Existence and Uniqueness of Positive Solution for Third-Order Three-Point Boundary Value Problems

Tongchun Hu, Yongping Sun

College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, China.
Department of Public Teaching, Hangzhou Polytechnic, Hangzhou, China.

DOI: 10.4236/apm.2014.46037                

Abstract

This paper is devoted to the study of the existence and uniqueness of the positive solution for a type of the nonlinear third-order three-point boundary value problem. Our results are based on an iterative method and the Leray-Schauder fixed point theorem.

Keywords

Positive Solution, Uniqueness and Existence, Third-Order Three-Point BVPs

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Guo, L., Sun, J. and Zhao, Y. (2007) Multiple Positive Solutions for Nonlinear Third-Order Three-Point Boundary-Value Problems. Electronic Journal of Differential Equations, 112, 1-7.
[2]	Guo, L., Sun, J. and Zhao, Y. (2008) Existence of Positive Solution for Nonlinear Third-Order Three-Point Boundary Value Problem. Nonlinear Analysis, Theory, Methods and Applications, 68, 3151-3158.
[3]	Anderson, D. (1998) Multiple Positive Solutions for a Three-Point Boundary Value Problem. Mathematical and Computer Modelling, 27, 49-57. http://dx.doi.org/10.1016/S0895-7177(98)00028-4
[4]	Graef, J.R. and Yang, B. (2005) Multiple Positive Silutions to a Three Point Third Order Boundary Value Problem. Discrete and Continuous Dynmical Systems, 1-8.
[5]	Palamides, P.K. and Palamides, A.P. (2008) A Third-Order 3-Point BVP. Applying Krasnosel’skii’s Theorem on the Plane without a Green’s Function. Electronic Journal of Differential Equations, 14, 1-15.
[6]	Palamides, A.P. and Stavrakakis, N.M. (2010) Existence and Uniqueness of a Positive Solution for a Third-Order Three-Point Boundary-Value Problem. Electronic Journal of Differential Equations, 155, 1-12.
[7]	Sun, J., Ren, Q. and Zhao, Y. (2010) The Upper and Lower Solution Method for Nonlinear Third-Order Three-Point Boundary Value Problem. Electronic Journal of Qualitative Theory of Differential Equations, 26, 1-8.
[8]	Sun, Y. (2005) Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems. Journal of Mathematical Analysis and Applications, 306, 589-603. http://dx.doi.org/10.1016/j.jmaa.2004.10.029
[9]	Sun, Y. (2009) Positive Solutions for Third-Order Three-Point Nonhomogeneous Boundary Value Problems. Applied Mathematics Letters, 22, 45-51. http://dx.doi.org/10.1016/j.aml.2008.02.002
[10]	Sun, Y. (2008) Existence of Triple Positive Solutions for a Third-Order Three-Point Boundary Value Problem. Journal of Computational and Applied Mathematics, 221, 194-201. http://dx.doi.org/10.1016/j.cam.2007.10.064
[11]	Torres, F.J. (2013) Positive Solutions for a Third-Order Three-Point Boundary-Value Problem. Electronic Journal of Differential Equations, 147, 1-11.
[12]	Yao, Q. (2009) Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems. Journal of Mathematical Analysis and Applications, 354, 207-212. http://dx.doi.org/10.1016/j.jmaa.2008.12.057
[13]	Zhang, X. and Liu, L. (2008) Nontrivial Solution of Third-Order Nonlinear Eigenvalue Problems (II). Applied Mathematics and Computation, 204, 508-512. http://dx.doi.org/10.1016/j.amc.2008.06.048
[14]	Yao, Q. and Feng, Y. (2002) The Existence of Solutions for a Third Order Two-Point Boundary Value Problem. Applied Mathematics Letters, 15, 227-232. http://dx.doi.org/10.1016/S0893-9659(01)00122-7
[15]	Feng, Y. and Liu, S. (2005) Solvability of a Third-Order Two-Point Boundary Value Problem. Applied Mathematics Letters, 18, 1034-1040. http://dx.doi.org/10.1016/j.aml.2004.04.016