I'm studying a variant of the discrete Nerlove Arrow model \begin{equation} x_{t} = \lambda x_{t-1} + a_t \end{equation} which can be easily shown to be decomposed as \begin{equation} x_{t} = \sum_{k=0}^\infty \lambda^k a_{t-k}. \end{equation} Here, $x_t\ge0$ can be regarded as goodwill, $a_t\in\{0,1\}$ advertising decisions, parameter $0<\lambda<1$ to model the decay or goodwill for each time period $t$.
I wonder if there is any similar or general solution to the following extension \begin{equation} x_{t} = \lambda_1 x_{t-1} + \lambda_2 x_{t-1}^2 + a_t. \end{equation} In the latter case, we have everything defined as before but this time $0<\lambda_1<1$ and $\lambda_2\in\mathbb{R}$.
I skimmed through some material online and thought that the $z$ transform could a be a possible approach to solve this type of difference equations but have yet to find material that directly answers my question without going into depth on $z$ transforms.
A closed form solution for such a non-linear recurrence is higly unlikely. The $z$ transform (or generating function approaches) work only for linear recurrences.
You can derive an asymptotic approximation by noting that if you disregard the linear terms, you get $y_{t + 1} = \lambda_2 y_t^2$, this hints at a solution roughly like $y_t = \lambda_2^{2^t - 1} y_0^{2^{t - 1}}$ if $\lambda_2 > 1$ and $y_0 \ge 1$. Guess your solution is $x_t = z_t y_t$, where $z_t$ is a more behaved correction factor, and work on that.