Asymptotic of rapid varying solutions of differential equations of the second order with rapid varying nonlinearities
Journal Title: Дослідження в математиці і механіці - Year 2015, Vol 20, Issue 2
Abstract
The question of the asymptotic behavior of the differential equations’ solution by 𝑡 ↑ 𝜔 𝑦′′ = 𝛼0𝑝(𝑡)𝜙(𝑦), where 𝛼0 ∈ {−1, 1}, 𝑝 : [𝑎, 𝜔[−→]0,+∞[ is a continuous function, −∞ < 𝑎 < 𝜔 6 +∞ , 𝜙 : Δ𝑦0 −→]0,+∞[ is a twice continuously differentiable function belonging to the advanced Khan’s class 𝛾𝑦0 (𝑧0), where 𝑦0 is either zero or Ѓ}∞, Δ𝑦0 — some onesided neighborhood 𝑦0 and 𝑧0 = lim𝑦→𝑦0 𝜙(𝑦) is investigated. Here, unlike most well known works on the subject, the function 𝜙 is the regularly varying one by 𝑦 → 𝑌0. Necessary and sufficient conditions are established for the given equation in this case, for the existence of 𝑃𝜔(𝑌0, 0)-solutions and asymptotic representations of these solutions and their derivatives of the first order by 𝑡 ↑ 𝜔. Such solutions are slowly varying functions when 𝑡 ↑ 𝜔.
Authors and Affiliations
A. G. Chernikova
Keywords
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