Positive solutions of singular third-order three-point boundary value problems. (English) Zbl 1169.34314

Yao, Qingliu

J. Math. Anal. Appl. 354, No. 1, 207-212 (2009).

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

Keywords:

ordinary differential equation; multi-point boundary value problem; singular nonlinearity; positive solution

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