Let $X$ by a connected metric space and $F: X\to X$ be a continuous map. Consider an autonomous difference equation $$ x_{n+1}=F(x_n),\ n=0,1,2,\dots. $$ {\it S. Elaydi} and {\it A.Yakubu} [J. Difference Equ. Appl. 8, No. 6, 537--549 (2002;

Zbl 1048.39002)] showed that if a $k$-cycle $c_k$ is globally asymptotically stable(GAS), then $c_k$ must be a fixed point. In this paper, the authors extend this result to periodic nonautonomous difference equation $$ x_{n+1}=F(n,x_n),\ n=0,1,2,\dots, \tag$*$ $$ via the concept of skew-product dynamical systems which comes from {\it G. R. Sell} [Topological Dynamics and ordinary differential equations (Van Nostrand- Reinhold, London) (1971;

Zbl 0212.29202)]. The authors show that for a $k$-periodic differential equation $(*)$, if a periodic orbit of period $r$ is GAS, then $r$ must be a divisor of $k$. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if $r$ divides $k$ they construct a non-autonomous dynamical system having minimum period $k$ and which has a GAS periodic orbit with minimum period $r$.
Then they apply these methods to prove a conjecture by {\it J. M. Cushing} and {\it S. M. Henson} [J. Differential Equations Appl. 8, No. 12, 1119--1120 (2002;

Zbl 1023.39013)] concerning a non-autonmous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates.