Open Access Research Article

Positive Solutions of Boundary Value Problems for System of Nonlinear Fourth-Order Differential Equations

Shengli Xie1* and Jiang Zhu2

Author Affiliations

1 Department of Mathematics, Suzhou College, Suzhou 234000, China

2 School of Mathematics Science, Xuzhou Normal University, Xuzhou 221116, China

For all author emails, please log on.

Boundary Value Problems 2007, 2007:076493 doi:10.1155/2007/76493


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/076493


Received:23 March 2006
Revisions received:8 October 2006
Accepted:5 December 2006
Published:30 January 2007

© 2007 Xie and Zhu

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.

Some existence theorems of the positive solutions and the multiple positive solutions for singular and nonsingular systems of nonlinear fourth-order boundary value problems are proved by using topological degree theory and cone theory.

References

  1. Gupta, CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis. 26(4), 289–304 (1988). Publisher Full Text OpenURL

  2. Gupta, CP: Existence and uniqueness results for the bending of an elastic beam equation at resonance. Journal of Mathematical Analysis and Applications. 135(1), 208–225 (1988). Publisher Full Text OpenURL

  3. Agarwal, RP: On fourth order boundary value problems arising in beam analysis. Differential Integral Equations. 2(1), 91–110 (1989)

  4. O'Regan, D: Solvability of some fourth (and higher) order singular boundary value problems. Journal of Mathematical Analysis and Applications. 161(1), 78–116 (1991). Publisher Full Text OpenURL

  5. Yang, YS: Fourth-order two-point boundary value problems. Proceedings of the American Mathematical Society. 104(1), 175–180 (1988). Publisher Full Text OpenURL

  6. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis. 59(1–4), 225–231 (1995)

  7. Wei, ZL, Zhang, ZT: A necessary and sufficient condition for the existence of positive solutions of singular superlinear boundary value problems. Acta Mathematica Sinica. 48(1), 25–34 (2005)

  8. Wei, ZL: Positive solutions to singular boundary value problems for a class of fourth-order sublinear differential equations. Acta Mathematica Sinica. 48(4), 727–738 (2005)

  9. Li, YX: Existence and multiplicity of positive solutions for fourth-order boundary value problems. Acta Mathematicae Applicatae Sinica. 26(1), 109–116 (2003)

  10. Zhou, YM: Positive solutions to fourth-order nonlinear eigenvalue problems. Journal of Systems Science and Mathematical Sciences. 24(4), 433–442 (2004)

  11. Guo, DJ: Nonlinear Functional Analysis, Science and Technology Press, Jinan, Shandong, China (1985)

  12. Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)

  13. Cheng, JG: Nonlinear singular boundary value problems. Acta Mathematicae Applicatae Sinica. 23(1), 122–129 (2000)