## Journal of Applied Mathematics

### Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales

#### Abstract

We study a system of second-order dynamic equations on time scales $\left({p}_{1}{u}_{1}^{\nabla }{\right)}^{\mathrm{\Delta }}\left(t\right)-{q}_{1}\left(t\right){u}_{1}\left(t\right)+\lambda {f}_{1}\left(t,{u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)=0,t\in \left({t}_{1},{t}_{n}\right),\left({p}_{2}{u}_{2}^{\nabla }{\right)}^{\mathrm{\Delta }}\left(t\right)-{q}_{2}\left(t\right){u}_{2}\left(t\right)+\lambda {f}_{2}\left(t,{u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)\right)=0$, satisfying four kinds of different multipoint boundary value conditions, ${f}_{i}$ is continuous and semipositone. We derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 679316, 12 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807910

Digital Object Identifier
doi:10.1155/2013/679316

Mathematical Reviews number (MathSciNet)
MR3039716

Zentralblatt MATH identifier
1266.34144

#### Citation

Wu, Gang; Li, Longsuo; Cong, Xinrong; Miao, Xiufeng. Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales. J. Appl. Math. 2013 (2013), Article ID 679316, 12 pages. doi:10.1155/2013/679316. https://projecteuclid.org/euclid.jam/1394807910

#### References

• D. R. Anderson and R. Ma, “Second-order $n$-point eigenvalue problems on time scales,” Advances in Difference Equations, vol. 2006, Article ID 59572, 17 pages, 2006.
• M. Feng, X. Zhang, and W. Ge, “Multiple positive solutions for a class of $m$-point boundary value problems on time scales,” Advances in Difference Equations, vol. 2009, Article ID 219251, 14 pages, 2009.
• S. G. Topal and A. Yantir, “Positive solutions of a second order $m$-point BVP on time scales,” Nonlinear Dynamics and Systems Theory, vol. 9, no. 2, pp. 185–197, 2009.
• C. Yuan and Y. Liu, “Multiple positive solutions of a second order nonlinear semipositone $m$-point boundary value problem on time scales,” Abstract and Applied Analysis, vol. 2010, Article ID 261741, 19 pages, 2010.
• X. Lin and Z. Du, “Positive solutions of $m$-point boundary value problem for second-order dynamic equations on time scales,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 851–864, 2008.
• H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508–524, 2004.
• Y. Pang and Z. Bai, “Upper and lower solution method for a fourth-order four-point boundary value problem on time scales,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2243–2247, 2009.
• F. M. Atici and S. G. Topal, “The generalized quasilinearization method and three point boundary value problems on time scales,” Applied Mathematics Letters, vol. 18, no. 5, pp. 577–585, 2005.
• S. Liang and J. Zhang, “The existence of countably many positive solutions for nonlinear singular $m$-point boundary value problems on time scales,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 291–303, 2009.
• S. Liang, J. Zhang, and Z. Wang, “The existence of three positive solutions of $m$-point boundary value problems for some dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1386–1393, 2009.
• L. Hu, “Positive solutions to singular third-order three-point boundary value problems on time scales,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 606–615, 2010.
• \.I. Yaslan, “Multiple positive solutions for nonlinear three-point boundary value problems on time scales,” Computers & Mathematics with Applications, vol. 55, no. 8, pp. 1861–1869, 2008.
• J. Li and J. Shen, “Existence results for second-order impulsive boundary value problems on time scales,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 4, pp. 1648–1655, 2009.
• J.-P. Sun, “Twin positive solutions of nonlinear first-order boundary value problems on time scales,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 68, no. 6, pp. 1754–1758, 2008.
• R. P. Agarwal, M. Bohner, and D. O'Regan, “Time scale boundary value problems on infinite intervals,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 27–34, 2002, Dynamic equations on time scales.
• H. Chen, H. Wang, Q. Zhang, and T. Zhou, “Double positive solutions of boundary value problems for $p$-Laplacian impulsive functional dynamic equations on time scales,” Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1473–1480, 2007.
• R. Ma and H. Luo, “Existence of solutions for a two-point boundary value problem on time scales,” Applied Mathematics and Computation, vol. 150, no. 1, pp. 139–147, 2004.
• D. R. Anderson, G. S. Guseinov, and J. Hoffacker, “Higher-order self-adjoint boundary-value problems on time scales,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 309–342, 2006.
• Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
• R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, vol. 141, Cambridge University Press, Cambridge, UK, 2001.
• D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, San Diego, Calif, USA, 1988.