Positive solutions of four-point boundary value problem for fourth order ordinary differential equation. (English) Zbl 1208.34027
The boundary value problem
\[ \begin{aligned} &y^{(4)}(t)-f(t,y(t),y''(t))=0,\quad 0\leq t\leq 1,\\ & y(0)=y(1)=0,\\ &ay''(\xi_1)-by'''(\xi_1)=0,\qquad cy''(\xi_2)+dy'''(\xi_2)=0\end{aligned} \]
is studied, where \(0\leq \xi_1<\xi_2\leq 1\). Some sufficient conditions guaranteeing the existence of a positive solution to the above four-point boundary value problem are obtained by using the Krasnoselskii fixed point theorem.
\[ \begin{aligned} &y^{(4)}(t)-f(t,y(t),y''(t))=0,\quad 0\leq t\leq 1,\\ & y(0)=y(1)=0,\\ &ay''(\xi_1)-by'''(\xi_1)=0,\qquad cy''(\xi_2)+dy'''(\xi_2)=0\end{aligned} \]
is studied, where \(0\leq \xi_1<\xi_2\leq 1\). Some sufficient conditions guaranteeing the existence of a positive solution to the above four-point boundary value problem are obtained by using the Krasnoselskii fixed point theorem.
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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