Classification of positive solutions of generalized functional differential equations

In this paper, we give a classification of nonoscillatory solutions of the quasilinear differential equation (ϑ(y′(t)))′ + f(t, y(g(t))) = 0, t ≥ a. (∗) Under suitable conditions on f and g, we obtain a classification of nonoscillatory solutions of (∗) according to their asymptotic behavior as t → ∞. We also afford some necessary and sufficient conditions to guarantee the existence of nonoscillatory and oscillatory solutions for the above equation.