Oscillations of functional differential equations generated by advanced arguments (Q809256)

We consider the functional differential equation \((1)\quad L_ nx(t)+\delta f(t,x[g_ 1(t)],...,x[g_ m(t)])=0,\) where \(n\geq 2\), \(\delta =\pm 1\), \(L_ 0x(t)=x(t)\), \(L_ kx(t)=(1/a_ k(t))(L_{k- 1}x(t))d/dt\), \(k=1,2,...,n\), \(a_ n=1\), \(a_ i: {\mathbb{R}}_+\to {\mathbb{R}}_+\setminus \{0\}\), \(i=1,2,...,n-1\), \(g_ i: {\mathbb{R}}_+\to {\mathbb{R}}_+=[0,\infty)\), \(i=1,2,...,m\), f: \({\mathbb{R}}_+\times {\mathbb{R}}^ m\to {\mathbb{R}}=(-\infty,\infty)\) are continuous and \(g_ i(t)\geq t\) on \({\mathbb{R}}_+\) for \(i=1,2,...,m\). We always assume \(\int^{\infty}a_ i(s)ds=\infty,\quad i=1,2,...,n-1.\) The main purpose of this paper is to establish some criteria for equation (1) to be oscillatory or almost oscillatory. The behavioural properties obtained for equation (1) are generated by the advanced arguments \(g_ i (i=1,2,...,m)\) and are not valid for the corresponding ordinary differential equations.