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Existence of triple positive solutions for a third-order three-point boundary value problem. (English) Zbl 1157.34311

Summary: We investigate the existence of triple positive solutions for the boundary value problem
\[ \begin{aligned} & u'''(t)=a(t)f(t,u(t),u'(t),u''(t)),\quad 0<t<1,\\ & u(0)=\delta u(\eta),\quad u'(\eta)=0,\quad u''(1)=0,\end{aligned} \]
where \(\delta\in (0,1)\), \(\eta\in [1/2,1)\) are constants. \(f:[0,1]\times [0,\infty)\times \mathbb{R}^2\to [0,\infty)\), \(q:(0,1)\to [0,\infty)\) are continuous. First, Green’s function for the associated linear boundary value problem is constructed, and then, by using a fixed-point theorem due to Avery and Peterson, we establish results on the existence of triple positive solutions to the boundary value problem.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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