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∈Rri(n)x(n−ρi)]]+∑i∈Ppi(n)x(n−τi)−∑i∈Qqi(n)x(n−σi)=f(n), where n∈N0≔N∪{0}n∈N0≔N∪{0}, R,P,QR,P,Q are bounded starting segments of positive integers, {ri(n)}n∈N0{ri(n)}n∈N0 and {f(n)}n∈N0{f(n)}n∈N0 are sequences of real numbers, {pj(n)}n∈N0{pj(n)}n∈N0 and {qk(n)}n∈N0{qk(n)}n∈N0 are non-negative sequences of real numbers, {a(n)}n∈N0{a(n)}n∈N0 is positive, and ρi,τj,σk≥0ρi,τj,σk≥0 are integers for all i∈R,j∈P,k∈Qi∈R,j∈P,k∈Q. Various ranges of the sequence {∑i∈Rri(n)}n∈N0{∑i∈Rri(n)}n∈N0 are considered. Also, illustrating examples are given to show the applicability of our results.