Levin's comparison theorems for nonlinear second order differential equations (Q1352358)

This article concerns Levin-type comparison theorems for nonlinear differential equations of type \((1)_i\) \(-(p_ig(y)|y'|^ay')'=q_if_i(y)\), \(i=1,2\), in a closed real interval \(I\), where \(a\) is a constant and \(f_i\), \(g\), \(p_i\), \(q_i\) are functions satisfying: \(f_i\in C(\mathbb{R},\mathbb{R})\cap C^1(\mathbb{R}\setminus\{0\})\), \(g\in C^1(\mathbb{R},\mathbb{R}_+)\), \(p_i\in C^1(I,\mathbb{R}_+)\), \(q_i\in C(I,\mathbb{R})\), and appropriate technical conditions. The theorems contain various sets of sufficient conditions for nontrivial solutions \(u,v\) of \((1)_1\), \((1)_2\), respectively, to satisfy \(|p_1g(u) |u'|^a u'/f_1(u) |> |p_2 g(v)|v' |^a v'/f_2(v) |\) pointwise on \(I\). The proofs employ a generalization of Osgood's inequality. Extensions and examples are included.