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I'm trying to prove the existence of a solution to the system of equations

$$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$

for $i\in\{1,2,....\}$ where $\sum c_i=1$.

I am also trying to prove that the implicit function $\Phi(\{c_i\})$ that solves for the $x_i$ is continuous.

Edit: I forgot to mention that $\gamma \in (0,1)$.

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What do we know about $x_j$? Only that it is a sequence of real numbers such that $\sum_i^\infty x_i < \infty$? –  Eric Auld Jul 20 '13 at 21:53
    
It a flow equation. $c_i$ is the entering population and $x_i$ is the steady state value. So $x_i$ should be positive. Also, since $\gamma\in(0,1)$ and $x_i>0$, $x_i<\frac{c_i}{\gamma}$ –  user68479 Jul 20 '13 at 22:17
    
How do we know $\sum_i^\infty x_i < \infty$? –  Eric Auld Jul 20 '13 at 22:28
    
Because $\sum x_i < \sum \frac{1}{\gamma} c_i = \frac{1}{\gamma} \sum c_i = \frac{1}{\gamma}$ –  user68479 Jul 20 '13 at 22:45

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