Oscillation of a class of difference equations of second order

In this paper, we investigate asymptotic behaviour of solutions of the following second-order difference equation: Δ [ a ( n ) Δ [ x ( n ) + ∑ i ∈ R r i ( n ) x ( n − ρ i ) ] ] + ∑ i ∈ P p i ( n ) x ( n − τ i ) − ∑ i ∈ Q q i ( n ) x ( n − σ i ) = f ( n ) , where n ∈ N 0 ≔ N ∪ { 0 } , R , P , Q are bounded starting segments of positive integers, { r i ( n ) } n ∈ N 0 and { f ( n ) } n ∈ N 0 are sequences of real numbers, { p j ( n ) } n ∈ N 0 and { q k ( n ) } n ∈ N 0 are non-negative sequences of real numbers, { a ( n ) } n ∈ N 0 is positive, and ρ i , τ j , σ k ≥ 0 are integers for all i ∈ R , j ∈ P , k ∈ Q . Various ranges of the sequence { ∑ i ∈ R r i ( n ) } n ∈ N 0 are considered. Also, illustrating examples are given to show the applicability of our results.