Open Access Research Article

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

Man Kam Kwong1,2* and James SW Wong3,4,5

Author Affiliations

1 Lucent Technologies Inc., Lisle, IL 60532, USA

2 Department of Mathematics, Statistics, Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA

3 Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

4 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

5 Chinney Investments Ltd., Hong Kong

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Boundary Value Problems 2007, 2007:064012 doi:10.1155/2007/64012


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/064012


Received:25 May 2007
Revisions received:20 August 2007
Accepted:23 August 2007
Published:28 November 2007

© 2007 Kwong and Wong

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.

References

  1. Kwong, MK: The shooting method and multiple solutions of two/multi-point BVPs of second-order ODE. Electronic Journal of Qualitative Theory of Differential Equations. 2006(6), 1–14 (2006)

  2. Krasnosel'skiĭ, MA: Positive Solutions of Operator Equations,p. 381. P. Noordhoff, Groningen, The Netherlands (1964)

  3. Baxley, JV, Haywood, LJ: Nonlinear boundary value problems with multiple solutions. Nonlinear Analysis: Theory, Methods & Applications. 47(2), 1187–1198 (2001). PubMed Abstract | Publisher Full Text OpenURL

  4. Il'in, VA, Moiseev, EL: Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Journal of Differential Equations. 23(7), 803–810 (1987)

  5. Il'in, VA, Moiseev, EL: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Journal of Differential Equations. 23, 979–987 (1987)

  6. Gupta, CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. Journal of Mathematical Analysis and Applications. 168(2), 540–551 (1992). Publisher Full Text OpenURL

  7. Gupta, CP: A note on a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications. 186(1), 277–281 (1994). Publisher Full Text OpenURL

  8. Marano, SA: A remark on a second-order three-point boundary value problem. Journal of Mathematical Analysis and Applications. 183(3), 518–522 (1994). Publisher Full Text OpenURL

  9. Constantin, A: On a two-point boundary value problem. Journal of Mathematical Analysis and Applications. 193(1), 318–328 (1995). Publisher Full Text OpenURL

  10. Avery, R: Existence of multiple positive solutions to a conjugate boundary value problem. Mathematical Sciences Research Hot-Line. 2(1), 1–6 (1998)

  11. Henderson, J, Thompson, HB: Multiple symmetric positive solutions for a second order boundary value problem. Proceedings of the American Mathematical Society. 128(8), 2373–2379 (2000). Publisher Full Text OpenURL

  12. Ma, R: Positive solutions for a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations. 1999(34), 1–8 (1999)

  13. Sun, W, Chen, S, Zhang, Q, Wang, C: Existence of positive solutions to -point nonhomogeneous boundary value problem. Journal of Mathematical Analysis and Applications. 330(1), 612–621 (2007). Publisher Full Text OpenURL

  14. Ma, R: Existence theorems for a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications. 212(2), 430–442 (1997). Publisher Full Text OpenURL

  15. Coffman, CV, Wong, JSW: Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations. Transactions of the American Mathematical Society. 167, 399–434 (1972)

  16. Raffoul, YN: Positive solutions of three-point nonlinear second order boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations. 2002(5), 1–11 (2002)

  17. Liu, B: Positive solutions of a nonlinear three-point boundary value problem. Computers & Mathematics with Applications. 44(1-2), 201–211 (2002). PubMed Abstract | Publisher Full Text OpenURL

  18. Liu, B: Positive solutions of a nonlinear three-point boundary value problem. Applied Mathematics and Computation. 132(1), 11–28 (2002). Publisher Full Text OpenURL

  19. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)

  20. Ma, R: Existence theorems for a second order -point boundary value problem. Journal of Mathematical Analysis and Applications. 211(2), 545–555 (1997). Publisher Full Text OpenURL

  21. Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics,p. v+122. American Mathematical Society, Providence, RI, USA (1979)

  22. Mawhin, J: Topological degree and boundary value problems for nonlinear differential equations. In: Furi M, Zecca P (eds.) Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics, vol. 1537, pp. 74–142. Springer, Berlin, Germany (1993).

  23. Gupta, CP, Ntouyas, SK, Tsamatos, PCh: Existence results for -point boundary value problems. Differential Equations and Dynamical Systems. 2(4), 289–298 (1994)

  24. Gupta, CP, Ntouyas, SK, Tsamatos, PCh: On an -point boundary-value problem for second-order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications. 23(11), 1427–1436 (1994). PubMed Abstract | Publisher Full Text OpenURL

  25. Gupta, CP, Trofimchuk, SI: Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear Analysis: Theory, Methods & Applications. 34(4), 489–507 (1998). PubMed Abstract | Publisher Full Text OpenURL

  26. Guo, YP, Shan, WR, Ge, WG: Positive solutions for second-order -point boundary value problems. Journal of Computational and Applied Mathematics. 151(2), 415–424 (2003). Publisher Full Text OpenURL

  27. Gupta, CP, Trofimchuk, SI: A sharper condition for the solvability of a three-point second order boundary value problem. Journal of Mathematical Analysis and Applications. 205(2), 586–597 (1997). Publisher Full Text OpenURL

  28. Gupta, CP: A generalized multi-point boundary value problem for second order ordinary differential equations. Applied Mathematics and Computation. 89(1–3), 133–146 (1998)

  29. Feng, W, Webb, JRL: Solvability of three point boundary value problems at resonance. Nonlinear Analysis: Theory, Methods & Applications. 30(6), 3227–3238 (1997). PubMed Abstract | Publisher Full Text OpenURL

  30. Feng, W, Webb, JRL: Solvability of -point boundary value problems with nonlinear growth. Journal of Mathematical Analysis and Applications. 212(2), 467–480 (1997). Publisher Full Text OpenURL

  31. Feng, W: On an -point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications. 30(8), 5369–5374 (1997). PubMed Abstract | Publisher Full Text OpenURL

  32. Cheung, W-S, Ren, J: Twin positive solutions for quasi-linear multi-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 62(1), 167–177 (2005). PubMed Abstract | Publisher Full Text OpenURL

  33. Naito, Y, Tanaka, S: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations. Nonlinear Analysis: Theory, Methods & Applications. 56(6), 919–935 (2004). PubMed Abstract | Publisher Full Text OpenURL

  34. Kong, Q: Existence and nonexistence of solutions of second-order nonlinear boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 66(11), 2635–2651 (2007). PubMed Abstract | Publisher Full Text OpenURL