Asymptotic representations of regularly varying pω(y0, y1, λ0)--solutions of a differential equation of the second order containing the product of different types of nonlinearities of the unknown function and its derivative

Abstract

We establish necessary and sufficient conditions for the existence of regularly varying solutions of the second-order differential equations whose right-hand sides contain the product of a regularly varying nonlinearity of the unknown function and a rapidly varying nonlinearity of the derivative of the unknown function as the arguments tend either to zero or to infinity. Asymptotic representations of these solutions and their first-order derivatives are also found.