = \sqrt{3} \approx 1.73$. Now, consider $f(x) = (-1,35,214), f(y) = (-20,24,180)$, then we have that $d(f(x),f(y)) \approx 40.4722$. Therefore, $f$ is NOT a contraction mapping. Thus, you should just solve the system of equations $$\begin{align} 4x_1+10x_2+5x_3 - 20 &= x_1\\ 4x_1+4x_2+3x_3 +24 &= x_2 \\ 15x_1+4x_2+15x_3 + 180 &= x_3\end{align}$$ directly. Note, this can be rewritten as $$\begin{align} 3x_1+10x_2+5x_3 &= 20\\ 4x_1+3x_2+3x_3 &= -24 \\ 15x_1+4x_2+14x_3 &= -180\end{align}$$ then we could find the associated (augmented) matrix's reduced row echelon form and the solution can be recovered from that.

EDIT

To beat a dead horse, the reduced form is $$\begin{bmatrix} 1 & 0 & 0 & -4.7273\\ 0 & 1 & 0 & 8.5333\\ 0 & 0 & 1 & -10.2303 \end{bmatrix}$$ hence the final column gives the solution.

David
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  • it looks like the same problem with this question http://math.stackexchange.com/questions/2208010/banach-contraction-mapping, but it uses fraction. they use banach contraction to solving the question. is it same with my question? – user273952 Apr 10 '17 at 04:46
  • Presumably not, I haven't verified it, but I'd assume their function is a contraction mapping. What I do know is that your function is not. – David Apr 10 '17 at 04:53
  • @user273952 I just checked. Their function is a contraction mapping. So no, the questions are fundamentally different. – David Apr 10 '17 at 04:55
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$f(x_1,x_2,x_3)=(x_1,x_2,x_3)$ provides three equations in three unknowns, so your usual techniques for solving simultaneous equations will work. The equations are linear, so there will be a single solution. Are you required to use Banach contraction? As presented, it doesn't look like a contraction to me. Some algebra could fix that, but it looks like overkill. Just solve the simultaneous equations.

Ross Millikan
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  • no, it must not use banach contraction. do you have another way to find the solution ? I only consider to find the solution using banach contraction – user273952 Apr 10 '17 at 04:21
  • The equation in my first line is a set of three simultaneous equations in there unknowns. These are unpacked in David's answer. Solve them. What is the problem? – Ross Millikan Apr 10 '17 at 04:31
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    @user273952 if you're in the middle of a course in topology, it stands to reason that you've taken a course in linear algebra? If so, Ross' answer appeals to the fact that solving this problem is tantamount to the most elementary problem (in linear algebra) that one can contrive. I've given more detail in my answer, but if you aren't able to see this immediately, I would recommend a review of the fundamentals of linear algebra. – David Apr 10 '17 at 04:38