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Boundary-value problems for third-order Lipschitz ordinary differential equations. (English) Zbl 1355.34048

In this paper, the authors consider a third-order boundary value problems, in which the nonlinear term satisfies a Lipschitz condition. The aim of this paper is to establish such results as “uniqueness implies uniqueness”,“uniqueness implies existence” and “optimal length subintervals of \((a,b)\) on which solutions are unique” et al. The main methods are shooting method and Pontryagin’s maximum principle.
Reviewer: Minghe Pei (Jilin)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] SIAM J. Math. Analysis 12 pp 602– (1982)
[2] Commun. Appl. Nonlin. Analysis 18 pp 89– (2011)
[3] Commentai,. Math. Univ. Carolinae 32 pp 495– (1991)
[4] Diff. Uravn. 17 pp 2160– (1981)
[5] DOI: 10.1016/j.jmaa.2006.08.087 · Zbl 1396.34011 · doi:10.1016/j.jmaa.2006.08.087
[6] DOI: 10.1016/j.jmaa.2005.09.017 · Zbl 1103.34004 · doi:10.1016/j.jmaa.2005.09.017
[7] Diff. Uravn. 10 pp 1630– (1974)
[8] Proc. Am. Math. Soc. 134 pp 3363– (2004)
[9] Lr, Diff. Uravn. 13 pp 1776– (1977)
[10] Z. Analysis Anwend. 13 pp 83– (1994)
[11] DOI: 10.1016/S0022-247X(03)00388-3 · Zbl 1048.34033 · doi:10.1016/S0022-247X(03)00388-3
[12] DOI: 10.1007/s12190-010-0470-z · Zbl 1303.34019 · doi:10.1007/s12190-010-0470-z
[13] DOI: 10.4171/ZAA/1118 · Zbl 1041.34009 · doi:10.4171/ZAA/1118
[14] Commun. Appl. Analysis 15 pp 195– (2011)
[15] Foundations of optimal control (1967)
[16] Discrete Contin. Dynam. Syst. 2001 pp 31– (2001)
[17] Colloq. Math. 18 pp 1– (1967)
[18] DOI: 10.1007/s00013-009-0023-6 · Zbl 1192.34033 · doi:10.1007/s00013-009-0023-6
[19] Zeszyty Naukowe UJ, Prace Matematyezne 12 pp 27– (1968)
[20] Diff. Uravn. 36 pp 1607– (2000)
[21] DOI: 10.1216/RMJ-1973-3-3-457 · Zbl 0268.34025 · doi:10.1216/RMJ-1973-3-3-457
[22] J. Diff. Eqns 9 pp 46– (1971)
[23] Differential Equations, Proc. 8th Fall Conf., Oklahoma State University, Stillwater, OK, 1979, pp 31– (1980)
[24] DOI: 10.1016/0022-0396(79)90052-4 · Zbl 0407.34018 · doi:10.1016/0022-0396(79)90052-4
[25] DOI: 10.1016/0022-0396(73)90002-8 · Zbl 0256.34018 · doi:10.1016/0022-0396(73)90002-8
[26] DOI: 10.1137/0124054 · Zbl 0237.34030 · doi:10.1137/0124054
[27] DOI: 10.1090/S0002-9939-04-07647-6 · Zbl 1061.34009 · doi:10.1090/S0002-9939-04-07647-6
[28] Panamer. Math. J. 3 pp 25– (1993)
[29] DOI: 10.1016/0022-0396(87)90135-5 · Zbl 0642.34005 · doi:10.1016/0022-0396(87)90135-5
[30] DOI: 10.1016/j.na.2010.11.048 · Zbl 1219.34024 · doi:10.1016/j.na.2010.11.048
[31] DOI: 10.1016/j.aml.2004.07.032 · Zbl 1092.34507 · doi:10.1016/j.aml.2004.07.032
[32] DOI: 10.1016/j.jmaa.2006.07.059 · Zbl 1119.76072 · doi:10.1016/j.jmaa.2006.07.059
[33] DOI: 10.1137/0518023 · Zbl 0668.34017 · doi:10.1137/0518023
[34] DOI: 10.1016/j.nonrwa.2012.01.028 · Zbl 1312.34069 · doi:10.1016/j.nonrwa.2012.01.028
[35] DOI: 10.1016/j.aml.2010.04.005 · Zbl 1197.34031 · doi:10.1016/j.aml.2010.04.005
[36] DOI: 10.1216/RMJ-1984-14-2-487 · Zbl 0542.34014 · doi:10.1216/RMJ-1984-14-2-487
[37] DOI: 10.1016/0362-546X(81)90058-4 · Zbl 0468.34010 · doi:10.1016/0362-546X(81)90058-4
[38] DOI: 10.1016/S0893-9659(03)90003-6 · Zbl 1054.34037 · doi:10.1016/S0893-9659(03)90003-6
[39] DOI: 10.1016/0022-0396(81)90058-9 · Zbl 0438.34015 · doi:10.1016/0022-0396(81)90058-9
[40] DOI: 10.1007/s000000050104 · Zbl 0907.34016 · doi:10.1007/s000000050104
[41] DOI: 10.1002/zamm.19730530304 · doi:10.1002/zamm.19730530304
[42] DOI: 10.1007/s11012-010-9402-0 · Zbl 1293.76179 · doi:10.1007/s11012-010-9402-0
[43] Trans. Am. Math. Soc. 154 pp 201– (1971)
[44] Algebraic topology (1966) · Zbl 0956.55004
[45] DOI: 10.1016/0022-0396(89)90151-4 · Zbl 0684.34024 · doi:10.1016/0022-0396(89)90151-4
[46] Nonlin. Analysis Model. Control 16 pp 231– (2011)
[47] Principles of optimal control theory (1978)
[48] Abstr. Am. Math. Soc. 6 pp 235– (1985)
[49] DOI: 10.4153/CMB-2011-117-0 · Zbl 1254.34029 · doi:10.4153/CMB-2011-117-0
[50] DOI: 10.1002/mana.200810190 · Zbl 1242.34033 · doi:10.1002/mana.200810190
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