ASYMPTOTIC REPRESENTATIONS OF SOLUTIONS WITH SLOWLY VARYING DERIVATIVES OF THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH THE PRODUCT OF DIFFERENT TYPES OF NONLINEARITIES

Signi cantly nonlinear non-autonomous di erential equations have begun to appear in practice from the second half of the nineteenth century in the study of real physical processes in atomic and nuclear physics, and also in astrophysics. The di erential equation, that contains in its right part the product of regularly and rapidly varying nonlinearities of an unknown function and its rst-order derivative is considered in the paper. Partial cases of such equations arise, rst of all, in the theory of combustion and in the theory of plasma. The rst important results on the asymptotic behavior of solutions of such equations have been obtained for a second-order di erential equation, that contains the product of power and exponential nonlinearities in its right part. For, no such equations have been obtained before. According to this, the study of the asymptotic behavior of solutions of nonlinear di erential equations of the second order of general case, that contain the product of regularly and rapidly varying nonlinearities as the argument tends either to zero or to in nity, is actual not only from the theoretical but also from the practical point of view. The asymptotic representations, as well as the necessary and su cient conditions of the existence of Pω(Y0, Y1,±∞)-solutions of such equations are investigated in the paper. This class of solutions is the one of the most di cult of studying due to the fact that, by the a priori properties of the functions of the class, their second-order derivatives aren't explicitly expressed through the rst-order derivative. The results obtained in this article supplement the previously obtained results for Pω(Y0, Y1,±∞)-solutions of the investigated equation concerning the su cient conditions of their existence and quantity.


Introduction
The following second order dierential equation y = α 0 p(t)ϕ 0 (y)ϕ 1 (y ) (1) ÓÄÊ 517.925 PHIH Mathematics Subject Classication: QReQRD QRgRID QRÅWWF c ghepok yF yFD PHPH is considered. In this equation α 0 ∈ {−1; 1}, functions p : [a, ω[→]0, +∞[, (−∞ < a < ω ≤ +∞) and ϕ i : we put y 0 i > 0 (y 0 i < 0). We also suppose that function ϕ 1 is a regularly varying as y → Y 1 function of index σ 1 ([9], p.10-15), function ϕ 0 is twice continuously dierentiable on ∆ Y 0 and satises the next conditions ϕ 0 (y) = 0 as y ∈ ∆ Y 0 , lim It follows from the above conditions (2) that the function ϕ 0 and its derivative of the rst order are rapidly varying functions as the argument tends to Y 0 ([9], p.15). Thus, the investigated dierential equation contains regularly and rapidly varying nonlinearities in its right-hand side. Partial cases of the equation (1), which contains both power-type and exponential-type nonlinearities in the right-hand side, are found in practice, in particular, in the theory of combustion and in the theory of plasma. For example, during investigations of distribution of electrostatic potential in a cylindrical plasma volume of combustion products it have been aroused the nonlinear dierential equation that can be reduced to the next one: The equation (3) is an equation of the type (1) (1) is a natural generalization of equation (3) and plays an important role in the development of a qualitative theory of dierential equations.
The main aim of the article is the investigation of conditions of the existence of following class of solutions of the equation (1). Denition 1. The solution y of the equation (1), that is dened on the interval , if the following conditions take place This class of solutions was dened in the work of V. M. Evtukhov [3] for the n-th order differential equations of Emden-Fowler type and was concretized for the second-order equation. Due to the asymptotic properties of functions in the class of P ω (Y 0 , Y 1 , λ 0 )-solutions [6], every such solution belongs to one of four non-intersecting sets according to the value of λ 0 : λ 0 ∈ R\{0, 1}, λ 0 = 0, λ 0 = 1, λ 0 = ±∞. In this article we consider the case λ 0 = ±∞ of such solutions, every P ω (Y 0 , Y 1 , ±∞)-solution and its derivative satisfy the following limit relations lim t↑ω π ω (t)y (t) y(t) = 1, lim t↑ω π ω (t)y (t) This class P ω (Y 0 , Y 1 , ±∞)-solutions for equations of the form (1) is one of the most dicult to study due to the fact that the second-order derivative is not explicitly expressed through the rst-order derivative. From (4) it means that the derivative of the rst order of each such solution is a slowly varying function as t ↑ ω.
From the conditions (2) it also follows that the function ϕ 0 and its rst-order derivative belong to the class Γ Y 0 (Z 0 ), that was introduced in the works of V. M. Evtukhov and A. G. Chernikova [4] as a generalization of the class Γ (L. Khan, see, for example, [1], p. 75). The properties of the class Γ Y 0 (Z 0 ) were used to get our results.
For the equation (1), in previous works [2] the necessary and sucient conditions for the existence of the investigated class of P ω (Y 0 , Y 1 , ±∞)-solutions were established in case of the existence of some innite limit. In this work we establish the sucient conditions for the existence of P ω (Y 0 , Y 1 , ±∞)-solutions of the equation (1) in case this limit equals nonzero real number. We also have found the asymptotic representations of such solutions and its rst order derivatives as t ↑ ω and indicated the number of such solutions. 1 Section with results To formulate the main results, we introduce the following denitions Denition 3. We say that a slowly varying as Condition S is satised, for example, for such functions as ln |y|, | ln |y|| µ (µ ∈ R), ln ln |y|.
The following theorem is obtained in our previous work [2] and contains a necessary conditions for the existence the P ω (Y 0 , Y 1 , ±∞)-solution of the equation (1).
where L : [t 0 , ω[→ R is twice continuously dierentiable on ∆ Y 0 and satises the next conditions Thus, in the case of the existence of a nite or innite limit the following relations take place Let's introduce the following denition.
Denition 4. We say that the condition N is satised for the equation (1) if for some which satises conditions (6)(7) and (8), the following representation takes place where To formulate the sucient conditions for the existence the P ω (Y 0 , Y 1 , ±∞)-solution of the equation (1) let's introduce some notations.
For each of such solutions the following asymptotic representations take place as t ↑ ω.
We apply an additional transformation to the system (20) Finally we have where From the (6)(8), (11) and (12) it follows that Then we must to nd the lim t↑ω c 11 (t). Let's consider .

Then we have
The rst term from (23) equals to zero according to (11) and (12), the second term equals to − 1 γ due to the condition (13). So, we have This equation has no roots which real part equals to zero. In the case α 0 µ 0 > 0(α 0 µ 0 = 1) the characteristic equation has two real roots with opposite signs. So, according to the theorem 2.2 from [7] the system (22) has one-parameter family of solutions that tend to zero as the argument tends to +∞.
In the case α 0 µ 0 < 0(α 0 µ 0 = −1) the characteristic equation has two real roots with or two complex roots with the real part of the same sign as γ.