Is it possible to solve the difference equation $K_{n+1}=aK_n+bK_n^{\theta}+c$?

Question

Is it possible to solve the difference equation $K_{n+1}=aK_n+bK_n^{\theta}+c$, where a, b, c are real numbers while $\theta\in (0,1)$? How about $\theta=\frac{1}{2}$?

Answer 1

In the case $\theta = 2$ you have the logistic map, for which there is a closed form solution only in a few special cases. With general $\theta$, I would think it is even more unlikely that you would have closed form solutions.