Asymptotic representations of the singular solutions of non-linear differential equations with regularly varying functions
Journal Title: Дослідження в математиці і механіці - Year 2016, Vol 21, Issue 1
Abstract
In the study of non-linear non-autonomous ordinary differential equations, one of the most important questions are existence of regular and singular solutions of the first and second kind, particularly, so-called ¡¡explosive solutions¿¿ (in the terminology by I. T. Kiguradze [1]). This paper deals with the differential equation 𝑦(𝑛) = Σ︀𝑚 𝑘=1 𝛼𝑘𝑝𝑘(𝑡) Π︀𝑛−1 𝑗=0 𝜙𝑘𝑗(𝑦(𝑗)), where 𝛼𝑘 ∈ {−1; 1} (𝑘 = 1,𝑚), 𝑝𝑘 : [𝑎, 𝜔[−→]0,+∞[ (𝑘 = 1,𝑚) are continuous functions, 𝜙𝑘𝑗 : △𝑌𝑗 −→]0,+∞[ (𝑘 = 1,𝑚; 𝑗 = 0, 𝑛 − 1) are continuous and regularly varying functions subject to 𝑦(𝑗) −→ 𝑌𝑗 , −∞ < 𝑎 < 𝜔 6 +∞, △𝑌𝑗 is an one-sided neighborhood of 𝑌𝑗 , 𝑌𝑗 equals either 0 or ±∞. We establish existence conditions of singular 𝑃𝑡* (𝑌0, 𝑌1, . . . , 𝑌𝑛−1, 𝜆0)–solutions, which are contained in the class of singular solutions of first and second kind as well as the asymptotic representations of such solutions and their derivatives up to 𝑛 − 1 order inclusively.
Authors and Affiliations
V. M. Evtukhov, A. M. Klopot
Keywords
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