Existence of Positive Solutions for Three-Point Boundary Value Problem with $p$-Laplacian

Abstract:

      By means of the Krasnoselskii's fixed-point theorem in cone, we study the existence of positive solution for the three-point boundary value problem with $p$-Laplacian operator $$\left\{\begin{array}{ll}(\phi_p(u'(t)))'+a(t)f(u(t))=0,\;\;\;t\in (0,1),\\ u(0)=\alpha u(\eta),\;u(1)=\beta u(\eta), \end{array}\right.$$ where $0<\alpha,\; \beta< 1,$ $0<\eta< 1$ and $\phi_p(z)=|z|^{p-2}z,\; p>1.$ Sufficient conditions are given which guarantee the existence of positive solutions of this problem.