Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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
Posted: 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

  • 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: http://dx.doi.org/10.1090/S0002-9939-06-08368-7
PII: S 0002-9939(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
Posted: 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 after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia