Asymptotic behavior of solutions of certain higher order nonlinear difference equation

In this paper, we consider the higher order difference equation y ( k + n ) + p 1 ( k ) y ( k + n - 1 ) + p 2 ( k ) y ( k + n - 2 ) + ⋯ + p n ( k ) y ( k ) (1) = f k , y ( k ) , y ( k + 1 ) , … , y ( k + n - 1 ) , ∑ s = k 0 k - 1 g ( k , s , y ( s ) , … , y ( s + n - 1 ) ) , k ∈ N ( k 0 ) = { k 0 , k 0 + 1 , k 0 + 2 , … } , k 0 ∈ { 1 , 2 , … } . With the aid of a discrete inequality, we obtain some sufficient conditions which guarantee that for every solution y ( k ) of (1) satisfies the equation as k → ∞ , y ( k ) = ∑ i = 1 n ( δ i + o ( 1 ) ) z i ( k ) , where δ i , i = 1 , 2 , … , n are constants, { z i ( k ) } i = 1 n are any independent solutions of the equation z ( k + n ) + p 1 ( k ) z ( k + n - 1 ) + p 2 ( k ) z ( k + n - 2 ) + ⋯ + p n ( k ) z ( k ) = 0 , k ∈ N ( k 0 ) .