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Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

Part of: Boundary value problems

Published online by Cambridge University Press:  28 October 2014

John R. Graef ,
Johnny Henderson ,
Rodrica Luca  and
Yu Tian
John R. Graef
Affiliation:
Department of Mathematics, University of Tennesseeat Chattanooga, Chattanooga, TN 37403-2598, USA, (john-graef@utc.edu)
Johnny Henderson
Affiliation:
Department of Mathematics, Baylor UniversityWaco, TX 76798-7328, USA, (johnny_henderson@baylor.edu)
Rodrica Luca
Affiliation:
Department of Mathematics, Gheorghe Asachi Technical University, Iasi 700506, Romania, (rluca@math.tuiasi.ro)
Yu Tian
Affiliation:
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China (, tianyu2992@163.com)
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Abstract

For the third-order differential equation y′″ = ƒ(t, y, y′, y″), where , questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

Keywords

boundary-value problemLipschitz equationuniquenessexistencePontryagin maximum principleoptimal length interval

MSC classification

Primary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 183 - 197
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Aiboudi, M. and Brighi, B., On the solutions of a boundary value problem arising in free convection with prescribed flux, Arch. Math. 93 (2009), 165–174.Google Scholar
2.Bartušek, M. and Graef, J. R., Some limit-point/limit-circle results for third order differential equations, Discrete Contin. Dynam. Syst. 2001 (2001), 31–38.Google Scholar
3.Baxley, J. V. and Ballard, G. M., Existence of solutions for a class of singular nonlinear third order autonomous boundary value problems, Commun. Appl. Analysis 15(2) (2011), 195–202.Google Scholar
4.Brighi, B., On a similarity boundary layer equation, Z. Analysis Anwend. 21 (2002), 931–948.Google Scholar
5.Cabada, A., Grossinho, M. Do Rasário and Minhós, F., On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. Math. Analysis Applic. 285 (2003), 174–190.Google Scholar
6.Chen, S., Multiple solutions of a nonlinear boundary layer problem, Z. Analysis Anwend. 13 (1994), 83–88.CrossRefGoogle Scholar
7.Clark, S. and Henderson, J., Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations, Proc. Am. Math. Soc. 134(11) (2004), 3363–3372.CrossRefGoogle Scholar
8.Clark, S. and Henderson, J., Optimal interval lengths for nonlocal boundary value problems associated with third order Lipschitz equations, J. Math. Analysis Applic. 322 (2006), 468–576.Google Scholar
9.Eloe, P. W. and Henderson, J., Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations, J. Math. Analysis Applic. 331 (2007), 240–247.Google Scholar
10.Eloe, P. W. and Henderson, J., Optimal intervals for uniqueness of solutions for nonlocal boundary value problems, Commun. Appl. Nonlin. Analysis 18(3) (2011), 89–97.Google Scholar
11.Eloe, P. W. and Henderson, J., Uniqueness implies existence and uniqueness conditions for a class of (k + j)-point boundary value problems for nth order differential equations, Math. Nachr. 284(2) (2011), 229–239.CrossRefGoogle Scholar
12.Eloe, P. W., Khan, R. A. and Henderson, J., Uniqueness implies existence and uniqueness conditions for a class of (k + j)-point boundary value problems for nth order differential equations, Can. Math. Bull. 55(2) (2012), 285–296.Google Scholar
13.Gamkrelidze, R., Principles of optimal control theory (Plenum, New York, 1978).Google Scholar
14.Hankerson, D. and Henderson, J., Optimality for boundary value problems for Lipschitz equations, J. Diff. Eqns 77 (1989), 392–404.CrossRefGoogle Scholar
15.Hartman, P., On n-parameter families and interpolation problems for nonlinear ordinary differential equations, Trans. Am. Math. Soc. 154 (1971), 201–226.Google Scholar
16.Heidel, J. W., A third order differential equation arising in fluid mechanics, Z. Angew. Math. Mech. 53(3) (1973), 167–170.Google Scholar
17.Henderson, J., Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations, J. Diff. Eqns 41 (1981), 218–227.CrossRefGoogle Scholar
18.Henderson, J., Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlin. Analysis 5 (1981), 989–1002.Google Scholar
19.Henderson, J., Existence and uniqueness of solutions of right focal point boundary value problems for third and fourth order equations, Rocky Mt. J. Math. 14 (1984), 487–497.Google Scholar
20.Henderson, J., Best interval lengths for boundary value problems for third order Lipschitz equations, SIAM J. Math. Analysis 18 (1987), 293–305.CrossRefGoogle Scholar
21.Henderson, J., Uniqueness implies existence for three-point boundary value problems for second order differential equations, Appl. Math. Lett. 18 (2005), 905–909.CrossRefGoogle Scholar
22.Henderson, J., Existence and uniqueness of solutions of (k + 2)-point nonlocal boundary value problems for ordinary differential equations, Nonlin. Analysis 74 (2011), 2576–2584.Google Scholar
