ПРО АСИМПТОТИЧНІ ЗОБРАЖЕННЯ ОДНОГО КЛАСУ РОЗВ’ЯЗКІВ ДИФЕРЕНЦІАЛЬНИХ РІВНЯНЬ ДРУГОГО ПОРЯДКУ

For the second-order differential equation of general form y′′ = f(t, y, y′), where f : [α, ω[ × ΔY0 × ΔY1 → R is continuous function, −∞ < α < ω ≤ +∞, ΔYi is a one-neighborhood of Yi, Yi ∊ {0, ±∞} (i ∊ {0, 1}) we study the question of the existence of solutions, for which limt↑ω y(i) (t) = Yi (i ∊ {0, 1}). Among the set of such solutions we separate a sufficiently wide class of so-called Pω(Y0, Y1, λ0)-solutions. Such a solution was previously introduced in the study of the two-term equation y′′ = α0p(t)ϕ0(y)ϕ1(y′), where α0 ∊ {-1, 1}, p : [α, ω[→]0, +∞[ is continuous function, ϕi : ΔYi →]0, +∞[ (i = 0, 1) are continuous regular varying as z → Yi (i = 0, 1) functions of orders σi (i = 0, 1), such that σ0 + σ1 ≠ 1. In this paper a condition under which the right-hand side of the equation as λ0 ∊ R∖{0, 1} and t↑ω in some sense close to the multiplication α0p(t)ϕ0(y) with rapidly varying ϕ0 function at y → Y0, is established. We have obtained necessary and also sufficient conditions of existence of Pω(Y0, Y1, λ0)-solutions, asymptotic representations of these solutions and their first-order derivative and number of parametric family of these solutions. An example is given.