Open Access

Existence of Symmetric Positive Solutions for an https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq1_HTML.gif -Point Boundary Value Problem

Boundary Value Problems20072007:079090

DOI: 10.1155/2007/79090

Received: 23 June 2006

Accepted: 11 March 2007

Published: 16 May 2007

Abstract

We study the second-order https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq2_HTML.gif -point boundary value problem https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq5_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq7_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq8_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq9_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq10_HTML.gif is continuous, symmetric on the interval https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq11_HTML.gif , and maybe singular at https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq14_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq15_HTML.gif is symmetric on the interval https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq16_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq17_HTML.gif and satisfies some appropriate growth conditions. By using Krasnoselskii's fixed point theorem in a cone, we get some existence results of symmetric positive solutions.

[123456789101112131415161718192021222324252627282930]

Authors’ Affiliations

(1)
Department of Electron and Information, Zhejiang University of Media and Communications

References

  1. Chen S, Hu J, Chen L, Wang C:Existence results for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq18_HTML.gif -point boundary value problem of second order ordinary differential equations. Journal of Computational and Applied Mathematics 2005,180(2):425-432. 10.1016/j.cam.2004.11.010MATHMathSciNetView Article
  2. Cheung W-S, Ren J:Positive solution for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq19_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2005,303(2):565-575. 10.1016/j.jmaa.2004.08.056MATHMathSciNetView Article
  3. Cheung W-S, Ren J: Twin positive solutions for quasi-linear multi-point boundary value problems. Nonlinear Analysis 2005,62(1):167-177. 10.1016/j.na.2005.03.018MATHMathSciNetView Article
  4. Dong S, Ge W:Positive solutions of an https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq20_HTML.gif -point boundary value problem with sign changing nonlinearities. Computers & Mathematics with Applications 2005,49(4):589-598. 10.1016/j.camwa.2004.07.018MATHMathSciNetView Article
  5. Guo Y, Shan W, Ge W:Positive solutions for second-order https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq21_HTML.gif -point boundary value problems. Journal of Computational and Applied Mathematics 2003,151(2):415-424. 10.1016/S0377-0427(02)00739-2MATHMathSciNetView Article
  6. Infante G, Webb JRL: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. Journal of Mathematical Analysis and Applications 2002,272(1):30-42. 10.1016/S0022-247X(02)00125-7MATHMathSciNetView Article
  7. Kong L, Kong Q: Second-order boundary value problems with nonhomogeneous boundary conditions—I. Mathematische Nachrichten 2005,278(1-2):173-193. 10.1002/mana.200410234MATHMathSciNetView Article
  8. Kosmatov N:Semipositone https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq22_HTML.gif -point boundary-value problems. Electronic Journal of Differential Equations 2004,2004(119):1-7.View Article
  9. Liu B: Solvability of multi-point boundary value problem at resonance.—part IV. Applied Mathematics and Computation 2003,143(2-3):275-299. 10.1016/S0096-3003(02)00361-2MATHMathSciNetView Article
  10. Liu B: Positive solutions of a nonlinear four-point boundary value problems. Applied Mathematics and Computation 2004,155(1):179-203. 10.1016/S0096-3003(03)00770-7MATHMathSciNetView Article
  11. Liu X:Nontrivial solutions of singular nonlinear https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq23_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2003,284(2):576-590. 10.1016/S0022-247X(03)00365-2MATHMathSciNetView Article
  12. Ma R:Existence results of a https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq24_HTML.gif -point boundary value problem at resonance. Journal of Mathematical Analysis and Applications 2004,294(1):147-157. 10.1016/j.jmaa.2004.02.005MATHMathSciNetView Article
  13. Ma R:Multiple positive solutions for nonlinear https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq25_HTML.gif -point boundary value problems. Applied Mathematics and Computation 2004,148(1):249-262. 10.1016/S0096-3003(02)00843-3MATHMathSciNetView Article
  14. Ma R, O'Regan D:Solvability of singular second order https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq26_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2005,301(1):124-134. 10.1016/j.jmaa.2004.07.009MATHMathSciNetView Article
  15. Sun Y:Positive solutions of nonlinear second-order https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq27_HTML.gif -point boundary value problem. Nonlinear Analysis 2005,61(7):1283-1294. 10.1016/j.na.2005.01.105MATHMathSciNetView Article
  16. Sun Y: Nontrivial solution for a three-point boundary-value problem. Electronic Journal of Differential Equations 2004,2004(111):1-10.MATH
  17. Sun Y, Liu L: Solvability for a nonlinear second-order three-point boundary value problem. Journal of Mathematical Analysis and Applications 2004,296(1):265-275. 10.1016/j.jmaa.2004.04.013MATHMathSciNetView Article
  18. Xu X:Multiple sign-changing solutions for some https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq28_HTML.gif -point boundary-value problems. Electronic Journal of Differential Equations 2004,2004(89):1-14.
  19. Xu X:Positive solutions for singular https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq29_HTML.gif -point boundary value problems with positive parameter. Journal of Mathematical Analysis and Applications 2004,291(1):352-367. 10.1016/j.jmaa.2003.11.009MATHMathSciNetView Article
  20. Zhang G, Sun J:Positive solutions of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq30_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2004,291(2):406-418. 10.1016/j.jmaa.2003.11.034MATHMathSciNetView Article
  21. Zhang Z, Wang J: On existence and multiplicity of positive solutions to singular multi-point boundary value problems. Journal of Mathematical Analysis and Applications 2004,295(2):502-512. 10.1016/j.jmaa.2004.03.057MATHMathSciNetView Article
  22. Avery RI, Henderson J: Three symmetric positive solutions for a second-order boundary value problem. Applied Mathematics Letters 2000,13(3):1-7. 10.1016/S0893-9659(99)00177-9MATHMathSciNetView Article
  23. 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 Article
  24. Li F, Zhang Y: Multiple symmetric nonnegative solutions of second-order ordinary differential equations. Applied Mathematics Letters 2004,17(3):261-267. 10.1016/S0893-9659(04)90061-4MathSciNetView Article
  25. Yao Q:Existence and iteration of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F79090/MediaObjects/13661_2006_Article_665_IEq31_HTML.gif symmetric positive solutions for a singular two-point boundary value problem. Computers & Mathematics with Applications 2004,47(8-9):1195-1200. 10.1016/S0898-1221(04)90113-7MATHMathSciNetView Article
  26. Kosmatov N: Symmetric solutions of a multi-point boundary value problem. Journal of Mathematical Analysis and Applications 2005,309(1):25-36. 10.1016/j.jmaa.2004.11.008MATHMathSciNetView Article
  27. Kosmatov N: A symmetric solution of a multipoint boundary value problem at resonance. Abstract and Applied Analysis 2006, 2006: 11 pages.MathSciNetView Article
  28. Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.MATHView Article
  29. Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands; 1964:381.
  30. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.MATH

Copyright

© Y. Sun and X. Zhang. 2007

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.