The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form $${[x\left(t\right)+cx\left(\tau \left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)f\left(x\left(\sigma \right(t\left)\right)\right)=0,t\ge t_0>0,$$ where $c$ is a real number with $\left|c\right|\ne 1$, $q\in C([t_0,\infty ),ℝ)$, $f\in C(ℝ,ℝ)$, $\tau ,\sigma \in C([t_0,\infty ),{ℝ}_{+})$ with $\tau \left(t\right)<t$ and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty$. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations...

Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation ${\sum }_{i,j=1}^N{D}_i\left[a_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^N{b}_i\left(x\right)D_{i}y+p\left(x\right)f\left(y\right)=0$ are obtained. Our results are extensions of those for ordinary differential equations and improve some known oscillation criteria in the literature. Several examples are given to show the significance of the results.

We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and ${\int }^{\infty }[1/r\left(t\right)]dt$ converges.

Some new oscillation criteria are obtained for second order elliptic differential equations with damping ${\sum }_{i,j=1}^n{D}_i\left[A_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^n{b}_i\left(x\right)D_{i}y+q\left(x\right)f\left(y\right)=0$, x ∈ Ω, where Ω is an exterior domain in ℝⁿ. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of subdomains of Ω ⊂ ℝⁿ, rather than on the whole exterior domain Ω. Our results are more natural in view of the Sturm Separation Theorem.

In this paper, the oscillation criteria for solutions of the neutral delay differential equation (NDDE) $${\left(y\left(t\right)-p\left(t\right)y\left(t-\tau \right)\right)}^{\left(n\right)}+\alpha Q\left(t\right)G\left(y\left(t-\sigma \right)\right)=f\left(t\right)$$ has been studied where $p\left(t\right)=1$ or $p\left(t\right)\le 0$, $\alpha =\pm 1$, $Q\in C\left([0,\infty ),R^{+}\right)$, $f\in C\left(\right[0,\infty ),R)$, $G\in C(R,R)$. This work improves and generalizes some recent results and answer some questions that are raised in [1].

In this note we consider the third order linear difference equations of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+\delta q\left(n\right)x\left(\tau \left(n\right)\right)=0,n\in N\left(n_0\right),\left(\mathrm{E}\right)$$ where $\delta =\pm 1$, $p,qN\left(n_0\right)\to {ℝ}_{+};$ $\sigma ,\tau N\left(n_0\right)\to \mathbb{N}$, ${lim}_{n\to \infty }\sigma \left(n\right)=\underset{n\to \infty }{lim}\tau \left(n\right)=\infty .$ We examine the following two cases: $$\begin{array}{ccc}\hfill \{0<p(n)& \le 1,\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l\},\{p\left(n\right)\hfill & \hfill >1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l\},\end{array}$$ where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.