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Positive Solutions for Third Order Three-Point Boundary Value Problems with $p$-Laplacian
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KeyWord:Positive solution, three-point boundary value problem, fixed point index, $p$-Laplacian operator
Author NameAffiliation
Xingfang Feng Department of Mathematics, Hebei Normal University, Shijiazhang 050024, China
Department of Scientific Culture, Shijiazhuang Branch, Army Engineering University of PLA, Shijiazhuang 050003, Hebei, China 
Hanying Feng Department of Mathematics, Nantong Institute of Technology, Nantong 226002, China 
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      In this paper, the existence of positive solutions of the following third-order three-point boundary value problem with $p$-Laplacian \begin{equation*} \begin{gathered} \ \ \left\{ \begin{array}{l}\displaystyle(\phi_{p}(u''(t)))'+f(t,u(t))=0,\ t\in (0,1),\u(0)=\alpha u(\eta), u(1)=\alpha u(\eta), u''(0)=0,\end{array} \right. \end{gathered} \end{equation*} is studied, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. By using the fixed point index method, we establish sufficient conditions for the existence of at least one or at least two positive solutions for the above boundary value problem. The main result is demonstrated by providing an example as an application.