Асимптотична поведiнка розв’язкiв диференцiальних рiвнянь другого порядку з нелiнiйностями рiзного виду

In this paper for the second-order differential equation which has a right-hand side containing the sum of the terms with regularly and rapidly varying nonlinearities the necessary and sufficient conditions of the existence so-called Pω(Y0,λ0) – solutions (Y0 is either 0, or ±∞, −∞ < a < ω ≤ +∞) in a special case when the parameter λ0 = ±∞ are established. The asymptotic representations when t ↑ ω for such solutions and their first-order derivatives also are established. The results of the work were obtained on the assumption that on each solution from the class under consideration the right-hand side of the differential equation being studied is equivalent when t ↑ ω to one term with a rapidly varying nonlinearity. This term must be considered as the principal one on the right side of the equation. The method of allocation of the main term was proposed by H. Hardy when studying the differential equation of the first order. Later, A.V. Kostin, V.M. Evtukhov, E.V. Shebanina used this method in studying the asymptotic properties of solutions of differential equations of n-th order with power nonlinearities. In the study of the asymptotic properties of the set Pω(Y0,λ0) - solutions that corresponds to this value of the parameter λ0, was used the method proposed by V.M. Evtukhov during the study together with A.G. Chernikova binomial differential equation with rapidly varying nonlinearity. The work has a theoretical nature. The results obtained and the method employed in the work can be used to construct an asymptotic theory of differential equations of a more general type containing the sum of the terms in the right-hand side with regularly and rapidly varying nonlinearities.