Growth, Oscillation and Comparison Theorems for Second Order Linear Difference Equations

Abstract

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 ngeq 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 ngeq 1, ] where, unlike earlier works, $f_{n}geq 0$ or $leq 0$ (but $ ot 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 ngeq 1. |*|