Integral conditions for nonoscillation of second order nonlinear differential equations

An extension of the Fubini theorem is presented and applied in studying the possible coexistence of nonoscillatory solutions for the differential equation(r(t)|x′|αsgnx′)′=q(t)|x|βsgnx.Some discrepancies among the cases α>βα>β, α=βα=β and α<βα<β are illustrated and compared with the linear case. In addition, when α=βα=β an application to the integral characterization of principal solutions, which answers an open problem stated in [O. Došlý, A. Elbert, Integral characterization of the principal solution of half-linear second order differential equations, Studia Sci. Math. Hungar. 36 (2000) 455–469], is given.