Asymptotic representations of solutions of second
order differential equations with nonlinearities close to regularly varying

M. A. Belozerova; G. A. Gerzhanovskaya

doi:10.15330/ms.44.2.204-214

The asymptotic representation, necessary and sufficient conditions of the existence of sufficient broad classes of the solutions are found for differential equations of the second order that are in some sense similar to equations with nonlinearities, that are regularly varying at the singular points.

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2. Evtukhov V.М. Asymtotic representaions of solutions of one class of nonlinear differential equations of the second order// Reports of Academy of sciences GSSR. – 1982. – V.106, №3. – P. 473–476. (in Russian)

3. Evtukhov V.М. Asymtotic properties of solutions of one class of differential equations of the second order// Math. Nachr. – 1984. – B.115. – P. 215–236.

4. Seneta Е. Regularly varying functions. – Lecture notes in Mathematics 508. SpringerVerlag, Berlin Heidelberg New York. – 1976, 113 p.

5. Evtukhov V.М. Asymptotic representations of solutions of nonautonomuous ordinary differential equations: thesis of a Doctor of sciences: 01.01.02 – Kyiv. – 1998. – 295 p. (in Russian)

6. Bilozerova M.O. Asymtotic properties of one class of solutions of essentialy nonlinear differential equations of the second order// Mat. Stud. – 2008. – V.29, №1. – P. 52–62. (in Russian)

7. Bilozerova M.O. Asymptotic representations of solutions of nonautomous differential equations of the second order with nonlinerities, that are in some sence near to the power nonlinearities// Nonlinear oscillations. – 2009. – V.12, №1. – P. 3–15. (in Russian)

8. Bilozerova M.O. Asymptotic representations of solutions of differential equations of the second order with nonlinerities, that are in some sence near to the power nonlinearities: thesis of a Candidate of sciences: 01.01.02 – Odessa. – 2009. – 121 p. (in Russian)

9. Maric V. Regular variation and differential equations. – Lecture notes in Mathematics 1726. SpringerVerlag, Berlin Heidelberg New York. – 2000. – 128 p.

10. Evtukhov V.М., Samojlenko A.M. Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point// Ukr. Math. J. – 2010. – V.62, №1. – P. 52–80. (in Russian)

11. Kusik L.I. Existence conditions and asymptotics of a class of solutions of differential equations of the second order// Mat. Stud. – 2014. – V.41, №2. – P. 184–197. (in Russian)

12. Bilozerova M.O. Asymptotic representations of solutions with slowly varying derivatives of essentially nonlinear second order differential equations// Bulletin of the Odessa National University. Mathematics and Mechanics. – 2015. – V.20, №1(25). – P. 7–19 (in Russian)

	Asymptotic representations of solutions of second order differential equations with nonlinearities close to regularly varying (in Ukrainian)
Author	M. A. Belozerova, G. A. Gerzhanovskaya hello_greta@mail.ru Odessa I.I.Mechnikov National University, Institute of Mathematics, Economics and Mechanics
Abstract	The asymptotic representation, necessary and sufficient conditions of the existence of sufficient broad classes of the solutions are found for differential equations of the second order that are in some sense similar to equations with nonlinearities, that are regularly varying at the singular points.
Keywords	asymptotic representation of solution; regularly varying nonlinearities
DOI	doi:10.15330/ms.44.2.204-214
Reference	1. Evtukhov V.М. About one nonlinear differential eqution of the second order// Roceedings of USSR Academy of sciences. – 1977. – V.233, №4. – P. 531–534. (in Russian) 2. Evtukhov V.М. Asymtotic representaions of solutions of one class of nonlinear differential equations of the second order// Reports of Academy of sciences GSSR. – 1982. – V.106, №3. – P. 473–476. (in Russian) 3. Evtukhov V.М. Asymtotic properties of solutions of one class of differential equations of the second order// Math. Nachr. – 1984. – B.115. – P. 215–236. 4. Seneta Е. Regularly varying functions. – Lecture notes in Mathematics 508. SpringerVerlag, Berlin Heidelberg New York. – 1976, 113 p. 5. Evtukhov V.М. Asymptotic representations of solutions of nonautonomuous ordinary differential equations: thesis of a Doctor of sciences: 01.01.02 – Kyiv. – 1998. – 295 p. (in Russian) 6. Bilozerova M.O. Asymtotic properties of one class of solutions of essentialy nonlinear differential equations of the second order// Mat. Stud. – 2008. – V.29, №1. – P. 52–62. (in Russian) 7. Bilozerova M.O. Asymptotic representations of solutions of nonautomous differential equations of the second order with nonlinerities, that are in some sence near to the power nonlinearities// Nonlinear oscillations. – 2009. – V.12, №1. – P. 3–15. (in Russian) 8. Bilozerova M.O. Asymptotic representations of solutions of differential equations of the second order with nonlinerities, that are in some sence near to the power nonlinearities: thesis of a Candidate of sciences: 01.01.02 – Odessa. – 2009. – 121 p. (in Russian) 9. Maric V. Regular variation and differential equations. – Lecture notes in Mathematics 1726. SpringerVerlag, Berlin Heidelberg New York. – 2000. – 128 p. 10. Evtukhov V.М., Samojlenko A.M. Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point// Ukr. Math. J. – 2010. – V.62, №1. – P. 52–80. (in Russian) 11. Kusik L.I. Existence conditions and asymptotics of a class of solutions of differential equations of the second order// Mat. Stud. – 2014. – V.41, №2. – P. 184–197. (in Russian) 12. Bilozerova M.O. Asymptotic representations of solutions with slowly varying derivatives of essentially nonlinear second order differential equations// Bulletin of the Odessa National University. Mathematics and Mechanics. – 2015. – V.20, №1(25). – P. 7–19 (in Russian)
Pages	204-214
Volume	44
Issue	2
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Asymptotic representations of solutions of second order differential equations with nonlinearities close to regularly varying (in Ukrainian)