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 $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q{y}_n=0,n\ge 1,$$ 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 $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q_{n}g\left(y_n\right)=f_{n-1},n\ge 1,$$ where, unlike earlier works, $f_n\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are...

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form $$y_i^{\text{'}}\left(t\right)-p_i\left(t\right)y_{i+1}\left(t\right)=0,i=1,2,\cdots ,n-2,y_{n-1}^{\text{'}}\left(t\right)-p_{n-1}\left(t\right)|y_n\left(h_n\left(t\right)\right){|}^{\alpha }\mathrm{s}gn\left[y_n\left(h_n\left(t\right)\right)\right]=0,y_n^{\text{'}}\left(t\right)\mathrm{s}gn\left[y_1\left(h_1\left(t\right)\right)\right]+p_n\left(t\right){\left|y_1\left(h_1\left(t\right)\right)\right|}^{\beta }\le 0,$$ where $n\ge 3$ is odd, $\alpha >0$, $\beta >0$.

In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form $$y_{n+3}+r_n{y}_{n+2}+q_n{y}_{n+1}+p_n{y}_n=0,n\ge 0.$$ These results are generalization of the results concerning difference equations with constant coefficients $$y_{n+3}+r{y}_{n+2}+q{y}_{n+1}+p{y}_n=0,n\ge 0.$$ Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.

This paper deals with oscillatory and asymptotic behaviour of solutions of second order quasilinear difference equation of the form $$\Delta (a_{n-1}|\Delta y_{n-1}{|}^{\alpha -1}\Delta y_{n-1})+F(n,y_n)=G(n,y_n,\Delta y_n),n\in N\left(n_0\right)\left(\mathrm{E}\right)$$ where $\alpha >0$. Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are also considered.