Existence of Positive Solutions for a Third-Order Multi-Point Boundary Value Problem

Abstract

By using Leray-Schauder nonlinear alternative, Banach contraction theorem and Guo-Krasnosel’skii theorem, we discuss the existence, uniqueness and positivity of solution to the third-order multi-point nonhomogeneous boundary value problem (BVP1): where for The interesting point lies in the fact that the nonlinear term is allowed to depend on the first order derivative .

Share and Cite:

A. Guezane-Lakoud and L. Zenkoufi, "Existence of Positive Solutions for a Third-Order Multi-Point Boundary Value Problem," Applied Mathematics, Vol. 3 No. 9, 2012, pp. 1008-1013. doi: 10.4236/am.2012.39149.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] D. Anderson and R. Avery, “Multiple Positive Solutions to Third-Order Discrete Focal Boundary Value Problem,” Acta Mathematicae Applicatae Sinica, Vol. 19, No. 1, 2003, pp. 117-122. doi:10.1007/s10255-003-0087-1
[2] D. R. Anderson, “Green’s Function for a Third-Order Generalized Right Focal Problem,” Journal of Mathematical Analysis and Applications, Vol. 288, No. 1, 2003, pp. 1-14. doi:10.1016/S0022-247X(03)00132-X
[3] A. Guezane-Lakoud and L. Zenkoufi, “Positive Solution of a Three-Point Nonlinear Boundary Value Problem for Second Order Differential Equations,” International Journal of Applied Mathematics and Statistics, Vol. 20, 2011, pp. 38-46.
[4] A. Guezane-Lakoud and S. Kelaiaia, “Solvability of a Three-Point Nonlinear Noundary-Value Problem,” Electronic Journal of Differential Equations, Vol. 2010, No. 139, 2010, pp. 1-9.
[5] A. Guezane-Lakoud, S. Kelaiaia, A. M. Eid, “A Positive Solution for a Non-local Boundary Value Problem,” International Journal of Open Problems in Computer Science and Mathematics, Vol. 4, No. 1, 2011, pp. 36-43.
[6] J. R. Graef and Bo Yang, “Existence and Nonexistence of Positive Solutions of a Nonlinear Third Order Boundary Value Problem,” Electronic Journal of Qualitative Theory of Differential Equations, No. 9, 2008, pp. 1-13.
[7] J. R. Graef and B. Yang, “Positive Solutions of a Nonlinear Third Order Eigenvalue Problem,” Dynamic Systems & Applications, Vol. 15, 2006, pp. 97-110.
[8] S. Li, “Positive Solutions of Nonlinear Singular ThirdOrder Two-Point Boundary Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 413-425. doi:10.1016/j.jmaa.2005.10.037
[9] B. Hopkins and N. Kosmatov, “Third-Order Boundary Value Problems with Sign-Changing Solutions,” Nonlinear Analysis, Vol. 67, No. 1, 2007, pp. 126-137. doi:10.1016/j.na.2006.05.003
[10] L. J. Guo, J. P. Sun and Y. H. Zhao, “Existence of Positive Solutions for Nonlinear Third-Order Three-Point Boundary Value Problem,” Nonlinear Analysis, Vol. 68, No. 10, 2008, pp. 3151-3158. doi:10.1016/j.na.2007.03.008
[11] Y. Sun, “Positive Solutions of Singular Third-Order ThreePoint Boundary Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 306, No. 2, 2005, pp. 589-603. doi:10.1016/j.jmaa.2004.10.029
[12] K. Deimling, “Nonlinear Functional Analysis,” Springer, Berlin, 1985. doi:10.1007/978-3-662-00547-7
[13] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” Academic Press, San Diego, 1988.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.