| Title:
|
Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian (English) |
| Author:
|
Miao, Chunmei |
| Author:
|
Zhao, Junfang |
| Author:
|
Ge, Weigao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
957-973 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we deal with the four-point singular boundary value problem $$ \begin {cases} (\phi _p(u'(t)))'+q(t)f(t,u(t),u'(t))=0,& t\in (0,1),\\ u'(0)-\alpha u(\xi )=0, \quad u'(1)+\beta u(\eta )=0, \end {cases} $$ where $\phi _p(s)=|s|^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q(t)>0$, $t\in (0,1)$, and $f\in C([0,1]\times (0,+\infty )\times \mathbb R,(0,+\infty ))$ may be singular at $u = 0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions. (English) |
| Keyword:
|
singular |
| Keyword:
|
four-point |
| Keyword:
|
positive solution |
| Keyword:
|
$p$-Laplacian |
| MSC:
|
34B10 |
| MSC:
|
34B16 |
| MSC:
|
34B18 |
| idZBL:
|
Zbl 1224.34053 |
| idMR:
|
MR2563569 |
| . |
| Date available:
|
2010-07-20T15:51:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140528 |
| . |
| Reference:
|
[1] Agarwal, R. P., O'Regan, D.: Nonlinear superlinear singular and nonsingular second order boundary value problems.J. Differ. Equations 143 (1998), 60-95. Zbl 0902.34015, MR 1604959, 10.1006/jdeq.1997.3353 |
| Reference:
|
[2] Agarwal, R. P., O'Regan, D.: Existence theory for single and multiple solutions to singular positone boundary value problems.J. Differ. Equations 175 (2001), 393-414. Zbl 0999.34018, MR 1855974, 10.1006/jdeq.2001.3975 |
| Reference:
|
[3] Agarwal, R. P., O'Regan, D.: Twin solutions to singular Dirichlet problems.J. Math. Anal. Appl. 240 (1999), 433-445. Zbl 0946.34022, MR 1731655, 10.1006/jmaa.1999.6597 |
| Reference:
|
[4] Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations.J. Differ. Equations 211 (2005), 282-302. Zbl 1074.34048, MR 2125544, 10.1016/j.jde.2004.10.031 |
| Reference:
|
[5] Ha, K., Lee, Y.: Existence of multiple positive solutions of singular boundary value problems.Nonlinear Anal. 28 (1997), 1429-1438. Zbl 0874.34016, MR 1428660, 10.1016/0362-546X(95)00231-J |
| Reference:
|
[6] Khan, R. A.: Positive solutions of four-point singular boundary value problems.Appl. Math. Comput. 201 (2008), 762-773. Zbl 1152.34016, MR 2431973, 10.1016/j.amc.2008.01.014 |
| Reference:
|
[7] Lan, K., Webb, J. L.: Positive solutions of semilinear differential equations with singularities.J. Differ. Equations 148 (1998), 407-421. Zbl 0909.34013, MR 1643199, 10.1006/jdeq.1998.3475 |
| Reference:
|
[8] Liu, Y., Qi, A.: Positive solutions of nonlinear singular boundary value problem in abstract space.Comput. Math. Appl. 47 (2004), 683-688. Zbl 1070.34079, MR 2051339, 10.1016/S0898-1221(04)90055-7 |
| Reference:
|
[9] Liu, B., Liu, L., Wu, Y.: Positive solutions for singular second order three-point boundary value problems.Nonlinear Anal. 66 (2007), 2756-2766. Zbl 1117.34021, MR 2311636, 10.1016/j.na.2006.04.005 |
| Reference:
|
[10] Ma, D., Han, J., Chen, X.: Positive solution of three-point boundary value problem for the one-dimensional $p$-Laplacian with singularities.J. Math. Anal. Appl. 324 (2006), 118-133. Zbl 1110.34016, MR 2262460, 10.1016/j.jmaa.2005.11.063 |
| Reference:
|
[11] Ma, D., Ge, W.: Positive solution of multi-point boundary value problem for the one-dimensional $p$-Laplacian with singularities.Rocky Mountain J. Math. 137 (2007), 1229-1249. Zbl 1139.34018, MR 2360295, 10.1216/rmjm/1187453108 |
| Reference:
|
[12] Ma, D., Ge, W.: The existence of positive solution of multi-point boundary value problem for the one-dimensional $p$-Laplacian with singularities.Acta Mech. Sinica (Beijing) 48 (2005), 1079-1088. Zbl 1124.34308, MR 2205048 |
| Reference:
|
[13] Rachůnková, I., Staněk, S., Tvrdý, M.: Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations.In: Handbook of Differential Equations. Ordinary Differential Equations, Vol. 3 A. Cañada, P. Drábek, A. Fonda Elsevier (2006), 607-723. MR 2457638 |
| Reference:
|
[14] Wei, Z., Pang, C.: Positive solutions of some singular $m$-point boundary value problems at non-resonance.Appl. Math. Comput. 171 (2005), 433-449. Zbl 1085.34017, MR 2192885, 10.1016/j.amc.2005.01.043 |
| Reference:
|
[15] Xu, X.: Positive solutions for singular $m$-point boundary value problems with positive parameter.J. Math. Anal. Appl. 291 (2004), 352-367. Zbl 1047.34016, MR 2034079, 10.1016/j.jmaa.2003.11.009 |
| Reference:
|
[16] Zhang, X., Liu, L.: Eigenvalue of fourth-order $m$-point boundary value problem with derivatives.Comput. Math. Appl. 56 (2008), 172-185. Zbl 1145.34315, MR 2427696, 10.1016/j.camwa.2007.08.048 |
| Reference:
|
[17] Zhang, X., Liu, L.: Positive solutions of fourth-order four-point boundary value problems with $p$-Laplacian operator.J. Math. Anal. Appl. 336 (2007), 1414-1423. Zbl 1125.34018, MR 2353024, 10.1016/j.jmaa.2007.03.015 |
| . |
