Global attractivity of periodic solutions in a higher order difference equation

Abstract

Consider the following higher order difference equation with periodic coefficients: x n + 1 = a n x n + F ( n , x n − k ) , n = 0 , 1 , … , where { a n } is a periodic sequence in ( 0 , 1 ] with period p and a n ≢ 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.