Nonoscillation theorems for functional differential equations of arbitrary order (Q2266158)

In this paper we are dealing with n-th order (n\(\geq 2)\) sublinear functional differential equations of the form \[ (1)\quad [r(t)x^{(n- \nu)}(t)]^{(\nu)}=f(t,x(t),x[g(t)]),\quad t\geq t_ 0 \] where \(1\leq \nu \leq n-1\), and the functions r,g: [t\({}_ 0,\infty)\to {\mathbb{R}}\) and f: [t\({}_ 0,\infty)\times {\mathbb{R}}^ 2\to {\mathbb{R}}\) are continuous. Furthermore, we assume that \(r(t)>0\) and \(\lim_{t\to \infty}g(t)=\infty\). We give sufficient conditions for all oscillatory solutions of equation (1) to converge to zero. We then prove a nonoscillation theorem for equation (1). A few intermediate results are also obtained.