Narahari Parhi, National Institute of Science Education and Research (NISER), I.O.P. Campus, Bhubaneswar-751005, Orissa, India, e-mail: parhi2002@rediffmail.com
Abstract: 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\geq1,
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_ng( y_n) = f_{n-1}, \quad n\geq1,
where, unlike earlier works, $f_n\geq0$ or $\leq0$ (but $\not\equiv0)$ 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_ny_{n+1}+ b_ny_n+ c_ny_{n-1}= g_{n-1}, \quad n\geq1.
Keywords: oscillation, non-oscillation, second order difference equation, third order difference equation, generalized zero
Classification (MSC 2010): 39A10, 39A12
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