# Existence of multiple positive solutions of $n\mathrm{th}$-order $m$-point boundary value problems

Mathematica Bohemica (2010)

• Volume: 135, Issue: 1, page 15-28
• ISSN: 0862-7959

top

## Abstract

top
The paper deals with the existence of multiple positive solutions for the boundary value problem $\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right){u}^{\left(n-1\right)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f\left(t,u\left(t\right),{u}^{\text{'}}\left(t\right),...,{u}^{\left(n-2\right)}\left(t\right)\right)=0,\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4pt}{0ex}}0 where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.

## How to cite

top

Liang, Sihua, and Zhang, Jihui. "Existence of multiple positive solutions of $n{\rm th}$-order $m$-point boundary value problems." Mathematica Bohemica 135.1 (2010): 15-28. <http://eudml.org/doc/38107>.

@article{Liang2010,
abstract = {The paper deals with the existence of multiple positive solutions for the boundary value problem $\{\left\lbrace \begin\{array\}\{ll\} (\varphi (p(t)u^\{(n-1)\})(t))^\{\prime \} + a(t)f(t, u(t), u^\{\prime \}(t), \ldots , u^\{(n-2)\}(t)) = 0, \quad \ 0 < t < 1, \\ u^\{(i)\}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^\{(n-2)\}(0) = \sum \_\{i=1\}^\{m-2\}\alpha \_iu^\{(n-2)\}(\xi \_i),\quad u^\{(n-1)\}(1) = 0, \end\{array\}\right.\}$ where $\varphi \colon \mathbb \{R\} \rightarrow \mathbb \{R\}$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.},
author = {Liang, Sihua, Zhang, Jihui},
journal = {Mathematica Bohemica},
keywords = {boundary-value problems; positive solutions; fixed-point theorem; cone; boundary-value problem; positive solution; fixed-point theorem; cone},
language = {eng},
number = {1},
pages = {15-28},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of multiple positive solutions of $n\{\rm th\}$-order $m$-point boundary value problems},
url = {http://eudml.org/doc/38107},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Liang, Sihua
AU - Zhang, Jihui
TI - Existence of multiple positive solutions of $n{\rm th}$-order $m$-point boundary value problems
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 1
SP - 15
EP - 28
AB - The paper deals with the existence of multiple positive solutions for the boundary value problem ${\left\lbrace \begin{array}{ll} (\varphi (p(t)u^{(n-1)})(t))^{\prime } + a(t)f(t, u(t), u^{\prime }(t), \ldots , u^{(n-2)}(t)) = 0, \quad \ 0 < t < 1, \\ u^{(i)}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^{(n-2)}(0) = \sum _{i=1}^{m-2}\alpha _iu^{(n-2)}(\xi _i),\quad u^{(n-1)}(1) = 0, \end{array}\right.}$ where $\varphi \colon \mathbb {R} \rightarrow \mathbb {R}$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
LA - eng
KW - boundary-value problems; positive solutions; fixed-point theorem; cone; boundary-value problem; positive solution; fixed-point theorem; cone
UR - http://eudml.org/doc/38107
ER -

## References

top
1. Agarwal, R. P., O'Regan, D., 10.1006/jdeq.2000.3808, J. Differ. Equations 170 142-156 (2001). (2001) MR1813103DOI10.1006/jdeq.2000.3808
2. Bai, C., Fang, J., 10.1016/S0096-3003(02)00227-8, Appl. Math. Comput. 140 (2003), 297-305. (2003) MR1953901DOI10.1016/S0096-3003(02)00227-8
3. Baxley, J. V., 10.1016/0022-247X(90)90388-V, J. Math. Anal. Appl. 147 (1990), 122-133. (1990) MR1044690DOI10.1016/0022-247X(90)90388-V
4. Deimling, K., Nonlinear Functional Analysis, Springer, New York (1985). (1985) Zbl0559.47040MR0787404
5. Feng, W., Webb, J. R. L., 10.1006/jmaa.1997.5520, J. Math. Anal. Appl. 212 (1997), 467-480. (1997) MR1464891DOI10.1006/jmaa.1997.5520
6. Il'in, V. A., Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differ. Equations 23 (1987), 979-987. (1987)
7. al, P. Kelevedjiev et, Another understanding of fourth-order four-point boundary-value problems, Electron. J. Differ. Equ., Paper No. 47 (2008), 1-15. (2008) MR2392951
8. Kong, L., Kong, Q., 10.1016/j.jmaa.2006.08.064, J. Math. Anal. Appl. 330 (2007), 1393-1411. (2007) Zbl1119.34009MR2308449DOI10.1016/j.jmaa.2006.08.064
9. Lan, K. Q., 10.1112/S002461070100206X, J. London Math. Soc. 63 (2001), 690-704. (2001) Zbl1032.34019MR1825983DOI10.1112/S002461070100206X
10. Liang, S. H., Zhang, J. H., 10.1016/j.cam.2007.02.020, J. Comput. Appl. Math. 214 (2008), 78-89. (2008) Zbl1183.34031MR2391674DOI10.1016/j.cam.2007.02.020
11. Liu, B. F., Zhang, J. H., 10.1016/j.jmaa.2004.09.036, J. Math. Anal. Appl. 309 (2005), 505-516. (2005) Zbl1086.34022MR2154132DOI10.1016/j.jmaa.2004.09.036
12. Liu, Yuji, Non-homogeneous boundary-value problems of higher order differential equations with $p$-Laplacian, Electron J. Differ. Equ., Paper No. 20 (2008), 1-43. (2008) Zbl1139.34012MR2383384
13. Wang, J. Y., 10.1090/S0002-9939-97-04148-8, Proc. Amer. Math. Soc. 125 (1997), 2275-2283. (1997) Zbl0884.34032MR1423340DOI10.1090/S0002-9939-97-04148-8
14. Wang, Y., Ge, W., Existence of multiple positive solutions for multi-point boundary value problems with a one-dimensional $p$-Laplacian, Nonlinear Anal., Theory Methods Appl. 67 (2007), 476-485. (2007) MR2317182
15. Wang, Y., Hou, C., 10.1016/j.jmaa.2005.09.085, J. Math. Anal. Appl. 315 (2006), 144-153. (2006) Zbl1098.34017MR2196536DOI10.1016/j.jmaa.2005.09.085
16. Webb, J. R. L., 10.1016/S0362-546X(01)00547-8, Nonlinear Anal. 47 (2001), 4319-4332. (2001) Zbl1042.34527MR1975828DOI10.1016/S0362-546X(01)00547-8
17. Zhou, Y. M., Su, H., Positive solutions of four-point boundary value problems for higher-order with $p$-Laplacian operator, Electron. J. Differ. Equ., Paper No. 05 1-14 (2007). (2007) Zbl1118.34021MR2278419

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.