Let $ B,y,z \in \mathbb R$. Is it possible to solve the equation \begin{align} x = -(e^{Bx}z-y)Be^{Bx} z \end{align} for $x \in \mathbb R$? Or can one show that a unique solution $x \in \mathbb R$ exists?
My claim for uniqueness stems from observing several Matlab plots of the right-hand-side, for different cases of $B,y,z$. However, it seems difficult for me to prove it.
(More generally, I consider this equation for a matrix $B$ and vectors $z,y$.)