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Uniqueness of solutions for fourth-order nonlocal boundary value problems

Abstract

Uniqueness implies uniqueness relationships are examined among solutions of the fourth-order ordinary differential equation,, satisfying-point,-point, and-point nonlocal boundary conditions.

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References

  1. Agarwal RP: Compactness condition for boundary value problems. Equadiff (Brno) 1997, 9: 1-23.

    Google Scholar 

  2. Clark S, Henderson J: Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations. to appear in Proceedings of the American Mathematical Society

  3. Conti M, Terracini S, Verzini G: Infinitely many solutions to fourth order superlinear periodic problems. Transactions of the American Mathematical Society 2004,356(8):3283-3300. 10.1090/S0002-9947-03-03514-1

    Article  MathSciNet  MATH  Google Scholar 

  4. Ehme J, Hankerson D: Existence of solutions for right focal boundary value problems. Nonlinear Analysis 1992,18(2):191-197. 10.1016/0362-546X(92)90093-T

    Article  MathSciNet  MATH  Google Scholar 

  5. Franco D, O'Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions. Journal of Computational and Applied Mathematics 2005,174(2):315-327. 10.1016/j.cam.2004.04.013

    Article  MathSciNet  MATH  Google Scholar 

  6. Graef JR, Qian C, Yang B: A three point boundary value problem for nonlinear fourth order differential equations. Journal of Mathematical Analysis and Applications 2003,287(1):217-233. 10.1016/S0022-247X(03)00545-6

    Article  MathSciNet  MATH  Google Scholar 

  7. Gupta CP: A Dirichlet type multi-point boundary value problem for second order ordinary differential equations. Nonlinear Analysis 1996,26(5):925-931. 10.1016/0362-546X(94)00338-X

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta CP, Ntouyas SK, Tsamatos PCh:Solvability of an-point boundary value problem for second order ordinary differential equations. Journal of Mathematical Analysis and Applications 1995,189(2):575-584. 10.1006/jmaa.1995.1036

    Article  MathSciNet  MATH  Google Scholar 

  9. Hartman P:Unrestricted-parameter families. Rendiconti del Circolo Matematico di Palermo. Serie II 1958, 7: 123-142. 10.1007/BF02854523

    Article  MathSciNet  MATH  Google Scholar 

  10. Hartman P:On-parameter families and interpolation problems for nonlinear ordinary differential equations. Transactions of the American Mathematical Society 1971, 154: 201-226.

    MathSciNet  MATH  Google Scholar 

  11. Henderson J: Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations. Journal of Differential Equations 1981,41(2):218-227. 10.1016/0022-0396(81)90058-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Henderson J: Existence of solutions of right focal point boundary value problems for ordinary differential equations. Nonlinear Analysis 1981,5(9):989-1002. 10.1016/0362-546X(81)90058-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Henderson J:Right boundary value problems for third order differential equations. Journal of Mathematical and Physical Sciences 1984,18(4):405-413.

    MathSciNet  MATH  Google Scholar 

  14. Henderson J:Existence theorems for boundary value problems forth-order nonlinear difference equations. SIAM Journal on Mathematical Analysis 1989,20(2):468-478. 10.1137/0520032

    Article  MathSciNet  MATH  Google Scholar 

  15. Henderson J: Uniqueness implies existence for three-point boundary value problems for second order differential equations. Applied Mathematics Letters 2005,18(8):905-909. 10.1016/j.aml.2004.07.032

    Article  MathSciNet  MATH  Google Scholar 

  16. Henderson J, Karna B, Tisdell CC: Existence of solutions for three-point boundary value problems for second order equations. Proceedings of the American Mathematical Society 2005,133(5):1365-1369. 10.1090/S0002-9939-04-07647-6

    Article  MathSciNet  MATH  Google Scholar 

  17. Henderson J, McGwier RW Jr.: Uniqueness, existence, and optimality for fourth-order Lipschitz equations. Journal of Differential Equations 1987,67(3):414-440. 10.1016/0022-0396(87)90135-5

    Article  MathSciNet  MATH  Google Scholar 

  18. Henderson J, Yin WKC: Existence of solutions for fourth order boundary value problems on a time scale. Journal of Difference Equations and Applications 2003,9(1):15-28.

    Article  MathSciNet  MATH  Google Scholar 

  19. Ill'in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations 1987,23(8):979-987.

    MathSciNet  MATH  Google Scholar 

  20. Jackson LK: Uniqueness of solutions of boundary value problems for ordinary differential equations. SIAM Journal on Applied Mathematics 1973,24(4):535-538. 10.1137/0124054

    Article  MathSciNet  MATH  Google Scholar 

  21. Jackson LK: Existence and uniqueness of solutions of boundary value problems for third order differential equations. Journal of Differential Equations 1973,13(3):432-437. 10.1016/0022-0396(73)90002-8

    Article  MathSciNet  MATH  Google Scholar 

  22. Jackson LK: Boundary value problems for Lipschitz equations. In Differential Equations. Edited by: Ahmed S, Keener M, Lazer A. Academic Press, New York; 1980:31-50.

