New oscillation criteria are given for the second order sublinear differential equation $${\left[a\left(t\right)\psi \left(x\left(t\right)\right)x^{\text{'}}\left(t\right)\right]}^{\text{'}}+q\left(t\right)f\left(x\left(t\right)\right)=0,t\ge t_0>0,$$ where $a\in C^1\left([t_0,\infty )\right)$ is a nonnegative function, $\psi ,f\in C\left(ℝ\right)$ with $\psi \left(x\right)\ne 0$, $xf\left(x\right)/\psi \left(x\right)>0$ for $x\ne 0$, $\psi$, $f$ have continuous derivative on $ℝ\setminus \left\{0\right\}$ with ${[f\left(x\right)/\psi \left(x\right)]}^{\text{'}}\ge 0$ for $x\ne 0$ and $q\in C\left([t_0,\infty )\right)$ has no restriction on its sign. This oscillation criteria involve integral averages of the coefficients $q$ and $a$ and extend known oscillation criteria for the equation $x^{\text{'}\text{'}}\left(t\right)+q\left(t\right)x\left(t\right)=0$.

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are...

By employing the matrix Riccati technique and the integral averaging technique, new interval oscillation criteria are established for second-order matrix differential systems of the form [P(t)Y']' + Q(t)Y = 0.

By using the generalized Riccati technique and the averaging technique, we establish new oscillation criteria for the second order self-adjoint matrix differential system with damping (P(t)Y'(t))' + r(t)P(t)Y'(t) + Q(t)Y(t) = 0, t ≥ t₀. The criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. In particular, our results complement a number of...

2000 Mathematics Subject Classification: 34C10, 34C15.It is the purpose of this paper to give oscillation criteria for the second order nonlinear differential equation with a damping term (a(t) y′(t))′ + p(t)y′(t) + q(t) |y(t)| α−1 y(t) = 0, t ≥ t0, where α ≥ 1, a ∈ C1([t0,∞);(0,∞)) and p,q ∈ C([t0,∞);R). Our results here are different, generalize and improve some known results for oscillation of second order nonlinear differential equations that are different from most known ones in the sencse...