# Article

 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 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. 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: 2013-09-21 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 .

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