Oscillation and non-oscillation of solutions of second order difference equations involving generalized difference

摘要

In this paper, sufficient conditions are obtained for oscillationon- oscillation of solutions of second order difference equations, involving generalized difference, of the form. Δa (p(n-1)~(Δa)y(n-1)) +q(n)y(n)=0,n≥1and~(Δa)(p(n-1) Δ~ay(n-1))+q(n)y(n)=f(n), n≥1,where Δ_a is defined by Δ_ay(n) = y(n + 1) - ay(n), a ≠ 0. Necessary and sufficient conditions are obtained for oscillation of the first equation with a > 0 and q(n) = q, a non-zero constant. The signs of q(n) and a play an important role for oscillationon-oscillation of these two equations.