Title:
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On the oscillatory integration of some ordinary differential equations (English) |
Author:
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Mustafa, Octavian G. |
Language:
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English |
Journal:
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Archivum Mathematicum |
ISSN:
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0044-8753 (print) |
ISSN:
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1212-5059 (online) |
Volume:
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44 |
Issue:
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1 |
Year:
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2008 |
Pages:
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23-36 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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Conditions are given for a class of nonlinear ordinary differential equations $x^{\prime \prime }+a(t)w(x)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x(t)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\prime }(t)-\frac{x(t)}{t}$. (English) |
Keyword:
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ordinary differential equation |
Keyword:
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asymptotic integration |
Keyword:
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prescribed asymptote |
Keyword:
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non-oscillation of solutions |
MSC:
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34C10 |
MSC:
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34D05 |
MSC:
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34E05 |
MSC:
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34K25 |
idZBL:
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Zbl 1212.34145 |
idMR:
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MR2431228 |
. |
Date available:
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2008-06-06T22:52:36Z |
Last updated:
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2013-09-19 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/108093 |
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Reference:
|
[1] Agarwal, R. P., Djebali, S., Moussaoui, T., Mustafa, O. G.: On the asymptotic integration of nonlinear differential equations.J. Comput. Appl. Math. 202 (2007), 352–376. Zbl 1123.34038, MR 2319962, 10.1016/j.cam.2005.11.038 |
Reference:
|
[2] Agarwal, R. P., Djebali, S., Moussaoui, T., Mustafa, O. G., Rogovchenko, Yu. V.: On the asymptotic behavior of solutions to nonlinear ordinary differential equations.Asymptot. Anal. 54 (2007), 1–50. MR 2356463 |
Reference:
|
[3] Agarwal, R. P., Mustafa, O. G.: A Riccatian approach to the decay of solutions of certain semi-linear PDE’s.Appl. Math. Lett. 20 (2007), 1206–1210. Zbl 1137.35356, MR 2384247, 10.1016/j.aml.2006.11.015 |
Reference:
|
[4] Atkinson, F. V.: On second order nonlinear oscillation.Pacific J. Math. 5 (1995), 643–647. MR 0072316, 10.2140/pjm.1955.5.643 |
Reference:
|
[5] Bartušek, M., Došlá, Z., Graef, J. R.: The nonlinear limit-point/limit-circle problem.limit-circle problem, Birkhäuser, Boston, 2004. Zbl 1052.34021, MR 2020682 |
Reference:
|
[6] Cecchi, M., Marini, M., Villari, G.: Integral criteria for a classification of solutions of linear differential equations.J. Differential Equations 99 (1992), 381–397. Zbl 0761.34009, MR 1184060, 10.1016/0022-0396(92)90027-K |
Reference:
|
[7] Cecchi, M., Marini, M., Villari, G.: Comparison results for oscillation of nonlinear differential equations.NoDEA Nonlinear Differential Equations Appl. 6 (1999), 173–190. Zbl 0927.34023, MR 1694795, 10.1007/s000300050071 |
Reference:
|
[8] Constantin, A.: Existence of positive solutions of quasilinear elliptic equations.Bull. Austral. Math. Soc. 54 (1996), 147–154. Zbl 0878.35040, MR 1402999, 10.1017/S0004972700015148 |
Reference:
|
[9] Constantin, A.: Positive solutions of quasilinear elliptic equations.J. Math. Anal. Appl. 213 (1997), 334–339. Zbl 0891.35033, MR 1469378, 10.1006/jmaa.1997.5541 |
Reference:
|
[10] Deng, J.: Bounded positive solutions of semilinear elliptic equations.J. Math. Anal. Appl. 336 (2007), 1395–1405. Zbl 1152.35367, MR 2353022, 10.1016/j.jmaa.2007.03.071 |
Reference:
|
[11] Djebali, S., Moussaoui, T., Mustafa, O. G.: Positive evanescent solutions of nonlinear elliptic equations.J. Math. Anal. Appl. 333 (2007), 863–870. Zbl 1155.35029, MR 2331699, 10.1016/j.jmaa.2006.12.004 |
Reference:
|
[12] Dugundji, J., Granas, A.: Fixed point theory I.Polish Sci. Publ., Warszawa, 1982. MR 0660439 |
Reference:
|
[13] Ehrnström, M.: Positive solutions for second-order nonlinear differential equations.Nonlinear Anal. 64 (2006), 1608–1620. Zbl 1101.34022, MR 2200162, 10.1016/j.na.2005.07.010 |
Reference:
|
[14] Ehrnström, M., Mustafa, O. G.: On positive solutions of a class of nonlinear elliptic equations.Nonlinear Anal. 67 (2007), 1147–1154. Zbl 1165.35016, MR 2325368, 10.1016/j.na.2006.07.002 |
Reference:
|
[15] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order.Springer-Verlag, Berlin, 2001. Zbl 1042.35002, MR 1814364 |
Reference:
|
[16] Hartman, P.: On non-oscillatory linear differential equations of second order.Amer. J. Math. 74 (1952), 389–400. Zbl 0048.06602, MR 0048667, 10.2307/2372004 |
Reference:
|
[17] Heidel, J. W.: A nonoscillation theorem for a nonlinear second order differential equation.Proc. Amer. Math. Soc. 22 (1969), 485–488. Zbl 0169.42203, MR 0248396, 10.1090/S0002-9939-1969-0248396-0 |
Reference:
|
[18] Hesaaraki, M., Moradifam, A.: On the existence of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains.Appl. Math. Lett. 20 (2007), 1227–1231. Zbl 1137.35429, MR 2384252, 10.1016/j.aml.2007.03.001 |
Reference:
|
[19] Kiguradze, I. T., Chanturia, T. A.: Asymptotic properties of solutions of nonautonomous ordinary differential equations.Kluwer, Dordrecht, 1993. Zbl 0782.34002 |
Reference:
|
[20] Mustafa, O. G.: Initial value problem with infinitely many linear-like solutions for a second order differential equation.Appl. Math. Lett. 18 (2005), 931–934. Zbl 1095.34505, MR 2152306, 10.1016/j.aml.2004.07.036 |
Reference:
|
[21] Mustafa, O. G.: On the existence of solutions with prescribed asymptotic behaviour for perturbed nonlinear differential equations of second order.Glasgow Math. J. 47 (2005), 177–185. Zbl 1072.34049, MR 2200965, 10.1017/S0017089504002228 |
Reference:
|
[22] Mustafa, O. G., Rogovchenko, Yu. V.: Global existence and asymptotic behavior of solutions of nonlinear differential equations.Funkcial. Ekvac. 47 (2004), 167–186. Zbl 1118.34046, MR 2108671, 10.1619/fesi.47.167 |
Reference:
|
[23] Mustafa, O. G., Rogovchenko, Yu. V.: Limit-point type solutions of nonlinear differential equations.J. Math. Anal. Appl. 294 (2004), 548–559. Zbl 1057.34023, MR 2061342, 10.1016/j.jmaa.2004.02.029 |
Reference:
|
[24] Mustafa, O. G., Rogovchenko, Yu. V.: Asymptotic integration of a class of nonlinear differential equations.Appl. Math. Lett. 19 (2006), 849–853. Zbl 1126.34339, MR 2240473, 10.1016/j.aml.2005.10.013 |
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