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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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    thanks for your post. Perhaps I couldnt explain very well. In your example we have such equations for example $(x-1)^2=1$, $(y-1)^2=1$, $(z-1)^2=1$, $(t-1)^2=1$. For example for the first it is first decreasing $x\in[0,1]$ and then it is increasing $x>1$. I must know that for positive $x$ it is only increasing and lets assume that the beginning point of increase is $0$, namely $f1(x=0,...)=0$.2012-11-07
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    @SeyhmusGüngören: I modified my answer slightly so it is increasing for positive $x$, but in any case the affine example will satisfy this condition. Subtract 1 from all functions above to satisfy $f_k(...) = 0$. Perhaps I am missing your point?2012-11-07
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    Ok. If the function is starting from $0$ and increasing, then it can pass through a positive number only once. Therefore there will be a unique solutions for any of the $f$ functions by the chosen parameters such as $x$ for $f_1$ and $y$ for $f_2$ etc.. Now my question is; does this guarrantee that the whole system has a unique solution for some $x,y,z,t$ or is it still possible that there exist other $x,y,z,t$ that can be another solution.2012-11-07
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    Hi @SeyhmusGüngören, I think I am missing your point. I added some elaboration above, which seems to meet all the conditions in the question and comments?2012-11-07
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    I think there is a mismatch in the understanding)) ok so for the first one $f_1(x,y,z,t)=x-\lambda$ and $f_1$ is increasing in $x$. Then $x=\lambda$ is the unique solution for $f_1$.2012-11-07
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    No, $x=\lambda$ is the unique solution to the equation $f_1(x,\lambda,\lambda,\lambda) = x-\lambda = 0$, but that is not what you are asking in the question.2012-11-07
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    I think I am asking? beucause I said when $3$ of them are fixed. And here $f_1(x,\lambda,\lambda,\lambda)$ three of them are fixed and equal to $\lambda$ and x is variable and increasing and we are looking for a solution in this case.2012-11-07
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    Are you trying to solve a system of 4 equations in 4 unknowns or not? If you are just trying to solve one equation that is strictly monotonic in one parameter, then of course there is a unique solution. Perhaps you should elaborate your question a little?2012-11-07
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    yes. right. I am trying to solve $4$ equations in $4$ unknowns. I also know that each of them is strictly increasing in one parameter given the others. This is true for all functions. However I dont know if this proves that the whole system of equations have a unique solution. Is it better now?2012-11-07
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    I don't know what to say. I have given you an example that satisfies all your requirements and clearly does not have a unique solution. So you can not prove uniqueness unless there is some other condition imposed.2012-11-07
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    Ok. you are right. we can choose different $x,y,z,t$ and we have still a solution to the system. If I have such non linear equations and If I want to show that the whole system has a unique solution, what must I do?2012-11-07
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    That is a difficult problem, in general. The inverse function theorem can be used to show 'local' uniqueness. Sometimes you can transform the equations so the solution is represented as a fixed point and use the contraction mapping theorem. However, I don't think there are satisfying conditions that guarantee uniqueness, in general.2012-11-07
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    Ok inverse function theorem and contraction mapping. I will check both. I guess in case I am able to reduce four equations to a single one with single parameter, then I can show else I dont have a generic method to apply. Thank you very much again.2012-11-07
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    Good luck with the problem!2012-11-07