Taiwanese Journal of Mathematics

EXISTENCE OF THREE POSITIVE SOLUTIONS FOR M-POINT BOUNDARY-VALUE PROBLEM WITH ONE-DIMENSIONAL P-LAPLACIAN

Han-Ying Feng and Wei-Gao Ge

Full-text: Open access

Abstract

In this paper, we consider the multipoint boundary value problem for the one-dimensional $p$-Laplacian $$\left(\phi_p(u'(t))\right)' + q(t) f\left(t,u(t),u'(t)\right) = 0,~~~~\ \ \ \ \ \ t \in (0,1), $$ subject to the boundary value conditions: $$ u(0) = \sum_{i=1}^{m-2} a_{i} u(\xi_{i}),\ \ \ \ \ \ u(1) = \sum_{i=1}^{m-2} b_{i} u(\xi_{i}), $$ where $\phi_{p}(s) = |s|^{p-2}s$, $p \gt 1$, $\xi_{i} \in (0,1)$ with $0 \lt \xi_{1} \lt \xi_{2} \lt \cdots \lt \xi_{m-2} \lt 1$ and $a_{i}, b_{i} \in [0,1)$, $0 \leq \sum\limits_{i=1}^{m-2} a_{i} \lt 1$, $0 \leq \sum\limits_{i=1}^{m-2} b_{i} \lt 1$. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term $f$ is involved with the first-order derivative explicitly.

Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 647-665.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405811

Digital Object Identifier
doi:10.11650/twjm/1500405811

Mathematical Reviews number (MathSciNet)
MR2655791

Zentralblatt MATH identifier
1205.34025

Subjects
Primary: 34B10: Nonlocal and multipoint boundary value problems 34B15: Nonlinear boundary value problems

Keywords
multipoint boundary value problem Avery-Peterson's fixed point theorem positive solution one-dimensional $p$-Laplacian

Citation

Feng, Han-Ying; Ge, Wei-Gao. EXISTENCE OF THREE POSITIVE SOLUTIONS FOR M-POINT BOUNDARY-VALUE PROBLEM WITH ONE-DIMENSIONAL P-LAPLACIAN. Taiwanese J. Math. 14 (2010), no. 2, 647--665. doi:10.11650/twjm/1500405811. https://projecteuclid.org/euclid.twjm/1500405811


Export citation

References

  • [1.] V. A. Il'in and E. I. Moiseev, Nonlocal boundary value problem of the second kind for a sturm-Liouville operator, Differential Equations., 23 (1987), 979-987.
  • [2.] C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equation, Appl. Math. Comput., 89 (1998), 33-146.
  • [3.] R. Ma, Positive solutions for multipoint boundary value problems with a one-dimensional $p$-Laplacian, Comput. Math. Appl., 42 (2001), 755-765.
  • [4.] W. Feng and J. R. L. Webb, Solvability of $m$-point boundary value problem with nonlinear growth, J. Math. Anal. Appl., 212 (1997), 467-480.
  • [5.] D. Ma, Z. Du and W. Ge, Existence and iteration of monotone positive solutions for multipoint boundary value problem with $p$-Laplacian operator, Comput. Math. Appl., 50 (2005), 729-739.
  • [6.] Y. Wang and C. Hou, Existence of multiple positive solutions for one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 315 (2006), 144-153.
  • [7.] Y. Wang and W. Ge, Positive solutions for multipoint boundary value problems with a one-dimensional $p$-Laplacian, Nonlinear Anal., 66 (2007), 1246-1256.
  • [8.] Z. Bai, Z. Gui and W. Ge, Multiple positive solutions for some $p$-Laplacian boundary value problems, J. Math. Anal. Appl. 300 (2004), 477-490.
  • [9.] Y. Wang and W. Ge, Triple positive solutions for two-point boundary value problems with one dimensional $p$-Laplacian, Appl. Anal., 84(8) (2005), 821-831.
  • [10.] R. I. Avery and A. C. Peterson, Three fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42 (2001), 313-322.
  • [11.] H. Lü, D. O'Regan and C. Zhong, Multiple positive solutions for the one-dimension singular $p$-Laplacian, Appl. Math. Comput., 46 (2002), 407-422.
  • [12.] F. Wong, Existence of positive solutions for $m$-Laplacian boundary value problems, Appl. Math.Lett., 12 (1999), 11-17.