Mertens’ product formula asserts that$$\prod _{p\le x}\left(1-\frac{1}{p}\right)logx\to e^{-\gamma }$$as $x\to \infty$. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

In the paper ordinary neutral differential equations with ?maxima? are considered. Sufficient conditions for oscillation of all solutions are obtained.

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form $$y_i^{\text{'}}\left(t\right)-p_i\left(t\right)y_{i+1}\left(t\right)=0,i=1,2,\cdots ,n-2,y_{n-1}^{\text{'}}\left(t\right)-p_{n-1}\left(t\right)|y_n\left(h_n\left(t\right)\right){|}^{\alpha }\mathrm{s}gn\left[y_n\left(h_n\left(t\right)\right)\right]=0,y_n^{\text{'}}\left(t\right)\mathrm{s}gn\left[y_1\left(h_1\left(t\right)\right)\right]+p_n\left(t\right){\left|y_1\left(h_1\left(t\right)\right)\right|}^{\beta }\le 0,$$ where $n\ge 3$ is odd, $\alpha >0$, $\beta >0$.

Oscillation criteria, extended Kamenev and Philos-type oscillation theorems for the nonlinear second order neutral delay differential equation with and without the forced term are given. These results extend and improve the well known results of Grammatikopoulos et. al., Graef et. al., Tanaka for the nonlinear neutral case and the recent results of Dzurina and Mihalikova for the neutral linear case. Some examples are considered to illustrate our main results.

We consider nonlinear neutral delay differential equations with variable coefficients. Finite and infinite integral conditions for oscillation are obtained. As an example, the neutral delay logistic differential equation is discussed.

2000 Mathematics Subject Classification: 34K15, 34C10.In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equation (x(t) − q(t) x(t − σ(t))) ′ +f(t,x( t − τ(t))) = 0, where σ, τ ∈ C([t0,∞),(0,∞)), q О C([t0,∞), [0,∞)) and f ∈ C([t0,∞) ×R,R). The obtained results extended and improve several of the well known previously results in the literature. Our results are illustrated with an example.

We establish some new oscillation criteria for the second order neutral delay differential equation $${\left[r\left(t\right)\right|[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}{{|}^{\alpha -1}{[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}]}^{\text{'}}+q\left(t\right)f\left(x\left[\sigma \left(t\right)\right]\right)=0.$$ The obtained results supplement those of Dzurina and Stavroulakis, Sun and Meng, Xu and Meng, Baculíková and Lacková. We also make a slight improvement of one assumption in the paper of Xu and Meng.

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.