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## Archive of Issues

Russia Moscow
Year
2016
Volume
26
Issue
2
Pages
207-214
 Section Mathematics Title On asymptotic behaviour of solutions with infinite derivative for regular second-order Emden-Fowler type differential equations with negative potential Author(-s) Dulina K.M.a Affiliations Lomonosov Moscow State Universitya Abstract In this paper we consider the second-order Emden-Fowler type differential equation with negative potential $y''-p(x, y, y') |y|^k \text{ sgn } y=0$ in case of regular nonlinearity $k>1$. We assume that the function $p(x, u, v)$ is continuous in $x$ and Lipschitz continuous in two last variables. We investigate asymptotic behaviour of non-extensible solutions to the equation above. We consider the case of a positive function $p(x, u, v)$ unbounded from above and bounded away from 0 from below. The conditions guaranteeing an existence of a vertical asymptote of all nontrivial non-extensible solutions to the equation are obtained. Also the sufficient conditions providing the following solutions' properties $\displaystyle \lim_{x \to a} |y'(x)| = +\infty$, $\displaystyle \lim_{x \to a} |y(x)| <+ \infty$, where $a < \infty$ is a boundary point, are obtained. Keywords second-order Emden-Fowler type differential equations, regular nonlinearity, asymptotic behaviour UDC 517.925.44 MSC 34C11, 34E10 DOI 10.20537/vm160206 Received 14 May 2016 Language Russian Citation Dulina K.M. On asymptotic behaviour of solutions with infinite derivative for regular second-order Emden-Fowler type differential equations with negative potential, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 207-214. References Kiguradze I.T., Chanturiya T.A. Asimptoticheskie svoistva reshenii neavtonomnykh obyknovennykh differentsial'nykh uravnenii (Asymptotic properties of solutions of nonautonomous ordinary differential equations), Moscow: Nauka, 1990, 432 p. Kondrat'ev V.A., Nikishkin V.A. On the positive solutions of the equation $y''=p(x)y$$k$ , Nekotorye voprosy kachestvennoi teorii differentsial'nykh uravnenii i teorii upravleniya dvizheniem (Some problems of the qualitative theory of differential equations and the theory of motion control), Saransk, 1980, pp. 131-141 (in Russian). Astashova I. On asymptotic behavior of solutions to a quasilinear second order differential equation, Functional Differential Equations, 2009, vol. 16, no. 1, pp. 93-115. Astashova I.V. Qualitative properties of solutions to quasilinear ordinary differential equations, Kachestvennye svoistva reshenii differentsial'nykh uravnenii i smezhnye voprosy spektral'nogo analiza (Qualitative properties of solutions to differential equations and related topics of spectral analysis), Moscow: Unity-Dana, 2012, pp. 22-288 (in Russian). Astashova I.V. On asymptotic classification of solutions to nonlinear third- and fourth- order differential equations with power nonlinearity, Vestn. Mosk. Gos. Tekh. Univ. Im. N. Eh. Baumana, Ser. Estestv. Nauki, 2015, no. 2, pp. 3-25 (in Russian). Dulina K.M., Korchemkina T.A. Asymptotic classification of solutions to second-order Emden-Fowler type differential equations with negative potential, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2015, no. 6 (128), pp. 50-56 (in Russian). Dulina K.M., Korchemkina T.A. Classification of solutions to singular nonlinear second-order Emden-Fowler type equations, Proceedings of the International Conference and School for Young Scientists “Information Technology and Nanotechnology”, Samara Research Centre of RAS, Samara, 2015, pp. 45-46 (in Russian). Astashova I. Application of dynamical systems to the study of asymptotic properties of solutions to nonlinear higher-order differential equations, Journal of Mathematical Sciences, 2005, vol. 126, no. 5, pp. 1361-1391. Astashova I.V. Uniform estimates of positive solutions to quasilinear differential equations, Izvestiya: Mathematics, 2008, vol. 72, no. 6, pp. 1141-1160. Dulina K.M., Korchemkina T.A. On existence of solutions to second-order Emden-Fowler type differential equations with prescribed domain, Proceedings of the International Conference “Qualitative theory of differential equations and applications”, Moscow State University of Economics, Statistics, and Informatics, 2014, pp. 19-27 (in Russian). Full text