Nonoscillatory criteria for singular higher order differential equations (Q2386100)

The authors consider higher-order singular differential equations of the form \[ (a(t) y^{(n-1)}(t))'= q(t) F(t,y(t)), \] where \(n\) is even, \(a\in C([t_0,\infty), \mathbb{R}_+)\), \(q\in L[t_0,\infty)\) with \(q(t)> 0\), and \(F: [t_0,\infty)\times (0,\infty)\to [0,\infty)\) is continuous. Here, the nonlinearity may be singular in both the independent and dependent variables. It is shown that under some assumptions, the equation has a solution \(y\) satisfying \[ \lim_{t\to\infty} y^{(j)}(t)= 0,\quad j= 0,\dots, n-2,\quad\text{and }\lim_{t\to\infty} a(t) y^{(n-1)}(t)= 0. \] A further result is obtained for the case when \(n= 2\). Examples are given to illustrate the results.