23.Henderson, J. and McGwier, R., Uniqueness, existence and optimality for fourth order Lipschitz equations, J. Diff. Eqns 67 (1987), 414–440.CrossRefGoogle Scholar
24.Henderson, J. and Pruet, S., Uniqueness of n-point boundary value problems, Panamer. Math. J. 3 (1993), 25–38.Google Scholar
25.Henderson, J., Karna, B. and Tisdell, C. C., Existence of solutions for three-point boundary value problems for second order equations, Proc. Am. Math. Soc. 133 (2005), 1365–1369.Google Scholar
26.Jackson, L. K., Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. Appl. Math. 24 (1973), 535–538.Google Scholar
27.Jackson, L. K., Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Diff. Eqns 13 (1973), 432–437.CrossRefGoogle Scholar
28.Jackson, L. K., Existence and uniqueness of solutions for boundary value problems for Lipschitz equations, J. Diff. Eqns 32 (1979), 76–90.Google Scholar
29.Jackson, L. K., Boundary value problems for Lipschitz equations, in Differential Equations, Proc. 8th Fall Conf., Oklahoma State University, Stillwater, OK, 1979, pp. 31–50 (Academic Press, 1980).Google Scholar
30.Jackson, L. K. and Schrader, K., Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Diff. Eqns 9 (1971), 46–54.Google Scholar
31.Klaasen, G., Existence theorems for boundary value problems for nth order ordinary differential equations, Rocky Mt. J. Math. 3 (1973), 457–472.CrossRefGoogle Scholar
32.Klokov, Yu. A., On upper and lower functions in boundary value problems for a third-order ordinary differential equation, Diff. Uravn. 36(12) (2000), 1607–1614.Google Scholar
33.Lasota, A. and Luczynski, M., A note on the uniqueness of two point boundary value problems, I, Zeszyty Naukowe UJ, Prace Matematyezne 12 (1968), 27–29.Google Scholar
34.Lasota, A. and Opial, Z., On the existence and uniqueness of solutions of a boundary value problem for an ordinary second order differential equation, Colloq. Math. 18 (1967), 1–5.CrossRefGoogle Scholar
35.Lee, E. and Markus, L., Foundations of optimal control (Wiley, 1967).Google Scholar
36.Liu, Z., Chen, H. and Liu, C., Positive solutions for singular third-order nonhomogeneous boundary value problems, J. Appl. Math. Comput. 38(1) (2012), 161–172.Google Scholar
37.Ma, D., Uniqueness implies uniqueness and existence for nonlocal boundary value problems for fourth order differential equations, Doctoral Dissertation, Baylor University, Waco, TX (2005).Google Scholar
38.Melentsova, Yu., A best possible estimate of the nonoscillation interval for a linear differential equation with coefficients bounded in Lr, Diff. Uravn. 13 (1977), 1776–1786.Google Scholar
39.Melentsova, Yu. and Mil'shtein, G., An optimal estimate of the interval on which a multipoint boundary value problem possesses a solution, Diff. Uravn. 10 (1974), 1630–1641.Google Scholar
40.Melentsova, Yu. and Mil'shtein, G., Optimal estimation of the nonoscillation interval for linear differential equations with bounded coefficients, Diff. Uravn. 17 (1981), 2160–2175.Google Scholar
41.Nieto, J. J., Periodic solutions for third order ordinary differential equations, Commentat. Math. Univ. Carolinae 32 (1991), 495–499.Google Scholar
42.Peterson, A. C., Existence-uniqueness for focal point boundary value problems, SIAM J. Math. Analysis 12 (1982), 602–610.Google Scholar
43.Peterson, D. E., Uniqueness, existence and comparison theorems for ordinary differential equations, Doctoral Dissertation, University of Nebraska, Lincoln, NE (1973).Google Scholar
44.Schrader, K., Uniqueness implies existence for solutions of nonlinear boundary value problems, Abstr. Am. Math. Soc. 6 (1985), 235.Google Scholar
45.Smirnov, S., On the third order boundary value problems with asymmetric nonlinearity, Nonlin. Analysis Model. Control 16(2) (2011), 231–241.Google Scholar
46.Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
47.Van Gorder, R. A., Vajravelu, K. and Pop, I., Hydromagnetic stagnation point flow of a viscous fluid over a stretching or shrinking sheet, Meccanica 47(1) (2012), 31–50.Google Scholar
48.Wang, J. and Zhang, Z., A boundary value problem from draining and coating flows involving a third-order ordinary differential equation, Z. Angew. Math. Phys. 49 (1998), 506–513.Google Scholar
49.Yang, G. C., Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory, Appl. Math. Lett. 16 (2003), 827–832.Google Scholar
50.Yu, H. and Pei, M., Solvability of a nonlinear third-order periodic boundary value problem, Appl. Math. Lett. 23(8) (2010), 892–896.Google Scholar
51.Zhang, Z., Concave solutions of a general self-similar boundary layer problem for power-law fluids, Nonlin. Analysis 13(6) (2012), 2708–2723.Google Scholar
52.Zhang, Z. and Wang, J., On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet, J. Math. Analysis Applic. 330(1) (2007), 207–220.CrossRefGoogle Scholar