Solving a system of equations

Question

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 $\gamma \in (0,1)$.