A maximum principle for second order nonlinear differential inequalities and its applications (Q1895125)

The main result concerns the differential inequality \[ y(t)\{(r(t) y'(t))'+ f(t, y(t))\}\leq 0, \] a.e. on \([a, b]\), where \(f\) satisfies an Osgood condition. Let \(y\) be a solution to the inequality such that \(y(t)\neq 0\) on \((a, b)\). If \(y(a)= 0\) or \(y(b)= 0\), then \(y'(a)\neq 0\) or \(y'(b)\neq 0\), respectively. Applications are given to maximum principles and to comparison principles for second-order differential equations.