Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations

Clark, Stephen; Henderson, Johnny

doi:10.1090/S0002-9939-06-08368-7

Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations
HTML articles powered by AMS MathViewer

by Stephen Clark and Johnny Henderson PDF

Proc. Amer. Math. Soc. 134 (2006), 3363-3372 Request permission

Abstract:

For the third order differential equation, $y''’ = f(x,y,y’,y''),$ we consider uniqueness implies existence results for solutions satisfying the nonlocal $4$-point boundary conditions, $y(x_1) = y_1,$ $y(x_2) = y_2,$ $y(x_3) - y(x_4) = y_3.$ Uniqueness of solutions of such boundary value problems is intimately related to solutions of the third order equation satisfying certain nonlocal $3$-point boundary conditions. These relationships are investigated as well.

References

Chuanzhi Bai and Jinxuan Fang, Existence of multiple positive solutions for nonlinear $m$-point boundary value problems, J. Math. Anal. Appl. 281 (2003), no. 1, 76–85. MR 1980075
David M. Goecke and Johnny Henderson, Uniqueness of solutions of right focal problems for third order differential equations, Nonlinear Anal. 8 (1984), no. 3, 253–259. MR 738010, DOI 10.1016/0362-546X(84)90047-6
Chaitan P. Gupta and Sergej I. Trofimchuk, Solvability of a multi-point boundary value problem and related a priori estimates, Canad. Appl. Math. Quart. 6 (1998), no. 1, 45–60. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1996). MR 1638415
Philip Hartman, Unrestricted $n$-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123–142. MR 105470, DOI 10.1007/BF02854523
Philip Hartman, On $N$-parameter families and interpolation problems for nonlinear ordinary differential equations, Trans. Amer. Math. Soc. 154 (1971), 201–226. MR 301277, DOI 10.1090/S0002-9947-1971-0301277-X
Johnny Henderson, Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations, J. Differential Equations 41 (1981), no. 2, 218–227. MR 630990, DOI 10.1016/0022-0396(81)90058-9
Johnny Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlinear Anal. 5 (1981), no. 9, 989–1002. MR 633013, DOI 10.1016/0362-546X(81)90058-4
Johnny Henderson, Existence theorems for boundary value problems for $n$th-order nonlinear difference equations, SIAM J. Math. Anal. 20 (1989), no. 2, 468–478. MR 982673, DOI 10.1137/0520032
Johnny Henderson, Focal boundary value problems for nonlinear difference equations. I, II, J. Math. Anal. Appl. 141 (1989), no. 2, 559–567, 568–579. MR 1009063, DOI 10.1016/0022-247X(89)90197-2
Johnny Henderson, Focal boundary value problems for nonlinear difference equations. I, II, J. Math. Anal. Appl. 141 (1989), no. 2, 559–567, 568–579. MR 1009063, DOI 10.1016/0022-247X(89)90197-2
Johnny Henderson, Uniqueness implies existence for three-point boundary value problems for second order differential equations, Appl. Math. Lett. 18 (2005), no. 8, 905–909. MR 2152302, DOI 10.1016/j.aml.2004.07.032
Johnny Henderson, Basant Karna, and Christopher C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1365–1369. MR 2111960, DOI 10.1090/S0002-9939-04-07647-6
Johnny Henderson and William Yin, Existence of solutions for fourth order boundary value problems on a time scale, J. Difference Equ. Appl. 9 (2003), no. 1, 15–28. In honour of Professor Allan Peterson on the occasion of his 60th birthday, Part II. MR 1958300, DOI 10.1080/1023619031000060954
Lloyd K. Jackson, Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. Appl. Math. 24 (1973), 535–538. MR 322253, DOI 10.1137/0124054
Lloyd K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Differential Equations 13 (1973), 432–437. MR 335925, DOI 10.1016/0022-0396(73)90002-8
Lloyd Jackson and Keith Schrader, Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Differential Equations 9 (1971), 46–54. MR 269920, DOI 10.1016/0022-0396(70)90152-X
Gene A. Klaasen, Existence theorems for boundary value problems of $n$th order ordinary differential equations, Rocky Mountain J. Math. 3 (1973), 457–472. MR 357944, DOI 10.1216/RMJ-1973-3-3-457
Andrzej Lasota and Marian Łuczyński, A note on the uniqueness of two point boundary value problems. I, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 12 (1968), 27–29. MR 224900
A. Lasota and Z. Opial, 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. MR 219792, DOI 10.4064/cm-18-1-1-5
Ruyun Ma, Existence theorems for a second order three-point boundary value problem, J. Math. Anal. Appl. 212 (1997), no. 2, 430–442. MR 1464888, DOI 10.1006/jmaa.1997.5515
Ruyun Ma, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl. Math. Lett. 15 (2002), no. 2, 211–216. MR 1880760, DOI 10.1016/S0893-9659(01)00120-3
Allan C. Peterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1981), no. 2, 173–185. MR 605428, DOI 10.1137/0512018
K. Schrader, Uniqueness implies existence for solutions of nonlinear boundary value problems, Abstracts Amer. Math. Soc. 6 (1985), 235.
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112