Title:
|
Continuability, boundedness, and convergence to zero of solutions of a perturbed nonlinear ordinary differential equation (English) |
Author:
|
Graef, John R. |
Author:
|
Spikes, Paul W. |
Language:
|
English |
Journal:
|
Czechoslovak Mathematical Journal |
ISSN:
|
0011-4642 (print) |
ISSN:
|
1572-9141 (online) |
Volume:
|
45 |
Issue:
|
4 |
Year:
|
1995 |
Pages:
|
663-683 |
. |
Category:
|
math |
. |
MSC:
|
34A34 |
MSC:
|
34C11 |
MSC:
|
34C15 |
MSC:
|
34D05 |
MSC:
|
34D10 |
idZBL:
|
Zbl 0851.34050 |
idMR:
|
MR1354925 |
DOI:
|
10.21136/CMJ.1995.128549 |
. |
Date available:
|
2009-09-24T09:51:39Z |
Last updated:
|
2020-07-29 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128549 |
. |
Reference:
|
[1] Z. S. Athanassov: Boundedness criteria for solutions of certain second order nonlinear differential equations.J. Math. Anal. Appl. 123 (1987), 461–479. Zbl 0642.34031, MR 0883702, 10.1016/0022-247X(87)90324-6 |
Reference:
|
[2] J. W. Baker: Stability properties of a second order damped and forced nonlinear differential equation.SIAM J. Appl. Math. 27 (1974), 159–166. Zbl 0289.34067, MR 0350138, 10.1137/0127013 |
Reference:
|
[3] R. J. Ballieu and K. Peiffer: Attractivity of the origin for the equation $\ddot{x} + f(t,x,\dot{x})\*|\dot{x}|^{\alpha } \dot{x} + g(x) = 0$.J. Math. Anal. Appl. 65 (1978), 321–332. MR 0506309 |
Reference:
|
[4] L. H. Erbe and Z. Liang: Qualitative behavior of a generalized Emden-Fowler differential system.Czech. Math. J. 41 (116) (1991), 454–466. MR 1117799 |
Reference:
|
[5] S. R. Grace and B. S. Lalli: Oscillation and convergence to zero of solutions of damped second order nonlinear differential equations.J. Math. Anal. Appl. 102 (1984), 539–548. MR 0755982, 10.1016/0022-247X(84)90191-4 |
Reference:
|
[6] J. R. Graef, L. Hatvani, J. Karsai, and P. W. Spikes: Boundedness and asymptotic behavior of solutions of second order nonlinear differential equations.Publ. Math. Debrecen 36 (1989), 85–99. MR 1047021 |
Reference:
|
[7] J. R. Graef and P. W. Spikes: Asymptotic behavior of solutions of a second order nonlinear differential equation.J. Differential Equations 17 (1975), 461–476. MR 0361275, 10.1016/0022-0396(75)90056-X |
Reference:
|
[8] J. R. Graef and P. W. Spikes: Asymptotic properties of solutions of a second order nonlinear differential equation.Publ. Math. Debrecen 24 (1977), 39–51. MR 0454188 |
Reference:
|
[9] J. R. Graef and P. W. Spikes: Boundedness and convergence to zero of solutions of a forced second-order nonlinear differential equation.J. Math. Anal. Appl. 62 (1978), 295–309. MR 0492527, 10.1016/0022-247X(78)90127-0 |
Reference:
|
[10] A. Halanay: Differential Equations: Stability, Oscillations, Time Lag.Academic Press, New York, 1966. MR 0216103 |
Reference:
|
[11] L. Hatvani: On the stability of the zero solution of certain second order non-linear differential equations.Acta Sci. Math. (Szeged) 32 (1971), 1–9. Zbl 0216.11704, MR 0306639 |
Reference:
|
[12] L. Hatvani: On the asymptotic behavior of the solutions of $(p(t)x^{\prime })^{\prime } + q(t)f(x) = 0$.Publ. Math. Debrecen 19 (1972), 225–237. MR 0326064 |
Reference:
|
[13] L. Hatvani: On the stability of the zero solution of nonlinear second order differential equations.Acta Sci. Math. (Szeged) (to appear). Zbl 0790.34046, MR 1243290 |
Reference:
|
[14] J. Karsai: Attractivity theorems for second order nonlinear differential equations.Publ. Math. Debrecen 30 (1983), 303–310. Zbl 0601.34039, MR 0739492 |
Reference:
|
[15] J. Karsai: Some attractivity results for second order nonlinear ordinary differential equations.in: Qualitative Theory of Differential Equations, B. Sz.-Nagy and L. Hatvani (eds.), Colloquia Mathematica Societatis János Bolyai, Vol. 53, North-Holland, Amsterdam, 1983, pp. 291–305. MR 1062654 |
Reference:
|
[16] Š. Kulcsár: Boundedness, convergence and global stability of solutions of a nonlinear differential equation of the second order.Publ. Math. Debrecen 37 (1990), 193–201. MR 1082298 |
Reference:
|
[17] Š. Kulcsár: Boundedness and stability of solutions of a certain nonlinear differential equation of the second order.Publ. Math. Debrecen 40 (1992), 57–70. MR 1154490 |
Reference:
|
[18] A. C. Lazer: A stability condition for the differential equation $y^{\prime \prime } + p(x)y = 0$.Michigan Math. J. 12 (1965), 193–196. MR 0176168, 10.1307/mmj/1028999309 |
Reference:
|
[19] K. S. Mamii and D. D. Mirzov: Properties of solutions of a second-order nonlinear differential equation on a half-axis.Differentsial’nye Uravneniya 7 (1971), 1330–1332. MR 0288349 |
Reference:
|
[20] S. N. Olekhnik: The boundedness of solutions of a second-order differential equation.Differentsial’nye Uravneniya 9 (1973), 1994–1999. (Russian) Zbl 0313.34031, MR 0333345 |
Reference:
|
[21] B. K. Sahoo: Asymptotic properties of solutions of a second order differential equation.Bull. Calcutta Math. Soc. 83 (1991), 209–226. Zbl 0755.34028, MR 1199402 |
Reference:
|
[22] F. J. Scott: New partial asymptotic stability results for nonlinear ordinary differential equations.Pacific J. Math. 72 (1977), 523–535. Zbl 0366.34040, MR 0466793, 10.2140/pjm.1977.72.523 |
Reference:
|
[23] F. J. Scott: On a partial asymptotic stability theorem of Willett and Wong.J. Math. Anal. Appl. 63 (1978), 416–420. Zbl 0383.34042, MR 0481295, 10.1016/0022-247X(78)90087-2 |
Reference:
|
[24] P. W. Spikes: Some stability type results for a nonlinear differential equation.Rend. Math. (6) 9 (1976), 259–271. Zbl 0346.34034, MR 0409980 |
Reference:
|
[25] N. Vornicescu: On the asymptotic behavior of the solutions of the differential equation $x^{\prime \prime } + fx = 0$.Bul. Stiint. Inst. Politehn. Cluj 14 (1971), 21–25. MR 0318608 |
Reference:
|
[26] P. X. Weng: Boundedness and asymptotic behavior of solutions of a second-order functional differential equation.Ann. Differential Equations 8 (1992), 367–378. Zbl 0765.34053, MR 1192175 |
Reference:
|
[27] D. Willett and J. S. W. Wong: Some properties of the solutions of $(p(t)x^{\prime })^{\prime } +q(t)f(x) = 0$.J. Math. Anal. Appl. 23 (1968), 15–24. MR 0226117 |
Reference:
|
[28] J. S. W. Wong: Some stability conditions for $x^{\prime \prime } + a(t) x^{2n - 1} = 0$.SIAM J. Appl. Math. 15 (1967), 889–892. MR 0221042, 10.1137/0115077 |
Reference:
|
[29] J. S. W. Wong: Remarks on stability conditions for the differential equation $x^{\prime \prime } +a(t)f(x) = 0$.J. Austral. Math. Soc. 9 (1969), 496–502. Zbl 0184.11905, MR 0241772, 10.1017/S144678870000745X |
Reference:
|
[30] E. H. Yang: Boundedness conditions for solutions of the differential equation $(a(t)x^{\prime })^{\prime } + f(t,x) = 0$.Nonlinear Anal. 8 (1984), 541–547. Zbl 0537.34030, MR 0741607, 10.1016/0362-546X(84)90092-0 |
. |