Existence of positive solutions for nonlinear third-order three-point boundary value problems. (English) Zbl 1141.34310
Summary: This paper is concerned with the following nonlinear third-order three-point boundary value problem:
\[ u'''(t)+a(t)f(u(t))=0,\quad t\in(0,1),\;u(0)=u'(0)=0,\;u'(1)=\alpha u'(\eta), \]
where \(0<\eta<1\) and \(1\leq\alpha<\frac 1\eta\). First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when \(f\) is superlinear or sublinear.
\[ u'''(t)+a(t)f(u(t))=0,\quad t\in(0,1),\;u(0)=u'(0)=0,\;u'(1)=\alpha u'(\eta), \]
where \(0<\eta<1\) and \(1\leq\alpha<\frac 1\eta\). First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when \(f\) is superlinear or sublinear.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
third-order three-point boundary value problem; Green’s function; positive solution; cone; fixed point theoremReferences:
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