Oscillatory properties of solutions of neutral differential systems (Q2757002)

Here, the author studies the neutral differential system NEWLINE\[NEWLINE[y_1(t)-a(t)y_1(g(t))]'=p(t)y_2(t),\;y_i'(t)=p(t)y_{i+1}(t),\;i=1,2,\dots,n-2;\;n\geq 3,\tag{S}NEWLINE\]NEWLINE NEWLINE\[NEWLINEy_{n-1}'(t)=p_{n-1}(t)y_n(t), \quad y_n'(t)=p_n(t)y_1(h(t)),\quad t\in \mathbb{R}_+=[0,\infty).NEWLINE\]NEWLINE Denote by \(W\) the set of all solutions \(y=(y_1,y_2,\cdots,y_n)\) to (S) which exist on some ray \([T_y,\infty), T_y\geq 0\). When \(n\) is odd or \(n\) is even, two sufficient conditions are obtained respectively for every solution \(y(t)\) to (S) to be either an oscillation or it holds \(\lim_{t\to\infty}|y_i(t)|=\infty\) or \(\lim_{t\to\infty} y_i(t)=0\), \(i=1,2,\dots,n\).