Is there a way to express $y_t$ in terms of $\delta$ and $p_t$ in the following system?
$$ \begin{cases} & \sum\limits_{t=0}^{\infty}\delta^t\cfrac{p_t}{p_t+(1-p_t)y_t}=K \\ & p_{t}=\cfrac{p_{t-1}}{p_{t-1}+(1-p_{t-1})y_{t-1}} \\ & p_0=\hat{p} \end{cases} $$ with $K>0$, $0<p<1$, $0<\delta<1$, $0\leq y_t\leq 1$, and $\hat{p}>0$ given.