Positive Solutions for Third Order Three-Point Boundary Value Problems with $p$-Laplacian

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.