Multiplicity of positive solutions for singular three-point boundary value problems at resonance. (English) Zbl 1188.34032

The paper deals with the problem of existence of positive solutions for the three-point boundary value problem
\[ x''+m^{2}x=f(t,x)+e(t),\quad t\in (0,1), \]
with conditions
\[ x'(0)= 0, \qquad x(\eta)=x(1), \] where \(m \in (0,\frac{\pi}{2})\) and \(\eta \in (0,1)\) are given.
The problem is assumed to be with singular nonlinear perturbations at resonance. The proof is based on a nonlinear Leray-Schauder principle and the fixed point theorem in cones for completely continuous operators.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text: DOI


[1] Infante, G.; Webb, J.R.L., Loss of positivity in a nonlinear scalar heat equation, Nodea nonlinear differential equations appl., 13, 249-261, (2006) · Zbl 1112.34017
[2] Il’in, V.; Moiseev, E., Non-local boundary value problem of the first kind for a strum – liouville operator in its differential and finite difference aspects, Differ. equ., 23, 803-810, (1987)
[3] Gupta, C., Existence theorems for a second order \(m\)-point boundary value problem at resonance, Int. J. math. math. sci., 18, 705-710, (1995) · Zbl 0839.34027
[4] Feng, M.; Ge, W., Positive solutions for a class of \(m\)-point singular boundary value problems, Math. comput. modelling, 46, 375-383, (2007) · Zbl 1142.34012
[5] Gupta, C., Solvability of a multi-point boundary value problem at resonance, Results math., 28, 270-276, (1995) · Zbl 0843.34023
[6] Khan, R.A.; Webb, J.R.L., Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear anal., 64, 1356-1366, (2006) · Zbl 1101.34005
[7] Ma, R., Positive solutions of some three-point boundary value problem, Electron. J. differential equations, 34, 1-8, (1999)
[8] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527
[9] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025
[10] Zhang, Q.; Jiang, D., Upper and lower solutions method and a second order three-point singular boundary value problem, Comput. math. appl., 56, 1059-1070, (2008) · Zbl 1155.34305
[11] Han, X., Positive solution for a three-point boundary value problem at resonance, J. math. anal. appl., 336, 556-568, (2007) · Zbl 1125.34014
[12] Infante, G.; Zima, M., Positive solutions of multi-point boundary value problems at resonance, Nonlinear anal., 69, 2458-2465, (2008) · Zbl 1203.34041
[13] Agarwal, R.P.; O’Regan, D., Existence theory for single and multiple solutions to singular positone boundary value problems, J. differential equations, 175, 393-414, (2001) · Zbl 0999.34018
[14] Chu, J.; Nieto, J.J., Impulsive periodic solutions of first-order singular differential equations, Bull. lond. math. soc., 40, 143-150, (2008) · Zbl 1144.34016
[15] Chu, J.; Torres, P.J.; Zhang, M., Periodic solutions of second order non-autonomous singular dynamical systems, J. differential equations, 239, 196-212, (2007) · Zbl 1127.34023
[16] del Pino, M.A.; Manásevich, R.F.; Montero, A., \(T\)-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect. A, 120, 231-243, (1992) · Zbl 0761.34031
[17] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 282-302, (2005) · Zbl 1074.34048
[18] Rachunková, I.; Tvrdý, M.; Vrkoc˘, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008
[19] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040
[20] Torres, P.J., Weak singularities may help periodic solutions to exist, J. differential equations, 232, 277-284, (2007) · Zbl 1116.34036
[21] Zhang, M., A relationship between the periodic and the Dirichlet equations, Proc. roy. soc. Edinburgh sect., 128A, 1099-1114, (1998) · Zbl 0918.34025
[22] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005
[23] O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Academic Dordrecht · Zbl 1077.34505
[24] Krasnosel’skii, M.A., Positive solution of operator equations, (1964), Noordhoff Groningen
[25] Lü, H.; O’Regan, D.; Agarwal, R.P., Upper and lower solutions for the singular \(p\)-Laplacian with sign changing nonlinearities and nonlinear boundary data, J. comput. appl. math., 181, 442-466, (2005) · Zbl 1082.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.