For a differential equation of the second order of the form $y''=\alpha_0 p(t)\varphi_0(y)|y'|^{\sigma_1},$ where $\alpha_0\in\{-1,1\}$, $p:[a,\omega[\longrightarrow]0,+\infty[$ is continuous function, $\varphi_0:\Delta_{Y_i}\longrightarrow ]0,+\infty[ \ $ is continuous regularly varying as $y\to Y_0 $ the function of $\sigma_0 $ order, and $\sigma_0+\sigma_1 = 1$, $\Delta_{Y_i} \ (i\in\{0,1\})$ is a one-side neighborhood of $Y_i$ and $Y_i \in\{0;\pm \infty \}\; (i\in\{0,1\})$, the question of the existence of solutions for which $\lim_{t\uparrow\omega} y^{(i)}(t)=Y_i$ $(i\in\{0,1\})$ is considered. Involvement in the 1980s in V.Marič, M. Tomič's works in the study of two-term second-order differential equations $y''=p(t)\varphi(y)$ with regularly varying nonlinearities in zero made it possible to find two-sides estimates of solutions tending to zero as $t\rightarrow+\infty$. Further study of two-term second-order differential equations with regularly varying nonlinearities, the right side of which preserves the sign in the neighborhood of singular point (both finite or equals $\pm \infty$) is carried out by Evtukhov V.M. on $P_\omega(\lambda_0)-$solutions, which arises in the study of generalized $n-$th order Emden - Fowler equations. Among the set of such solutions of equation under study we distinguish a fairly wide class of so-called $P_\omega(Y_0,Y_1,\lambda_0)$-solutions (generalization of $P_\omega(\lambda_0)-$solutions). The set of all $P_\omega(Y_0,Y_1,\lambda_0)-$solutions by its asymptotic properties separate into 4 disjoint classes of solutions corresponding to the values of $\lambda_0$: $\lambda_0\in \mathbb{R}\setminus\{0,1\}$ is nonsingular case, $\lambda_0=0,$ $\lambda_0=1$, $\lambda_0=\pm \infty$ are particular cases. This type of solution was previously introduced in the study of the two-term equation $ y''=\alpha_0p(t)\varphi_0(y)\varphi_1(y'),$ where, $\alpha_0\in\{-1,1\}$, $p:[a,\omega[\longrightarrow]0,+\infty[$ is continuous function, $\varphi_i:\Delta_{Y_i}\longrightarrow ]0,+\infty[ \ (i=0,1)$ are regularly varying as$z\to Y_i \ (i=0,1)$ functions of $\sigma_i \ (i=0,1)$ orders, and $\sigma_0+\sigma_1 \neq 1$. The case $\sigma_0+\sigma_1 =1$ corresponds to the so-called semilinear differential equations, which have a number of properties of both linear and nonlinear differential equations. Thus, for an equation $y''=p(t)|y|^{1-\lambda}|y'|^\lambda \mbox{sgn}\;y$ with some constraints on a function $p$ (in particular, if the function preserves the sign, it is locally absolutely continuous and $\int_a^\omega p^{\frac{1}{2-\lambda}}(t)\;dt=+\infty, $ $ \lim_{t\rightarrow \omega}p'(t)p^{\frac{\lambda-3}{2-\lambda}}(t)=l_0 $ $(|l_0|\leq+\infty),$ asymptotic representations are found as $t\rightarrow \omega$ for all types of proper solutions of this equation by Evtukhov V.M.. Here, for the equation we are studying, the necessary as well as sufficient conditions for the existence of $P_\omega(Y_0,Y_1,\lambda_0)$- solutions are found, asymptotic representations of such solutions and their first-order derivatives are established, and the number of parametric families of such solutions is indicated.

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