Oscillation Results for Fourth-Order Nonlinear Neutral Differential Equations with Positive and Negative Coefficients

Unbounded oscillation and asymptotic behaviors of a class of nonlinear fourth-order neutral differential equations with positive and negative coefficients of the form

$$ (r(t)(y(t)+p(t)y(t-\tau ){)}^{\prime\prime}{)}^{\prime\prime}+q(t)G(y(t-\alpha ))-h(t)H(y(t-\beta ))=0 $$

and

$$ (r(t)(y(t)+p(t)y(t-\tau ){)}^{\prime\prime}{)}^{\prime\prime}+q(t)G(y(t-\alpha ))-h(t)H(y(t-\beta ))=f(t) \qquad \qquad (E)$$

are investigated under the assumption that

$$ \int\limits_0^{\infty } {\frac{t}{r(t) }dt} <\infty $$

for various ranges of p(t). Sufficient conditions are obtained for the existence of positive bounded solutions of (E).