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.

The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equations of the form (r(t)(x(t) - px(t-τ))')' - q(t)f(x(σ(t))) = 0 to be oscillatory and to compare some existing results.

Oscillatory properties of solutions to the system of first-order linear difference equations $$\begin{array}{cc}\hfill \Delta u_k& =q_k{v}_k\hfill \\ \hfill \Delta v_k& =-p_k{u}_{k+1},\hfill \end{array}$$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

The paper deals with the existence of positive ω-periodic solutions for a class of nonlinear delay differential equations. For example, such equations represent the model for the survival of red blood cells in an animal. The sufficient conditions for the exponential stability of positive ω-periodic solution are also considered.

An extension of a result of R. Conti is given from which some integro-differential inequalities of the Gronwall-Bellman-Bihari type and a criterion for the continuation of solutions of a system of ordinary differential equations are deduced.

Some Wintner and Nehari type oscillation criteria are established for the second-order linear delay differential equation.

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $${\left(r\left(t\right)\Phi \left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(u\left(t\right)\right)=0,$$ where (i) $r,c\in C([t_0,\infty )$, $ℝ:=(-\infty ,\infty ))$ and $r\left(t\right)>0$ on $[t_0,\infty )$ for some $t_0\ge 0$; (ii) $\Phi \left(u\right)={\left|u\right|}^{p-2}u$ for some fixed number $p>1$. We also generalize some results of Hille-Wintner, Leighton and Willet.

In this paper there are generalized some results on oscillatory properties of the binomial linear differential equations of order $n(\ge 3$) for perturbed iterative linear differential equations of the same order.