On the Dynamics of a Higher-Order Difference Equation

Abstract

This paper deals with the investigation of the following more general rational difference equation: 𝑦𝑛+1=𝛼𝑦𝑛∑/(𝛽+𝛾𝑘𝑖=0𝑦𝑝𝑛−(2𝑖+1)∏𝑘𝑖=0𝑦𝑛−(2𝑖+1)), 𝑛=0,1,2,…, where 𝛼,𝛽,𝛾,𝑝∈(0,∞) with the initial conditions 𝑥0,𝑥−1,…,𝑥−2𝑘,𝑥−2𝑘−1∈(0,∞). We investigate the existence of the equilibrium points of the considered equation and then study their local and global stability. Also, some results related to the oscillation and the permanence of the considered equation have been presented.