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On the asymptotic behavior of monotone solutions of nonlinear differential equations of Emden-Fowler type. (Russian) Zbl 0834.34036

We consider the second-order nonlinear differential equation \[ y''= \alpha_0 p(t)|y|_{\sigma}|y'|_{\lambda},\tag{1} \] where \(\alpha_0\in \{-1, 1\}\), \(\sigma, \lambda\in \mathbb{R}\), \((\sigma+ \lambda- 1)(1- \lambda)\neq 0\) and \(p: [a, \omega)\to (0, +\infty)\) \((- \infty< a< \omega\leq+ \infty)\) is a locally summable function.
In earlier studies of the asymptotic behavior of the positive solution \(y\) of equation (1) defined in a left neighborhood of \(\omega\) and satisfying the conditions \(\lim_{t\to \infty} y(t)=\) either 0 or \(+\infty\) and \(\lim_{t\to \omega} y'(t)=\) either 0 or \(\pm\infty\) in both the case \(\lambda= 0\) and the case \(\lambda\neq 0\) it was assumed that \(p\in C_k([a, \omega))\), where \(k\geq 1\). We try to remove this constraint on the smoothness of the function \(p\). We use a method of study that differs in a number of important points from methods applied earlier.

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
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