On the asymptotics of solutions of the second-order differential equations with rapidly varying nonlinearities
In the article it is considered the following differential equation \[ y''=\alpha_0 p(t)\varphi(y), \tag{1} \] where \(\alpha_{0}\in\{-1;1\}\), \(p:[a,\omega[~\to~]0,+\infty[\) \((-\infty<a<\omega \leq +\infty)\) is a continuous function, \(\varphi_0:\Delta_{Y_0}~\to~]0,+\infty[\) -- is a twice continuously differentiable on \(\Delta_{Y_0}\) function for which \[ \varphi'_0(y)\neq0 \text{ as } y \in~\Delta_{Y_0}, \ \ \lim_{\substack{y \rightarrow Y_0 \\ y \in \Delta_{Y_0}}}\varphi_0(y)\in\{0,+\infty\},\quad\lim_{\substack{y \rightarrow Y_0 \\ y \in\, \Delta_{Y_0}}} \frac{\varphi_0(y)\varphi_0''(y)}{\left(\varphi_0'( y)\right)^2}=1.\tag{2} \] Here \(Y_0\in\{0, \pm \infty \}\), \(\Delta_{Y_0}\) -- is some one-sided neighborhood of the \(Y_0\). From conditions (2) it follows that the functions \(\varphi_0\) and its derivatives of the first order are rapidly varying functions as \(y \rightarrow Y_0\). For \(Y_0=+\infty\) and \(Z_0=+\infty\), the function \(\varphi_0\) and its first-order derivative belong to the class of functions \(\Gamma\) introduced by Hahn (see, for example, [\textit{N. H. Bingham} et al., Regular variation. Cambridge University Press, Cambridge (1987; Zbl 0617.26001)]). By a change of variables, the authors extend the class \(\Gamma\) to a class \(\Gamma_0(Y_0)\) and formulate the properties of functions of such classes. It is clear that the function \(\varphi_0\) and its first-order derivative belong to the class of functions \(\Gamma_0(Y_0)\) as \(y \rightarrow Y_0\). The authors establish necessary and sufficient conditions for the existence of \(P_{\omega}(Y_0,\lambda_0)\)-solutions of equation (1) in a special case when \(\lambda_0 \in \mathbb{R}\backslash \{0,1\}\), as well as the asymptotic representations of these solutions and their first derivatives as \(t \uparrow \omega\) using the a priori asymptotic properties of these solutions. The main results are given in the Theorems 2.1, 2.2. In Theorem 2.1, necessary conditions for the existence of \(P_\omega (Y_0,\lambda_0)\)-solutions of the equation (1) are obtained. In Theorem 2.2 sufficient conditions for the existence of \(P_\omega (Y_0,\lambda_0)\)-solutions of the equation (1) and the number of such solutions are found. The third section is devoted to examples that illustrate the results of Theorems 2.1 and 2.2. In my opinion the results presented in the article are very important for the further study of asymptotic representations of solutions of second and higher-order differential equations with rapidly varying nonlinearities.
scientific article; zbMATH DE number 7439583