Abstract and Applied Analysis

Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

Peiguo Zhang, Lishan Liu, and Yonghong Wu

Full-text: Open access

Abstract

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 640183, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393443517

Digital Object Identifier
doi:10.1155/2013/640183

Mathematical Reviews number (MathSciNet)
MR3121401

Zentralblatt MATH identifier
07095203

Citation

Zhang, Peiguo; Liu, Lishan; Wu, Yonghong. Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 640183, 7 pages. doi:10.1155/2013/640183. https://projecteuclid.org/euclid.aaa/1393443517


Export citation

References

  • A. V. Bicadze and A. A. Samarskiĭ, “Some elementary generalizations of linear elliptic boundary value problems,” Doklady Akademii Nauk SSSR, vol. 185, pp. 739–740, 1969.
  • V. A. II'in and E. I. Moiseev, “Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects,” Differential Equations, vol. 23, pp. 803–810, 1987.
  • V. A. II'in and E. I. Moiseev, “Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator,” Differential Equations, vol. 23, pp. 979–987, 1987.
  • C. P. Gupta, “Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 540–551, 1992.
  • P. W. Eloe and B. Ahmad, “Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol. 18, no. 5, pp. 521–527, 2005.
  • X. Hao, L. Liu, and Y. Wu, “Positive solutions for nonlinear $n$th-order singular nonlocal boundary value problems,” Boundary Value Problems, vol. 2007, Article ID 74517, 2007.
  • J. R. Graef and T. Moussaoui, “A class of $n$th-order BVPs with nonlocal conditions,” Computers & Mathematics with Applications, vol. 58, no. 8, pp. 1662–1671, 2009.
  • C. Pang, W. Dong, and Z. Wei, “Green's function and positive solutions of $n$th order $m$-point boundary value problem,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1231–1239, 2006.
  • J. Yang and Z. Wei, “Positive solutions of $n$th order $m$-point boundary value problem,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 715–720, 2008.
  • Y. Guo, Y. Ji, and J. Zhang, “Three positive solutions for a nonlinear $n$th-order $m$-point boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3485–3492, 2008.
  • M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.
  • M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.
  • G. Zhang and J. Sun, “Positive solutions of $m$-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 406–418, 2004.
  • M. Feng and W. Ge, “Existence results for a class of $n$th order $m$-point boundary value problems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 8, pp. 1303–1308, 2009.
  • X. Hao, L. Liu, and Y. Wu, “On positive solutions of an $m$-point nonhomogeneous singular boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2532–2540, 2010.
  • W. Jiang, “Multiple positive solutions for $n$th-order $m$-point boundary value problems with all derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 5, pp. 1064–1072, 2008.
  • J. R. Graef and B. Yang, “Positive solutions to a multi-point higher order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 409–421, 2006.
  • M. Zhang, Y. Yin, and Z. Wei, “Positive solution of singular higher-order $m$-point boundary value problem with nonlinearity that changes sign,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 678–687, 2008.
  • J. R. Graef, L. Kong, and B. Yang, “Existence of solutions for a higher order multi-point boundary value problem,” Results in Mathematics, vol. 53, no. 1-2, pp. 77–101, 2009.
  • Y. Ji and Y. Guo, “The existence of countably many positive solutions for some nonlinear $n$th order $m$-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 187–200, 2009.
  • X. Zhang, M. Feng, and W. Ge, “Multiple positive solutions for a class of $m$-point boundary value problems,” Applied Mathematics Letters, vol. 22, no. 1, pp. 12–18, 2009.
  • J. Zhao and W. Ge, “Existence results of $m$-point boundary value problem of Sturm-Liouville type with sign changing nonlinearity,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 946–954, 2009.
  • S. Liang and J. Zhang, “Existence of countably many positive solutions of $n$th-order $m$-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 527–537, 2009.
  • H. Su and X. Wang, “Positive solutions to singular semipositone $m$-point $n$-order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 187–200, 2011.
  • J. Henderson and R. Luca, “Existence and multiplicity for positive solutions of a multi-point boundary value problem,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10572–10585, 2012.
  • P. Zhang, “Iterative solutions of singular boundary value problems of third-order differential equation,” Boundary Value Problems, vol. 2011, Article ID 483057, 10 pages, 2011.
  • D. Guo, Semi-Ordered Method in Nonlinear Analysis, Shandong Scientific Technical Press, Jinan, China, 2000, Chinese.
  • D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.
  • D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.