Asymptotic behaviour of solutions of one class of the fourth-order nonlinear differential equations

Abstract

The asymptotic behaviour as t↑ω of one of the possible types of Pω(Y0, λ0)-solutions of a binomial non-autonomous fourth-order ordinary differential equation with exponential nonlinearity and a continuous and non-zero in some left neighbourhood ω(ω≤+∞) coefficient p(t) is investigated. First, using a priori asymptotic properties of the considered Pω(Y0, λ0)-solutions, necessary conditions for their existence are established, as well as asymptotic representations of these solutions and their derivatives up to the third order. The question of the actual existence of solutions with the obtained asymptotic representations is solved by reducing it to the question of the existence of solutions that vanish at a specific point of a system of quasilinear differential equations. This system is obtained as a result of some transformations of the original equation, taking into account the kind of established asymptotic representations. In addition, the question of the number of solutions with found asymptotic representations is also resolved.