Can anybody help me to find a system with 3 equations and 3 unknowns and a bounded domain D = [a,b]x[c,d]x[e,f] such that the system has an unique solution in D? Also, i need nice equations, because I have to make some calculations by hand.

Well, I know that if we write $f_1(x_1,x_2,x_3) = 0$, $f_2(x_1,x_2,x_3) = 0$, $f_3(x_1,x_2,x_3) = 0$ the system, or, even more compact: $F(x_1,x_2,x_3)=0$, where $F=(f_1,f_2,f_3)$, then $Im F$ should be included in $D$, should be continuous, and the partial derivatives of $f_1,f_2,f_3$ should be continuous and should satisfy the following property: $\exists k < 1$ such that $\left| \frac{\partial f_i(x)}{\partial x_j} \right| \le \frac{k}{3}, \forall i,j \in 1,2,3$.

A system using sin and cos is easy to obtain, but, the calculations are too complicated (in fact, impossible, by hand).

Thank you!


1 Answer 1


How about:

$$ \eqalign{x^2 + y &= 0\cr y^2 + z &= 0\cr x^2 - y + z &= 1\cr}$$ with $D = \{(x,y,z): x > 0\}$?

  • $\begingroup$ I forgot to mention that D is an bounded domain! :( Something like: D=[a,b]x[c,d]x[e,f] ) $\endgroup$
    – npatrat
    Commented Apr 26, 2016 at 21:57
  • $\begingroup$ OK, so choose some bounded subset of this domain containing the solution. $\endgroup$ Commented Apr 26, 2016 at 22:48

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