Global attractivity of a higher order nonlinear difference equation with decreasing terms

Abstract

In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation x ( n + 1 ) = a x ( n ) + b f ( x ( n ) ) + c f ( x ( n − k ) ) , n = 0 , 1 , … where a , b and c are constants with 0 < a < 1 , 0 ≤ b < 1 , 0 ≤ c < 1 and a + b + c = 1 , f ∈ C [ [ 0 , ∞ ) , [ 0 , ∞ ) ] with f ( x ) > 0 for x > 0 , and k is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.