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I was trying to solve a problem and end up with the following non-linear system $$\left\{\begin{array}{lll} a_{11}e^{x_1}+ \ldots+ a_{1n}e^{x_n}=f_1(x_1,\ldots,x_n)\\ \vdots\\ a_{n1}e^{x_1}+\ldots+a_{nn}e^{x_n}=f_n(x_1,\ldots,x_n) \end{array}\right.$$ where $f_i:\mathbb{R}^n \to \mathbb{R}$ is smooth, for all $i=1,\ldots,n$. I know that the real matrix $A:=(a_{ij})$ is invertible and I was wandering if this system has real solution. I am searching for methods of solving non-linear system, if somebody could give me some tip about this particular problem I would appreciate very much!

A.K.O
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    Insufficient data! Note that (assuming a real solution is sought) the change of variables $y_k = \log x_k$ would lead to a "quasi-linear" system formulated for positive real solutions. However more information about the $f_i$ and about matrix $A$ would be necessary to perform significant analysis for the existence of any solutions. – hardmath Jul 18 '20 at 20:26
  • $A$ is a real matrix and each $f_i$ is a function that has an exponential decay. – A.K.O Jul 18 '20 at 20:30
  • Please say more about this exponential decay. Also, how do you know $A$ is invertible? Is it symmetric positive definite? Is it diagonally dominant? – hardmath Jul 18 '20 at 21:20
  • For each $i=1,\ldots,n$ we have $|f_i| \leq \sum_{i=1}^n e^{\lambda x_i}$ for some $\lambda<0$. I know that $A$ is invertible because specifics of the problem context. I don't know if is symmetric or diagonal. – A.K.O Jul 18 '20 at 22:02
  • I guess that it is not symmetric ou diagonal. – A.K.O Jul 18 '20 at 22:02
  • The "specifics of the problem" will have to be exposed in order to provide any measure of help. Is it possible to present a model version of the problem in two unknowns? – hardmath Jul 19 '20 at 15:29
  • Does the system $Ae^X=0$ have solutions ? –  Jul 19 '20 at 17:16
  • I am working in a $n$-dimensional kernel of linear operator D that I am identifying with $\mathbb{R}^n$. If I assume that $A$ is not invertible I can produce an element of $\ker D$ that make a contradiction of how these elements look like. – A.K.O Jul 19 '20 at 21:15
  • For each $i=1,\ldots,n$, $f_i=c\langle e_i, g(x_1,\ldots,x_n)\rangle$, where $c$ is a constante, ${e_1,\ldots,e_n}$ is a base for $\ker D$ – A.K.O Jul 19 '20 at 21:23
  • and the only helpful information that I can summarize about $g$ is that $|g| \leq \sum_{i=1}^n e^{\lambda x_i}$ for some $\lambda <0$. – A.K.O Jul 19 '20 at 21:24
  • @YvesDaoust I do not understand your question, could you please explain to me a little bit more? – A.K.O Jul 19 '20 at 21:46

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Your system has the form

$$\left\{\begin{array}{lll} g_1(x_1,\ldots,x_n)=0\\ \vdots\\ g_n(x_1,\ldots,x_n)=0 \end{array}\right.$$ and is just a general nonlinear system, for which there is no general method. Unless your $f_n$ have something very special (smoothness tells little).

  • What kind of information about $f_n$ would help? – A.K.O Jul 18 '20 at 20:43
  • In my case, $f_n$ are small errors, and when each $x_n \to \infty$ we have $f_n \to 0$, this the only information that a have about them. – A.K.O Jul 18 '20 at 20:49
  • @A.K.O: small compared to what ? Are the $a_{ij}$ of both signs ? –  Jul 19 '20 at 17:10
  • Yes, the $a_{ij}$ can have both signs and how they are small I explain in the comments of the question above. – A.K.O Jul 19 '20 at 21:27