Global attractivity of periodic solutions in a higher order difference equation

Consider the following higher order difference equation with periodic coefficients: xn+1=anxn+F(n,xn−k),n=0,1,…, where {an} is a periodic sequence in (0,1] with period p and an≢1, F(n,x):{0,1,…}×[0,∞)→(0,∞) is a continuous function in x and a periodic function in n with period p, and k is a nonnegative integer. We obtain a sufficient condition such that every positive solution of the equation converges to a positive periodic solution. Applications to some difference equations derived from mathematical biology are also given.