Nonlinear discrete time systems

Question

Is it possible for discrete-time parameter $a(k)$ with an update rule like $a(k+1)=f(a(k))$ & always $|f(a(k))|<= c|a(k)|$ where $0<c<0.5$ to converge from the initial value $c_1$ to final value $c_2$, where $c_1>c_2>1$? (I mean: is it possible for such a discrete parameter $a(k)$ with mentioned condition not to converge to zero?If yes,I need an example of such update rule.