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

- Man Kam Kwong
^{1, 2}Email author and - James SW Wong
^{3, 4, 5}

**2007**:064012

https://doi.org/10.1155/2007/64012

© M.K. Kwong and J.S.W. Wong 2007

**Received: **25 May 2007

**Accepted: **23 August 2007

**Published: **28 November 2007

## Abstract

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.

[12345678910111213141516171819202122232425262728293031323334]

## Authors’ Affiliations

## References

- 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,**2006**(6):1–14.Google Scholar - Krasnosel'skiĭ MA:
*Positive Solutions of Operator Equations*. P. Noordhoff, Groningen, The Netherlands; 1964:381.Google Scholar - Baxley JV, Haywood LJ:
**Nonlinear boundary value problems with multiple solutions.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(2):1187–1198. 10.1016/S0362-546X(01)00257-7MATHMathSciNetView ArticleGoogle Scholar - 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*1987,**23**(7):803–810.MATHMathSciNetGoogle Scholar - Il'in VA, Moiseev EL:
**Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.***Journal of Differential Equations*1987,**23:**979–987.MATHGoogle Scholar - Gupta CP:
**Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.***Journal of Mathematical Analysis and Applications*1992,**168**(2):540–551. 10.1016/0022-247X(92)90179-HMATHMathSciNetView ArticleGoogle Scholar - Gupta CP:
**A note on a second order three-point boundary value problem.***Journal of Mathematical Analysis and Applications*1994,**186**(1):277–281. 10.1006/jmaa.1994.1299MATHMathSciNetView ArticleGoogle Scholar - Marano SA:
**A remark on a second-order three-point boundary value problem.***Journal of Mathematical Analysis and Applications*1994,**183**(3):518–522. 10.1006/jmaa.1994.1158MATHMathSciNetView ArticleGoogle Scholar - Constantin A:
**On a two-point boundary value problem.***Journal of Mathematical Analysis and Applications*1995,**193**(1):318–328. 10.1006/jmaa.1995.1238MATHMathSciNetView ArticleGoogle Scholar - Avery R:
**Existence of multiple positive solutions to a conjugate boundary value problem.***Mathematical Sciences Research Hot-Line*1998,**2**(1):1–6.MATHMathSciNetGoogle Scholar - Henderson J, Thompson HB:
**Multiple symmetric positive solutions for a second order boundary value problem.***Proceedings of the American Mathematical Society*2000,**128**(8):2373–2379. 10.1090/S0002-9939-00-05644-6MATHMathSciNetView ArticleGoogle Scholar - Ma R:
**Positive solutions for a nonlinear three-point boundary-value problem.***Electronic Journal of Differential Equations*1999,**1999**(34):1–8.Google Scholar - Sun W, Chen S, Zhang Q, Wang C:
**Existence of positive solutions to-point nonhomogeneous boundary value problem.***Journal of Mathematical Analysis and Applications*2007,**330**(1):612–621. 10.1016/j.jmaa.2006.08.022MATHMathSciNetView ArticleGoogle Scholar - Ma R:
**Existence theorems for a second order three-point boundary value problem.***Journal of Mathematical Analysis and Applications*1997,**212**(2):430–442. 10.1006/jmaa.1997.5515MATHMathSciNetView ArticleGoogle Scholar - Coffman CV, Wong JSW:
**Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations.***Transactions of the American Mathematical Society*1972,**167:**399–434.MATHMathSciNetView ArticleGoogle Scholar - Raffoul YN:
**Positive solutions of three-point nonlinear second order boundary value problem.***Electronic Journal of Qualitative Theory of Differential Equations*2002,**2002**(5):1–11.View ArticleGoogle Scholar - Liu B:
**Positive solutions of a nonlinear three-point boundary value problem.***Computers & Mathematics with Applications*2002,**44**(1–2):201–211. 10.1016/S0898-1221(02)00141-4MATHMathSciNetView ArticleGoogle Scholar - Liu B:
**Positive solutions of a nonlinear three-point boundary value problem.***Applied Mathematics and Computation*2002,**132**(1):11–28. 10.1016/S0096-3003(02)00341-7MATHMathSciNetView ArticleGoogle Scholar - Guo D, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar - Ma R:
**Existence theorems for a second order-point boundary value problem.***Journal of Mathematical Analysis and Applications*1997,**211**(2):545–555. 10.1006/jmaa.1997.5416MATHMathSciNetView ArticleGoogle Scholar - Mawhin J:
*Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics*.*Volume 40*. American Mathematical Society, Providence, RI, USA; 1979:v+122.Google Scholar - Mawhin J:
**Topological degree and boundary value problems for nonlinear differential equations.**In*Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics*.*Volume 1537*. Edited by: Furi M, Zecca P. Springer, Berlin, Germany; 1993:74–142. 10.1007/BFb0085076View ArticleGoogle Scholar - Gupta CP, Ntouyas SK, Tsamatos PCh:
**Existence results for-point boundary value problems.***Differential Equations and Dynamical Systems*1994,**2**(4):289–298.MATHMathSciNetGoogle Scholar - Gupta CP, Ntouyas SK, Tsamatos PCh:
**On an-point boundary-value problem for second-order ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*1994,**23**(11):1427–1436. 10.1016/0362-546X(94)90137-6MATHMathSciNetView ArticleGoogle Scholar - 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*1998,**34**(4):489–507. 10.1016/S0362-546X(97)00584-1MATHMathSciNetView ArticleGoogle Scholar - Guo YP, Shan WR, Ge WG:
**Positive solutions for second-order-point boundary value problems.***Journal of Computational and Applied Mathematics*2003,**151**(2):415–424. 10.1016/S0377-0427(02)00739-2MATHMathSciNetView ArticleGoogle Scholar - 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*1997,**205**(2):586–597. 10.1006/jmaa.1997.5252MATHMathSciNetView ArticleGoogle Scholar - Gupta CP:
**A generalized multi-point boundary value problem for second order ordinary differential equations.***Applied Mathematics and Computation*1998,**89**(1–3):133–146.MATHMathSciNetView ArticleGoogle Scholar - Feng W, Webb JRL:
**Solvability of three point boundary value problems at resonance.***Nonlinear Analysis: Theory, Methods & Applications*1997,**30**(6):3227–3238. 10.1016/S0362-546X(96)00118-6MATHMathSciNetView ArticleGoogle Scholar - Feng W, Webb JRL:
**Solvability of-point boundary value problems with nonlinear growth.***Journal of Mathematical Analysis and Applications*1997,**212**(2):467–480. 10.1006/jmaa.1997.5520MATHMathSciNetView ArticleGoogle Scholar - Feng W:
**On an-point boundary value problem.***Nonlinear Analysis: Theory, Methods & Applications*1997,**30**(8):5369–5374. 10.1016/S0362-546X(97)00360-XMATHMathSciNetView ArticleGoogle Scholar - Cheung W-S, Ren J:
**Twin positive solutions for quasi-linear multi-point boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2005,**62**(1):167–177. 10.1016/j.na.2005.03.018MATHMathSciNetView ArticleGoogle Scholar - 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*2004,**56**(6):919–935. 10.1016/j.na.2003.10.020MATHMathSciNetView ArticleGoogle Scholar - Kong Q:
**Existence and nonexistence of solutions of second-order nonlinear boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(11):2635–2651. 10.1016/j.na.2006.03.045MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.