Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Stephen Clark and Johnny Henderson
Journal: Proc. Amer. Math. Soc. 134 (2006), 3363-3372
MSC (2000): Primary 34B15; Secondary 34B10
DOI: https://doi.org/10.1090/S0002-9939-06-08368-7
Published electronically: May 18, 2006
MathSciNet review: 2231921
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. C. Bai and J. Fang, Existence of multiple positive solutions for $ m$-point boundary value problems, J. Math. Anal. Appl. 281 (2003), 76-85. MR 1980075 (2004b:34035)
  • 2. D. M. Goecke and J. Henderson, Uniqueness of solutions of right focal problems for third order differential equations, Nonlin. Anal. 8 (1984), 253-259.MR 0738010 (85i:34006)
  • 3. C. P. Gupta and S. I. Trofimchuk, Solvability of a multi-point boundary value problem and a priori estimates, Canad. Appl. Math. Quart. 6 (1998), 45-60. MR 1638415 (99f:34020)
  • 4. P. Hartman, Unrestricted $ n$-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. MR 0105470 (21:4211)
  • 5. P. Hartman, On $ n$-parameter families and interpolation problems for nonlinear ordinary differential equations, Trans. Amer. Math. Soc. 154 (1971), 201-226. MR 0301277 (46:435)
  • 6. J. Henderson, Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations, J. Diff. Eqs. 41 (1981), 218-227. MR 0630990 (83g:34018)
  • 7. J. Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlin. Anal. 5 (1981), 989-1002. MR 0633013 (82j:34015)
  • 8. J. Henderson, Existence theorems for boundary value problems for $ nth$ order nonlinear difference equations, SIAM J. Math. Anal. 20 (1989), 468-478. MR 0982673 (90a:39003)
  • 9. J. Henderson, Focal boundary value problems for nonlinear difference equations, I, J. Math. Anal. Appl. 141 (1989), 559-567. MR 1009063 (90g:39003)
  • 10. J. Henderson, Focal boundary value problems for nonlinear difference equations, II, J. Math. Anal. Appl. 141 (1989), 568-579. MR 1009063 (90g:39003)
  • 11. J. Henderson, Uniqueness implies existence for three-point boundary value problems for second order differential equations, Appl. Math. Lett. 18 (2005), 905-909.MR 2152302
  • 12. J. Henderson, B. Karna and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005), 1365-1369. MR 2111960 (2005j:34026)
  • 13. J. Henderson and W. K. C. Yin, Existence of solutions for fourth order boundary value problems on a time scale, J. Differ. Eqs. Appl. 9 (2003), 15-28. MR 1958300 (2003k:34041)
  • 14. L. K. Jackson, Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. Appl. Math. 24 (1973), 535-538. MR 0322253 (48:615)
  • 15. L. K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Diff. Eqs. 13 (1973), 432-437. MR 0335925 (49:703)
  • 16. L. K. Jackson and K. Schrader, Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Diff. Eqs. 9 (1971), 46-54. MR 0269920 (42:4813)
  • 17. G. Klaasen, Existence theorems for boundary value problems for $ nth$ order ordinary differential equations, Rocky Mtn. J. Math. 3 (1973), 457-472. MR 0357944 (50:10409)
  • 18. A. Lasota and M. Luczynski, A note on the uniqueness of two point boundary value problems I, Zeszyty Naukowe UJ, Prace Matematyezne 12 (1968), 27-29. MR 0224900 (37:499)
  • 19. 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 0219792 (36:2871)
  • 20. R. Ma, Existence theorems for a second-order three-point boundary value problem, J. Math. Anal. Appl. 212 (1997), 430-442.MR 1464888 (98h:34041)
  • 21. R. Ma, Existence and uniqueness of solutions to first order three-point boundary value problems, Appl. Math. Lett. 15 (2002), 211-216.MR 1880760 (2002k:34032)
  • 22. A. C. Peterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1982), 173-185.MR 0605428 (83h:34017)
  • 23. K. Schrader, Uniqueness implies existence for solutions of nonlinear boundary value problems, Abstracts Amer. Math. Soc. 6 (1985), 235.
  • 24. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.MR 0210112 (35:1007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34B15, 34B10

Retrieve articles in all journals with MSC (2000): 34B15, 34B10


Additional Information

Stephen Clark
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798-7328
Address at time of publication: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, Missouri 65409
Email: sclark@umr.edu

Johnny Henderson
Affiliation: Department of Mathematics, Baylor University Waco, Texas 76798-7328
Email: Johnny\underlineHenderson@baylor.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08368-7
Keywords: Boundary value problem, uniqueness, existence, nonlocal
Received by editor(s): February 18, 2005
Received by editor(s) in revised form: May 20, 2005, and June 11, 2005
Published electronically: May 18, 2006
Additional Notes: Research for the first author was partially supported by NSF Grant DMS-0405528, as well as by a Baylor University Visiting Professorship during the Fall of 2004.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society