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Given, say $4$ non linear equations with $4$ positive parameters,

$f_1(x,y,z,t)=a,\quad f_2(x,y,z,t)=b,\quad f_3(x,y,z,t)=c,\quad f_4(x,y,z,t)=d$

for given $a,b,c,d$, If I am able to show that when the other $3$ variables are fixed, if $f_1$ is increasing with $x$ and $f_2$ is increasing with $y$ and $f_3$ is increasing with $z$ and $f_4$ is increasing with $t$ and all functions have at least a positive point.

Can I claim that this equation system has a unique solution for positive $x,y,z,t$?

Many Thanks.

1 Answers 1

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If I understand the problem correctly, this is a counterexample.

Take $f_1(x,y,z,t) = x-y+1$, $f_2(x,y,z,t) = y-z+1$, $f_3(x,y,z,t) = z-t+1$, $f_4(x,y,z,t) = t-x+1$.

Then with $x=y=z=t=1$, we have all functions equal to $1$, but this is also true for $x=y=z=t=2$.

Actually my example is affine, replace the terms $x-y$ for example by $x^2-y^2$ to make it 'more nonlinear'.

Additional elaboration:

Take $f_1(x,y,z,t) = x-y$, $f_2(x,y,z,t) = y-z$, $f_3(x,y,z,t) = z-t$, $f_4(x,y,z,t) = t-x$. $f_1$ is increasing in $x$, $f_2$ is increasing in $y$, $f_3$ is increasing in $z$ and $f_4$ in increasing in $t$.

Then for all $\lambda$ we have $f_1(\lambda,\lambda,\lambda,\lambda) = 0$, $f_2(\lambda,\lambda,\lambda,\lambda) = 0$, $f_3(\lambda,\lambda,\lambda,\lambda) = 0$ and $f_4(\lambda,\lambda,\lambda,\lambda) = 0$, so the solution is not unique by any means.

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    Good luck with the problem!2012-11-07