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Oscillation and nonoscillation of solutions of second-order difference equations involving generalized difference. (English) Zbl 1262.39017
Summary: Sufficient conditions are obtained for oscillation/nonoscillation of solutions of second-order difference equations, involving the generalized difference, of the form $\Delta_a(p(n-1)\Delta_ay(n-1))+q(n)y(n)=0, \quad n \geqslant 1,$ and $\Delta_a(p(n-1)\Delta_ay(n-1))+q(n)y(n)=f(n), \quad n \geqslant 1,$ where $$\Delta _{a}$$ is defined by $$\Delta _{a}y(n) = y(n + 1) - ay(n)$$, $$a \neq 0$$. Necessary and sufficient conditions are obtained for the 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 the oscillation/nonoscillation of these two equations.

##### MSC:
 39A21 Oscillation theory for difference equations 39A10 Additive difference equations 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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