## Abstract and Applied Analysis

### Existence of Nontrivial Solutions for Unilaterally Asymptotically Linear Three-Point Boundary Value Problems

Hongyu Li

#### Abstract

Using fixed point theorems in ordered Banach spaces with the lattice structure, we consider the existence of nontrivial solutions under the condition that the nonlinear term can change sign and study the existence of sign-changing solutions for some second order three-point boundary value problems. Our results improve and generalize on those in the literatures.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 263042, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412276983

Digital Object Identifier
doi:10.1155/2014/263042

Mathematical Reviews number (MathSciNet)
MR3212405

Zentralblatt MATH identifier
07022040

#### Citation

Li, Hongyu. Existence of Nontrivial Solutions for Unilaterally Asymptotically Linear Three-Point Boundary Value Problems. Abstr. Appl. Anal. 2014 (2014), Article ID 263042, 7 pages. doi:10.1155/2014/263042. https://projecteuclid.org/euclid.aaa/1412276983

#### References

• S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, NY, USA, 1961.
• 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.
• J. R. L. Webb, “Positive solutions of some three point boundary value problems via fixed point index theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4319–4332, 2001.
• G. W. Zhang and J. X. Sun, “Positive solutions of $m$-point boun-dary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 406–418, 2004.
• G. W. Zhang and J. X. Sun, “Multiple positive solutions of singular second-order $m$-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 442–447, 2006.
• B. M. Liu, L. S. Liu, and Y. H. Wu, “Positive solutions for singular second order three-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 12, pp. 2756–2766, 2007.
• R. Ma and N. Castaneda, “Existence of solutions of nonlinear $m$-point boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556–567, 2001.
• B. M. Liu, “Positive solutions of a nonlinear three-point boundary value problem,” Applied Mathematics and Computation, vol. 132, no. 1, pp. 11–28, 2002.
• X. Xu, “Multiplicity results for positive solutions of some semi-positone three-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 673–689, 2004.
• X. Xu, D. O'Regan, and J. X. Sun, “Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition,” Mathematical and Computer Modelling, vol. 45, no. 1-2, pp. 189–200, 2007.
• Y. P. Guo, W. G. Ge, and S. J. Dong, “Two positive solutions for second order three point boundary value problems with sign changing nonlinearities,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 3, pp. 522–529, 2004.
• T. Bartsch and Z. Q. Wang, “On the existence of sign changing solutions for semilinear Dirichlet problems,” Topological Methods in Nonlinear Analysis, vol. 7, no. 1, pp. 115–131, 1996.
• X. Xu and J. X. Sun, “On sign-changing solution for some three-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 4, pp. 491–505, 2004.
• K. Zhang and X. J. Xie, “Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2796–2805, 2009.
• Y. J. Cui, Y. M. Zou, and H. Y. Li, “Existence of sign-changing solutions for nonlinear operator equations and its applications,” Journal of Systems Science and Mathematical Sciences, vol. 29, no. 8, pp. 1094–1101, 2009.
• J. X. Sun, Nonlinear Functional Analysis and Applications, Sci-ence Press, Beijing, China, 2008.
• B. P. Rynne, “Second-order, three-point boundary value problems with jumping non-linearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3294–3306, 2008.
• B. P. Rynne, “Spectral properties and nodal solutions for sec-ond-order, $m$-point, boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3318–3327, 2007.
• D. L. Bai and H. F. Feng, “Eigenvalue for a singular second order three-point boundary value problem,” Journal of Applied Mathematics and Computing, vol. 38, no. 1-2, pp. 443–452, 2012.
• Y. Z. Lin, J. Niu, and M. G. Cui, “A numerical solution to non-linear second order three-point boundary value problems in the reproducing kernel space,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7362–7368, 2012.
• X. Y. Liu and J. X. Sun, “Computation of topological degree of unilaterally asymptotically linear operators and its applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 96–106, 2009.
• J. X. Sun and X. Y. Liu, “Computation of topological degree for nonlinear operators and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4121–4130, 2008.
• J. X. Sun and X. Y. Liu, “Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 348, no. 2, pp. 927–937, 2008.
• H. Y. Li and F. Sun, “Existence of solutions for integral boun-dary value problems of second-order ordinary differential equations,” Boundary Value Problems, vol. 2012, article 147, 7 pages, 2012.
• H. X. Lu, L. Sun, and J. X. Sun, “Existence of positive solutions to a non-positive elastic beam equation with both ends fixed,” Boundary Value Problems, vol. 2012, article 56, 10 pages, 2012.
• D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, China, 2nd edition, 2001. \endinput