We give a sufficient condition for the oscillation of linear homogeneous second order differential equation $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0$, where $p(x), q(x)\in C[\alpha ,\infty )$ and $\alpha $ is positive real number.
This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasiiinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as t → ∞.
Qualitative comparison of the nonoscillatory behavior of the equations \[ L_ny(t) + H(t,y(t)) = 0 \] and \[ L_ny(t) + H(t,y(g(t))) = 0 \] is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form
\[ L_n = \frac{1}{p_n(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{1}{p_{n-1}(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \ldots \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{\cdot }{p_0(t)}. \] Both canonical and noncanonical forms of $L_n$ have been studied.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation \[ (r(t)\Phi (u^{\prime }(t)))^{\prime }+c(t)\Phi (u(t))=0, \] where (i) $r,c\in C([t_{0}, \infty )$, $\mathbb{R}:=(-\infty , \infty ))$ and $r(t)>0$ on $[t_{0},\infty )$ for some $t_{0}\ge 0$; (ii) $\Phi (u)=|u|^{p-2}u$ for some fixed number $p> 1$. We also generalize some results of Hille-Wintner, Leighton and Willet.
Necessary and sufficient conditions have been found to force all solutions of the equation \[ (r(t)y^{\prime }(t))^{(n-1)} + a(t)h(y(g(t))) = f(t), \] to behave in peculiar ways. These results are then extended to the elliptic equation \[ |x|^{p-1} \Delta y(|x|) + a(|x|)h(y(g(|x|))) = f(|x|) \] where $ \Delta $ is the Laplace operator and $p \ge 3$ is an integer.