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Keywords:
oscillation; non-oscillation; second order difference equation; third order difference equation; generalized zero
Summary:
In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $\Delta (p_{n-1}\Delta y_{n-1}) + q y_{n} =0 , \quad n\geq 1,$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $\Delta (p_{n-1}\Delta y_{n-1}) + q_{n}g( y_{n}) = f_{n-1}, \quad n\geq 1,$ where, unlike earlier works, $f_{n}\geq 0$ or $\leq 0$ (but $\not \equiv 0)$ for large $n$. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form $y_{n+2}+ a_{n}y_{n+1}+ b_{n}y_{n}+ c_{n}y_{n-1}= g_{n-1}, \quad n\geq 1.$
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