We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type. In the equations considered, the diffusion coefficient is linear and depends on the current state, and the drift term is a convolution integral which is in some sense mean reverting towards the zero equilibrium. The state dependent restoring force in the integral can be nonlinear. In broad terms, we show that when the restoring force is of linear or lower order in the neighbourhood of the equilibrium, or if the kernel decays more slowly than a critical noise-dependent rate, then there is a zero crossing almost surely. On the other hand, if the kernel decays more rapidly than this critical rate, and the restoring force is globally superlinear, then there is a positive probability that the solution remains of one sign for all time, given a sufficiently small initial condition. Moreover, the probability that the solution remains of one sign tends to unity as the initial condition tends to zero.

The aim of this paper is to study the dynamics of predator-prey interaction in a chemostat to determine whether including a discrete delay to model the time between the capture of the prey and its conversion to viable biomass can introduce oscillatory dynamics even though there is a globally asymptotically stable equilibrium when the delay is ignored. Hence, Holling type I response functions are chosen so that no oscillatory behavior is possible when there is no delay. It is proven that unlike the analogous model for competition, as the parameter modeling the delay is increased, Hopf bifurcations can occur.

The oscillatory behavior of the solutions of the second-order linear nonautonomous equation $x\left(n+1\right)=a\left(n\right)x\left(n\right)-b\left(n\right)x\left(n-1\right),n\in {\mathbb{N}}_0,$ where $a,b:{\mathbb{N}}_0\to ℝ$, is studied. Under the assumption that the sequence $b(n)$ dominates somehow $a(n)$, the amplitude of the oscillations and the asymptotic behavior of its solutions are also analized.

Consider the second-order linear delay differential equation $x^{\prime \prime }\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0$, $t\ge t_0$, where $p\in C\left(\left[t_0,\infty \right),{ℝ}^{+}\right)$, $\tau \in C\left(\left[t_0,\infty \right),ℝ\right)$, $\tau \left(t\right)$ is nondecreasing, $\tau \left(t\right)\le t$ for $t\ge t_0$ and ${\lim }_{t\to \infty }\tau \left(t\right)=\infty$, the (discrete analogue) second-order difference equation ${\Delta }^{2}x\left(n\right)+p\left(n\right)x\left(\tau \left(n\right)\right)=0$, where $\Delta x\left(n\right)=x\left(n+1\right)-x\left(n\right)$, ${\Delta }^2=\Delta \circ \Delta$, $p:\mathbb{N}\to {ℝ}^{+}$, $\tau :\mathbb{N}\to \mathbb{N}$, $\tau \left(n\right)\le n-1$, and ${\lim }_{n\to \infty }\tau \left(n\right)=+\infty$, and the second-order functional equation $x\left(g\left(t\right)\right)=P\left(t\right)x\left(t\right)+Q\left(t\right)x\left(g^2\left(t\right)\right)$, $t\ge t_0$, where the functions $P$, $Q\in C\left(\left[t_0,\infty \right),{ℝ}^{+}\right)$, $g\in C\left(\left[t_0,\infty \right),ℝ\right)$, $g\left(t\right)\not\equiv t$ for $t\ge t_0$, ${\lim }_{t\to \infty }g\left(t\right)=\infty$, and $g^2$ denotes the 2th iterate of the function $g$, that is, $g^0\left(t\right)=t$, $g^2\left(t\right)=g\left(g\left(t\right)\right)$, $t\ge t_0$. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where $\lim {\inf }_{t\to \infty }{\displaystyle {\int }_{\tau \left(t\right)}^t\tau \left(s\right)}p\left(s\right)ds\le 1/e$ and for the second-order linear delay differential equation, and and , for the second-order functional equation, are presented.

We are concerned with the oscillation of the forced second-order neutral nonlinear differential equations with delayed argument in the form ${\left(r\left(t\right){\left(x\left(t\right)+a\left(t\right)x\left(\sigma \left(t\right)\right)\right)}^{\prime }\right)}^{\prime }+p\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right)+{\sum }_{i=1}^n{q}_i\left(t\right){\left|x\left(t\right)\right|}^{{\lambda }_i}\text{s}\text{n}\text{g}x\left(t\right)=e\left(t\right)$. No restriction is imposed on the potentials $p\left(t\right)$, $q_i\left(t\right)$, and$e\left(t\right)$ to be nonnegative. Our methodology is somewhat different from those of previous authors.

For a neutral differential equation with positive and changeable sign coefficients ${\left[x\left(t\right)-a\left(t\right)x\left(\delta \left(t\right)\right)\right]}^{\prime }+p\left(t\right)F\left(x\left(\tau \left(t\right)\right)\right)-q\left(t\right)G\left(x\left(\sigma \left(t\right)\right)\right)=0$, oscillation criteria are established, where q(t) is not required as nonnegative. Several new results are obtained.

Picone-type inequality is established for a class of half-linear elliptic equations with forcing term, and oscillation results are derived on the basis of the Picone-type inequality. Our approach is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems for ordinary half-linear differential equations.

We present new oscillation criteria for the even order neutral delay differential equations with distributed deviating argument ${\left[r\left(t\right)\psi \left(x\left(t\right)\right)Z^{\left(n-1\right)}\left(t\right)\right]}^{\prime }+{\displaystyle {\int }_a^{b}p\left(t,\xi \right)}f\left[x\left(g\left(t,\xi \right)\right)\right]d\sigma \left(\xi \right)=0,t\ge t_0$, where $Z\left(t\right)=x\left(t\right)+q\left(t\right)x\left(t-\tau \right)$. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those by Wang et al. (2005).