On an extension of the Fubini theorem and its applications in ODEs

Abstract

The paper deals with the change of integration for integrals ∫ 0 ∞ b(t) ∫ 0 t A(s) ds m dt, ∫ 0 ∞ A(t) ∫ t ∞ b(s) ds 1/m dt, where m>0 and A, b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equations (a(t)|x′| α sgn x′)′+b(t)|x| β sgn x=0, where 0< β⩽ α and a, b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations.