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.

By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order ${\sum }_{i,j=1}^N{D}_i[A_{ij}{\left(x\right)\left|Dy\right|}^{p-2}{D}_{j}y]+p\left(x\right)f\left(y\right)=0$ are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

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...

In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form $${\Delta }^2\left(r\left(n\right){\Delta }^2(y\left(n\right)+p\left(n\right)y(n-m))\right)+q\left(n\right)G\left(y(n-k)\right)=0$$ is studied under the assumption $$\sum _{n=0}^{\infty }\frac{n}{r\left(n\right)}<\infty .$$ New oscillation criteria have been established which generalize some of the existing results in the literature.

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.

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.

Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${div\left(A\left(x\right)\right|\left|\nabla y\right||}^{p-1}{\nabla y)+⟨b\left(x\right),|\left|\nabla y\right||}^{p-1}\nabla y⟩+C(x,y)=e\left(x\right)$, ⎨ ⎩ $C(x,y)={c\left(x\right)\left|y\right|}^{p-1}y+{\sum }_{i=1}^m{c}_i\left(x\right){\left|y\right|}^{p_i-1}y$ $$under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with mixed nonlinearities, and include some known oscillation results in the literature

Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations $${(-1)}^{m+1}\frac{d^m{y}_i\left(t\right)}{d{t}^m}+\sum _{j=1}^n{q}_{ij}{y}_j(t-h_{jj})=0,m\ge 1,i=1,2,...,n,$$ to be oscillatory, where $q_{ij}\epsilon (-\infty ,\infty )$, $h_{jj}\in (0,\infty )$, $i,j=1,2,...,n$. Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations $${(-1)}^{m+1}\frac{d^m}{d{t}^m}(y_i\left(t\right)+c{y}_i(t-g))+\sum _{j=1}^n{q}_{ij}{y}_j(t-h)=0,$$ where $c$, $g$ and $h$ are real constants and $i=1,2,...,n$.

In the two-parameter setting, we say a function belongs to the mean little BMO if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the present author in relation to the multiplier algebra of the product BMO of Chang-Fefferman. We prove that the Cotlar-Sadosky space $bmo{(}^N)$ of functions of bounded mean oscillation is a strict subspace of the mean little BMO.

Oscillation criteria are obtained for nonlinear homogeneous third order differential equations of the form $y^{\text{'}\text{'}\text{'}}+q\left(t\right)y^{\text{'}}+p\left(t\right)y^{\alpha }=0$ and y”’ + q(t)y’ + p(t)f(y) = 0, where p and q are real-valued continuous functions on [a,∞), f is a real-valued continuous function on (-∞, ∞) and α > 0 is a quotient of odd integers. Sign restrictions are imposed on p(t) and q(t). These results generalize some of the results obtained earlier in this direction.