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 ∆(pn−1∆yn−1) + qyn = 0, n ≥ 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 ∆(pn−1∆yn−1) + qng(yn) = fn−1, n ≥ 1, where, unlike earlier works, fn > 0 or 6 0 (but 6≡ 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 yn+2 + anyn+1 + bnyn + cnyn−1 = gn−1, n ≥ 1.
In this paper, oscillation and asymptotic behaviour of solutions of
\[ y^{\prime \prime \prime} + a(t)y^{\prime \prime}+b(t)y^{\prime} + c(t)y=0 \] have been studied under suitable assumptions on the coefficient functions $a,b,c\in C([\sigma ,\infty),R)$, $ \sigma \in R$, such that $a(t)\ge 0$, $b(t) \le 0$ and $c(t) < 0$.
In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm{(*)}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \infty $. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
Necessary and sufficient conditions are obtained for every solution of
\[ \Delta (y_{n}+p_{n}y_{n-m})\pm q_{n}G(y_{n-k})=f_{n} \] to oscillate or tend to zero as $n\rightarrow \infty $, where $p_{n}$, $q_{n}$ and $f_{n}$ are sequences of real numbers such that $q_{n}\ge 0$. Different ranges for $p_{n}$ are considered.
In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of $n$th order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.
In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form yn+3 + rnyn+2 + qnyn+1 + pnyn = 0, n ≥ 0. These results are generalization of the results concerning difference equations with constant coefficients yn+3 + ryn+2 + qyn+1 + pyn = 0, n ≥ 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.
Necessary and sufficient conditions are obtained for oscillation of all bounded solutions of (∗) [y(t) − y(t − τ )](n) + Q(t)G(y(t − σ)) = 0, t ≥ 0, where n ≥ 3 is odd. Sufficient conditions are obtained for all solutions of (∗) to oscillate. Further, sufficient conditions are given for all solutions of the forced equation associated with (∗) to oscillate or tend to zero as t → ∞. In this case, there is no restriction on n.