I have a process $\{X_t\}_t$, where $x_t$ in $[0,1]$, and a Lyapunov function $V(x)=x$ such that the drift $$E[X_{t+1}|x_t] - x_t < -k(1-x_t)$$
for $x_t \in (1-\epsilon,1]$, where $k>0$. Thus the drift is negative on $(1-\epsilon,1]$ though not uniformly upper bounded. Is there any equivalent Foster-Lyapunov theorem that ensures stability of the process on $[0,1-\epsilon]$?
I have more structure in the process such that if $x_t = \alpha$, then $x_{t+1}\in \left((1-\gamma) \alpha, (\gamma+(1-\gamma)\alpha\right)$, where $\gamma<1$. Thus the process can not directly jump to 1, but can only do that in the limit (1 is the absorbing point of the process).