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.Investigations of the asymptotic behavior of solutions of the generalized Emden-Fowler equation (see [1, pp. 326-402] (Chap. V) and [2][3][4][5][6][7][8]) and first-order differential equations whose right-hand sides contain the sum of terms with power nonlinearities (see [9, pp. 113-127] (Chap. 5) and [10,11]) were important prerequisites for the development of methods for establishing the asymptotic behavior of regular nonoscillating solutions of ordinary differential equations of higher order with Emden-Fowler-type nonlinearities of the formfunctions, and -∞ < a < ω ≤ + ∞, carried out in [1,[12][13][14][15].In general, the problem of the asymptotic behavior of solutions of differential equations with nonlinearities of a more general form remains open even for the two-term equationsFor n = 2, equations of this type were studied in [16,17] and some other papers.In the present paper, we consider the differential equation1) where α k ∈ { -1; 1 }, k = 1, … , m, p k : [ a, ω [ → ] 0, + ∞ [, k = 1, … , m, are continuously differentiable functions, r k : [ a, ω [ → R, k = 1, … , m, are continuous functions satisfying the conditions lim ( ) t k r t ↑ω = 0, k = 1, … , m, (0.2)

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UDC 517. 925 We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.

STATEMENT OF THE MAIN RESULTSWe consider the differential equation1) where α 0 ∈ {−1, 1}, p : [a, ω[ → ]0, +∞[ (−∞ < a < ω ≤ +∞) is a continuous function, and ϕ : ]0, y 0 ] → ]0, +∞[ is a twice continuously differentiable function satisfying the conditions lim y→+0 ϕ(y) = either 0 or + ∞, lim y→+0 yϕ (y) ϕ (y) = σ ∈ {0, ±∞}. (1.2) By conditions (1.2), lim y→+0 yϕ (y) ϕ(y) = σ + 1, (1.3)and hence the function ϕ(y) admits the representationIf ϕ(y) = |y| σ+1 sgn y (σ = 0), then Eq. (1.1) is called a generalized Emden-Fowler equation. The problem on the asymptotics of all possible types of its nonoscillating solutions was considered in detail in [1][2][3][4][5][6][7].For α 0 = 1, σ > 0, and ω = +∞, sufficient conditions for the existence of solutions of Eq. (1.1) decaying at infinity were obtained in [8,9] for the case in which ϕ(y) is different from |y| σ+1 sgn y and in a sense close to functions of the form (1.4). For any solution y(t) of this kind, two-sided asymptotic estimates for the ratio y(t)/ϕ(y(t)) as t → +∞ were also obtained.A solution y :The aim of the present paper is to establish [under conditions (1.2) and for arbitrary α 0 and ω] not only sufficient but also necessary conditions for the existence of P 0 ω (λ 0 )-solutions of Eq. (1.1) such that λ 0 ∈ {0, 1, ±∞} and find exact asymptotic formulas for y(t)/ϕ(y(t)) and y (t) as t → ω−0.In what follows, we use the auxiliary notation π ω (t) = t for ω = +∞ t − ω for ω < +∞, I(t) = t Aω p(τ )π ω (τ )dτ, Φ(y) = y Y0 dz ϕ(z) , (1.6)