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## Archive of Issues

Russia Moscow
Year
2016
Volume
26
Issue
2
Pages
215-220
 Section Mathematics Title Asymptotic behaviour of solutions to nonlinear differential equations with exponentially equivalent right-hand sides Author(-s) Zabolotskiy S.A.a Affiliations Lomonosov Moscow State Universitya Abstract Nonlinear $n$-th order differential equations with lower term are considered. With the help of the contraction mapping principle an asymptotic equivalence of solutions to these equations is investigated in the case of exponentially equivalent right-hand sides. Obtained sufficient conditions for asymptotic equivalence of solutions extend and generalize results stated in previous author’s papers. The result, describing the asymptotic behaviour of all tending to zero at infinity solutions to second order differential equations with regular Emden-Fowler type nonlinearity and zero right-hand side appearing while investigating quasilinear elliptic equations, is stated. On the basis of this result the asymptotic behaviour of solutions to a corresponding equation with nonzero right-hand side is described. Keywords nonlinear ordinary differential equations, asymptotic equivalence UDC 517.928.1 MSC 34C41, 34E10 DOI 10.20537/vm160207 Received 17 May 2016 Language Russian Citation Zabolotskiy S.A. Asymptotic behaviour of solutions to nonlinear differential equations with exponentially equivalent right-hand sides, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 215-220. References Bellman R. Stability theory of differential equations, New York: McGraw-Hill, 1953, 166 p. 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. 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 equivalence of nonlinear differential equations, Differ. Uravn., 1996, vol. 32, no. 6, p. 855 (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. Zabolotskiy S.A. On asymptotic equivalence of Lane-Emden type differential equations and some generalizations, Functional Differential Equations, 2015, vol. 22, no. 3-4, pp. 169-177. Zabolotskiy S.A. On asymptotic equivalence of solutions to Lane-Emden type equations with power coefficient, Differ. Uravn., 2015, vol. 51, no. 6, p. 832 (in Russian). Astashova I.V. On asymptotic equivalence of $n$-th order nonlinear differential equations, Tatra Mt. Math. Publ., 2015, vol. 63, pp. 31-38. Egorov Yu.V., Kondrat'ev V.A., Oleinik O.A. Asymptotic behaviour of the solutions of non-linear elliptic and parabolic systems in tube domains, Sbornik: Mathematics, vol. 189, no. 3, pp. 45-68 (in Russian). Reinfelds A. Asymptotic equivalence of difference equations in Banach space, Theory and Applications of Difference Equations and Discrete Dynamical Systems, Eds.: Z. AlSharawi, J.M. Cushing, S. Elaydi, Springer, 2014, vol. 102, pp. 215-222. Full text