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 (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) 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 Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.