Asymptotic behavior of unbounded solutions of second-order differential equations with general nonlinearities (Q2307733)

The author considers the following second-order differential equation: \[ y''=p(x,y,y')|y|^{k_0}|y'|^{k_1}\operatorname{sign} (y,y'), \quad k_0,k_1\in \mathbb{R}, \,k_0>0,\, k_1>0, \tag{1} \] where the function $p(x,u,v)$ is continuous in all variables jointly and satisfies the Lipschitz condition in $u$, $v$ and the following inequality: \[ 0<m\leq p(x,u,v)\leq M<+\infty. \tag{2} \] The article consists of 4 parts. In the introduction, the author describes the object and formulates the main aim of the article -- to describe the qualitative behavior of solutions of Equation (1). In the second part, the author indicates all possible types of solutions of Equation (1). The third part of the article is devoted to establis the asymptotic behavior in the case of the following equation \[ y''=p_{0}|y|^{k_0} |y'|^{k_1} \operatorname{sign}(y,y'), \quad p_0>0, \quad k_0+k_1\neq1, \quad k_1\neq2.\tag{3} \] In Theorem 3 and Theorem 4, the author obtains conditions for the existence of Euation (3) of increasing singular solutions in the right neighborhood of some point as well as asymptotic representations of such solutions and their derivatives of the first order. It should be noted that the same results were obtained in the works of \textit{V. M. Evtukhov} (see, for example [Differ. Uravn. 28, No. 6, 1076--1078 (1992; Zbl 0834.34036)]) for generalized differential equations of Emden-Fowler type of the form \[ y''=\alpha_{0} p(t)|y|^{\sigma} |y'|^{\lambda} \operatorname{sign} y, \] where $\alpha_{0} \in \{ -1,1\} $, $p:[a,\omega \, [\to \, ]0,+\infty ]$ is a continuous function, $-\infty <\omega \le +\infty$. In the works of V. M. Evtukhov, under some constraints on the function $p$ ($p$ can also be a constant), there was described the asymptotic behavior for $ t \uparrow \omega $ of all possible types of monotone and various singular non-oscillating solutions in the case of $\sigma + \lambda \ne 1 $, and for arbitrary $ \sigma $ and $ \lambda $. Later, the results were generalized for equations containing on the right side the product of more general classes of nonlinearities than power ones (see, for example [\textit{M. A. Belozerova}, Nonlinear Oscil., N.Y. 12, No. 1, 1--14 (2009; Zbl 1277.34061); translation from Nelinijni Kolyvannya 12, No. 1, 3--15 (2009)]). Therefore, the results of Part 3 of the present article are not new. The results of Part 4, in which the general Equation (1) is considered, are the most interesting. Here, in Lemmas 1--5, the estimates of the solutions of Equation (1), whose modules tend to infinity as its argument approaches the right boundary of its domain, are proved. In Theorem 6, in the case $k_0+k_1\neq1\), \(k_1<2\) and $\alpha=\frac{2-k_1}{k_0+k_1-1}>0$, the asymptotic representations of increasing first-order singular solutions and their first-order derivatives in the right neighborhood of some point are obtained. Theorems 7 and 8 formulate similar results for the cases $k_0=k_1<0\), \(\alpha<-1$ and $k_1>2\), \(-1<\alpha<0$, respectively. It should also be noted that ``I.~T.~Kiguradze'' is correct in the list of references. In my opinion, the last results presented in the article are very important for the further study of asymptotic representations of solutions of second-order differential equations with nonlinearities of various types.