On second order nonlinear nonoscillation (Q2756175)

The author considers the generalized Emden-Fowler equation NEWLINE\[NEWLINEy''+g(x)y^\gamma=0, \quad x\geq 0,\tag{*}NEWLINE\]NEWLINE where \(\gamma=2n-1,\) with \(n>1\) an integer and \(g(x)>0.\) It is shown that equation (*) is nonoscillatory if the following conditions are satisfied: \(g'(x)\) is eventually nonincreasing, \(\int^\infty_0 x^{\frac{\gamma+1}{2}}g(x) dx<\infty,\) and for each sufficiently large \(a,\) the corresponding \(Q(t)\equiv\frac{\gamma+3}{2}+\frac{(t-a)g'(t)}{g(t)}\) is eventually of one sign. For a bounded positive solution \(y\) to equation (*) with \(y(a)=0,\) we have \(G_y(x)\equiv\frac{2}{\gamma+1}\int_a^x Q(t)g(t)y^{\gamma+1}(t) dt>0\) for all \(x\in (a,\infty).\)