Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II

Abstract

517.925We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden -Fowler type.Consider the differential equationare continuous functions satisfying the conditions lim ( ) t k r t ↑ω = 0, k = 1, … , m, (0.2) -∞ < a < ω ≤ + ∞, and ϕ k : [ y 0 , + ∞ [ → ] 0, + ∞ [, k = 1, … , m, 0 < y 0 < + ∞, are twice continuously differentiable functions such that lim ( ) y k y →+ ∞ ϕ = ϕ k 0 = const ≠ 0 for k = 1, … , m 1 , (0.3) lim ( ) y k y →+ ∞ ϕ = either or 0, + ∞ ⎧ ⎨ ⎪ ⎩ ⎪ for k = m 1 + 1, … , m, 1 (0.4) and, furthermore, ′ ϕ k ≠ 0 for y ≥ y 0 and lim ( ) ( ) y k k y y y →+ ∞ ′′ ′ ϕ ϕ = σ k = const (0.5) if k belongs to { 1, … , m } and differs from k ∈ { 1, … , m 1 } for which ϕ k ( y ) ≡ ϕ k 0 . 1 Here and in what follows, we set m 1 = 0 ( m 1 = m ) if only condition (0.4) [only condition (0.3)] is satisfied.