Necessary existence conditions of Pω(Y0, Y1, λ0)-solutions of second-order differential equation with rapidly varying nonliniarity

Abstract

We consider a differential equation of the second order of the general form y’’ = f(t, y, y’), where f : [a, ω] x ΔY0 x ΔY1 → R – a continuous function, −∞ < a < ω ≤ +∞, ΔYi – one-side neighborhood of Yi, Yi ∈ {0, ±∞} (i ∈ {0, 1}). Under certain conditions for the function f, this equation can be represented close to the two-term differential equation, namely y’’ = α0p(t)ϕ1(y’)(1 + o(1)) at t↑ω, where ϕ1 is a rapidly varying function at y’ → Y1. Found the necessary conditions for the existence of solutions for which lim<sup>t↑ω</sup> y(i) (t) + Yi (i ∈ {0,1}), lim<sup>t↑ω</sup> [y’(t)]2/y(t)y’’(t) = λ0, so called Pω(Y0, Y1, λ0)-solutions. This type of solution was previously presented in works by Evtukhov V. M., Belozerova M. O. when studying the two-term equation y’’ = α0p(t)ϕ0(y)ϕ1(y’), where α0 ∈ {−1, 1}, p : [a, ω[→]0, +∞[–continuous function, ϕi : ΔYi →]0, +∞[(i = 0, 1)–continuous regularly variables for z → Yi (i = 0, 1) functions of orders σi (i = 0, 1), and σ0 + σ1 ≠ 1. Further, in the studies of V. M. Evtukhov, A. G. Chernikova for equation y’’ = α0p(t)ϕ0(y) necessary and sufficient conditions are established existence, as well as asymptotic at t↑ω representations Pω(Y0, Y1, λ0)-solutions in the case when ϕ0 is a rapidly varying function at y → Y0.