Positive solutions for m-point boundary value problem

Volume 9, Issue 3, pp 1193--1201 http://dx.doi.org/10.22436/jnsa.009.03.45

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1413 Downloads
• 2201 Views

Authors

Hua Su - School of economics, Shandong University, 250014, Jinan, China. - School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, 250014, Jinan, China. Xinjun Wang - School of economics, Shandong University, 250014, Jinan, China.

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear m-point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity: \[ \begin{cases} (\phi(u'))'+a(t) f(t,u(t))=0,\,\,\,\,\, 0<t<1,\\ u'(0)=\Sigma^{m-2}_{i=1}a_i u'(\xi_i), \,\,u(1)=\Sigma^{k}_{i=1}b_i u(\xi_i)- \Sigma^{s}_{i=k+1}b_i u(\xi_i)-\Sigma^{m-2}_{i=s+1}b_i u'(\xi_i), \end{cases} \] where \(\phi: R \rightarrow R\) is an increasing homeomorphism and homomorphism and \(\phi(0) = 0\). The nonlinear term f may change sign. As an application, an example to demonstrate our results has given. The conclusions in this paper essentially extend and improve the known results.

Share and Cite

Keywords

• m-point boundary value problem
• positive solutions
• fixed-point theorem.

MSC

•  34B16

References

• [1] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, Inc., Boston (1988)


• [2] C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168 (1992), 540-551.


• [3] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differ. Equ., 23 (1987), 979-987.


• [4] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differ. Equ., 23 (1987), 803-810.


• [5] R. Ma, Positive solutions for a nonlinear m-point boundary value problems , Comput. Math. Appl., 42 (2001), 755-765.


• [6] D. X. Ma, Z. J. Du ,W. G. Ge, Existence and iteration of monotone positive solutions for multipoint boundary value problems with p-Laplacian operator, Comput. Math. Appl., 50 (2005), 729-739.


• [7] M. Moshinsky, Sobre los problems de condiciones a la frontiera en una dimension, Bol. Soc. Math. Mexicana, 7 (1950), 1-25.


• [8] H. Su, Positive Solutions for n-order m-point p-Laplacian Operator Singular Boundary Value Problems, Appl. Math. Comput., 199 (2008), 122-132.


• [9] H. Su, B. Wang, Z. Wei, X. Zhang, Positive Solutions of four-Point Boundary Value Problems for Higher-order p-Laplacian operator, J. Math. Anal. Appl., 330 (2007), 836-851.


• [10] H. Su, Z. Wei, B. Wang, The existence of positive solutions for a nonlinear four-point singular boundary value problems with a p-Laplacian operator, Nonlinear Anal., 66 (2007), 2204-2217.


• [11] H. Su, Z. Wei, F. Xu, The existence of positive solutions for nonlinear singular boundary value system with p-Laplacian, J. Appl. Math. Comp., 325 (2007), 826-836.


• [12] H. Su, Z. Wei, F. Xu, The wxistence of countably many positive solutions for a system of nonlinear singular boundary value problems with The p-Laplacian operator, J. Math. Anal. Appl., 325 (2007), 319-332.


• [13] S. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York (1961)


• [14] Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math.Anal.Appl., 315 (2006), 144-153.


• [15] F. Xu, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian operator , J. Appl. Math. Comput., 194 (2007), 366-380.


• [16] X. Zhang, Positive solutions for three-point semipositone boundary value problems with convex nonlinearity, J. Appl. Math. Comp., 30 (2009), 349-367.


• [17] X. Zhang, L. Liu, Positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian, J. Appl. Math. Comp., 37 (2011), 523-531.


• [18] X. Zhang, L. Liu, Y. Wu, The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives, Appl. Math. Comput., 218 (2012), 8526-8536.


• [19] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a singular fractional differential system involving derivatives, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1400-1409.


• [20] X. Zhang, L. Liu, Y.Wu, B. Wiwatanapataphee, The spectral analysis for a singular fractional differential equation with a signed measure, Appl. Math. Comput., 257 (2015), 252-263.