Global asymptotic stability and oscillation of a family of difference equations

Abstract

We consider the family of difference equations of the form x n+1 = ∑ i=0 i≠j,j−1 k x n−i +x n−j+1 x n−j +1 ∑ i=0 k x n−i , j=1,2,…,k, where n ∈{0,1,…}, k ∈{1,2,…} and the initial values x − k , x − k +1 ,…, x 0 are positive real numbers. For these difference equations, we investigate the oscillatory behavior of the positive solutions and prove that the unique equilibrium x ̄ =1 is globally asymptotically stable.