Geodesic
Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations
Mathematica Bohemica, Tome 136 (2011) no. 3, pp. 241-258

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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 \[ y_{n+2}+ a_{n}y_{n+1}+ b_{n}y_{n}+ c_{n}y_{n-1}= g_{n-1}, \quad n\geq 1. \]
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 \[ y_{n+2}+ a_{n}y_{n+1}+ b_{n}y_{n}+ c_{n}y_{n-1}= g_{n-1}, \quad n\geq 1. \]
DOI : 10.21136/MB.2011.141647
Classification : 39A06, 39A10, 39A12, 39A21
Keywords: oscillation; non-oscillation; second order difference equation; third order difference equation; generalized zero 
Parhi, N. Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations. Mathematica Bohemica, Tome 136 (2011) no. 3, pp. 241-258. doi: 10.21136/MB.2011.141647
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