Oscillation of solutions of second-order nonlinear self-adjoint differential equations

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation <f>(a(t)x′)′+b(t)g(x)=0</f>. Here <f>g(x)</f> satisfied the signum condition <f>xg(x)>0</f> if <f>x≠0</f>, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on <f>g(x)</f> play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Lie´nard type. We also explain an analogy between our results and an oscillation criterion of Kneser–Hille type for linear differential equations. [Copyright &y& Elsevier]