Positivity of a Simple Nonlinear Recursive Equation

by Fianra   Last Updated August 03, 2018 22:20 PM

For $k=0,1,...$, take $|w_k| \leq 1$ and $\alpha \in [0,1]$. Now consider an recursive equation given by $$ x_{k+1} = x_k + \alpha (1+w_{k-1})w_k x_{k-1} $$ with initial conditons $x_0 = x_1 = 1 >0$ and $x_i = 0$ for all $i < 0.$ I wonder is it possible to find a condition (with a proof) which assures that $x_k>0$ for all $k>0.$ Or is there any related literature I could consult with? It seems to me the question is closely related to the concept called invariant set (i.e., a set $S$ is invariant if, use the definiton of the recrusion above, $x_k \in C \Rightarrow x_{j} \in C\;\; \forall j > k$).

Indeed, I ran many simulations and form a belief that when $\alpha \leq 1/2$ might work, but keep failing to prove it. So I would very appreciate for having any suggestion or comment.

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