×

zbMATH — the first resource for mathematics

Positive solutions of singular third-order three-point boundary value problems. (English) Zbl 1169.34314
Summary: We consider the existence of multiple positive solutions for the following nonlinear third-order three-point boundary value problem
\[ \begin{cases} u'''(t)=h(t)f(t,u(t)), \quad 0<t<1,\\ u(0)=u'(\eta)=u''(1)=0.\end{cases}\tag{P} \]
Positive solutions are established by using the Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type. The nonlinear term is allowed to be singular. Main results show that this class of problems can have \(n\) positive solutions provided that the conditions on the nonlinear term on some bounded sets are appropriate.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, D.R., Multiple positive solutions for a three-point boundary value problem, Math. comput. modelling, 27, 49-57, (1998) · Zbl 0906.34014
[2] Anderson, D.R.; Davis, J.M., Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. math. anal. appl., 267, 135-157, (2002) · Zbl 1003.34021
[3] Yao, Q., The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta math. appl. sin. engl. ser., 19, 117-122, (2003) · Zbl 1048.34031
[4] Yao, Q., Existence of positive solution for a third-order three-point boundary value problem with semipositone nonlinearity, J. math. res. exposition, 23, 591-596, (2003) · Zbl 1050.34030
[5] Du, Z.; Ge, W.; Lin, X., A class of third-order multi-point boundary value problems, J. math. anal. appl., 294, 104-112, (2004) · Zbl 1053.34017
[6] Haiyan, H.; Liu, Y., Multiple positive solutions to third-order multi-point singular boundary value problems, Proc. Indian acad. sci. math. sci., 114, 4, 104-112, (2004)
[7] Sun, Y., Positive solutions of singular third-order three-point boundary value problem, J. math. anal. appl., 306, 589-603, (2005) · Zbl 1074.34028
[8] Yao, Q., Solutions and positive solutions to a class of nonlinear third-order three-point boundary value problems, J. math. (Wuhan), 27, 704-708, (2007), (in Chinese) · Zbl 1150.34331
[9] Eloe, W.P.; Henderson, J., Singular nonlinear \((n - k, k)\) conjugate boundary value problems, J. differential equations, 133, 136-151, (1997) · Zbl 0870.34031
[10] Hai, D.D., Positive solutions for semilinear elliptic equations in annular domains, Nonlinear anal., 37, 1051-1058, (1999) · Zbl 1034.35044
[11] Agarwal, R.P.; O’Regan, D., Twin solutions to singular boundary value problems, Proc. amer. math. soc., 128, 2085-2094, (2000) · Zbl 0946.34020
[12] Yao, Q., Positive solutions of a class of singular sublinear two-point boundary value problems, Acta math. appl. sin. Chinese ser., 24, 522-526, (2001), (in Chinese) · Zbl 0998.34017
[13] Yao, Q., Existence of n solutions and/or positive solutions to a semipositone elastic beam equation, Nonlinear anal., 66, 138-150, (2007) · Zbl 1113.34013
[14] Yao, Q., Positive solutions of nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right, Nonlinear anal., 69, 1570-1580, (2008) · Zbl 1217.34039
[15] Yao, Q., Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends, Nonlinear anal., 69, 2683-2694, (2008) · Zbl 1157.34018
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.