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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
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