    Google Scholar 

  23. Karna B: Extremal points for the fourth order boundary value problems. Mathematical Sciences Research Journal 2003,7(10):382-393.

    MathSciNet  MATH  Google Scholar 

  24. Klaasen GA:Existence theorems for boundary value problems forth order ordinary differential equations. The Rocky Mountain Journal of Mathematics 1973, 3: 457-472. 10.1216/RMJ-1973-3-3-457

    Article  MathSciNet  MATH  Google Scholar 

  25. Kong L, Kong Q:Positive solutions of nonlinear-point boundary value problems on a measure chain. Journal of Difference Equations and Applications 2003,9(6):615-627. 10.1080/1023619031000060963

    Article  MathSciNet  MATH  Google Scholar 

  26. Lasota A, Łuczyński M: A note on the uniqueness of two point boundary value problems I. Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Matematyczne 1968, 12: 27-29.

    MathSciNet  MATH  Google Scholar 

  27. Lasota A, Opial Z: On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation. Colloquium Mathematicum 1967, 18: 1-5.

    MathSciNet  MATH  Google Scholar 

  28. Liu Y, Ge W: Existence theorems of positive solutions for fourth-order four point boundary value problems. Analysis and Applications (Singapore) 2004,2(1):71-85.

    Article  MathSciNet  MATH  Google Scholar 

  29. Lü H, Yu H, Liu Y: Positive solutions for singular boundary value problems of a coupled system of differential equations. Journal of Mathematical Analysis and Applications 2005,302(1):14-29. 10.1016/j.jmaa.2004.08.003

    Article  MathSciNet  MATH  Google Scholar 

  30. Ma R: Positive solutions for second-order three-point boundary value problems. Applied Mathematics Letters 2001,14(1):1-5. 10.1016/S0893-9659(00)00102-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Ma R:Existence of positive solutions for superlinear semipositone-point boundary-value problems. Proceedings of the Edinburgh Mathematical Society. Series II 2003,46(2):279-292. 10.1017/S0013091502000391

    Article  MathSciNet  MATH  Google Scholar 

  32. Ma R, Castaneda N:Existence of solutions of nonlinear-point boundary-value problems. Journal of Mathematical Analysis and Applications 2001,256(2):556-567. 10.1006/jmaa.2000.7320

    Article  MathSciNet  MATH  Google Scholar 

  33. Ma R: Multiple positive solutions for a semipositone fourth-order boundary value problem. Hiroshima Mathematical Journal 2003,33(2):217-227.

    MathSciNet  MATH  Google Scholar 

  34. Ma R: Some multiplicity results for an elastic beam equation at resonance. Applied Mathematics and Mechanics 1993,14(2):193-200. 10.1007/BF02453362

    Article  MathSciNet  MATH  Google Scholar 

  35. Ma JJ, Zhang FR: Solvability of a class of fourth-order two-point boundary value problems. Heilongjiang Daxue Ziran Kexue Xuebao 2001,18(2):19-22.

    MathSciNet  MATH  Google Scholar 

  36. Marchant TR: Higher-order interaction of solitary waves on shallow water. Studies in Applied Mathematics 2002,109(1):1-17. 10.1111/1467-9590.00001

    Article  MathSciNet  MATH  Google Scholar 

  37. Palamides PK:Multi point boundary-value problems at resonance for-order differential equations: positive and monotone solutions. Electronic Journal of Differential Equations 2004,2004(25):1-14.

    MathSciNet  MATH  Google Scholar 

  38. Pei MH, Chang SK: Existence and uniqueness theorems of solutions for a class fourth-order boundary value problems. Kyungpook Mathematical Journal 2001,41(2):299-309.

    MathSciNet  MATH  Google Scholar 

  39. Peterson AC: Focal Green's functions for fourth-order differential equations. Journal of Mathematical Analysis and Applications 1980,75(2):602-610. 10.1016/0022-247X(80)90104-3

    Article  MathSciNet  MATH  Google Scholar 

  40. Peterson DE: Uniqueness, existence, and comparison theorems for ordinary differential equations, Doctoral dissertation. , Nebraska; 1973.

    Google Scholar 

  41. Spanier EH: Algebraic Topology. McGraw-Hill, New York; 1966:xiv+528.

    MATH  Google Scholar 

  42. Vidossich G: On the continuous dependence of solutions of boundary value problems for ordinary differential equations. Journal of Differential Equations 1989,82(1):1-14. 10.1016/0022-0396(89)90164-2

    Article  MathSciNet  MATH  Google Scholar 

  43. Webb JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Analysis 2001,47(7):4319-4332. 10.1016/S0362-546X(01)00547-8

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang Z, Wang J: Positive solutions to a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications 2003,285(1):237-249. 10.1016/S0022-247X(03)00396-2

    Article  MathSciNet  MATH  Google Scholar 

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Henderson, J., Ma, D. Uniqueness of solutions for fourth-order nonlocal boundary value problems. Bound Value Probl 2006, 23875 (2006). https://doi.org/10.1155/BVP/2006/23875

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