
Positive Solutions for Third Order ThreePoint Boundary Value Problems with $p$Laplacian 

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KeyWord:Positive solution, threepoint boundary value problem, fixed point index, $p$Laplacian operator 
Author Name  Affiliation  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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Abstract: 
In this paper, the existence of positive solutions of the following thirdorder threepoint 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 $\phi_{p}(s)=s^{p2}s$, $p>1$. 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. 


