ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF TWO-PART DIFFERENTIAL EQUATIONS WITH EXPONENTIAL NONLINEARITY

Abstract

For a two-term nonautonomous ordinary differential equation of the fourth order with an exponential nonlinearity of the form y(4) = α0p0(t)[1 + r(t)]eσy where α0 ∈ {−1,1}, σ ≠ 0, the function p0(t) is continuous or continuously differentiable and nonzero in some left neighbourhood of ω (ω ≤ +∞), r(t) is a continuous function such that limt↑ωr(t) = 0, the asymptotic behaviour at t↑ω of one class of Pω(Y0, λ0)-solutions is studied. For this equation, in [1], the necessary and sufficient conditions for the existence of such solutions were obtained in the case when λ0 ∈ R∖ {0, 1/2, 2/3, 1}. The proof of sufficient conditions for existence was carried out under some additional conditions that are quite strict. The aim of this paper is to improve the results obtained in [1] for sufficient conditions of existence. An attempt is made to extend the results of this paper to conditions that are less stringent. In contrast to [1], the proof of the main result in this paper assumes that there is a finite or infinite limit limt↑ω πω(t)q’(t). The equation under study is reduced to a system of equations for which it is necessary to determine the existence of solutions vanishing at infinity. This fact is established using the known results of [2]. The question of the number of solutions of the equation with the found asymptotic images is also solved.