Volume 37 • Number 4 • 2014   • Existence of Positive Solutions for a Three-Point Boundary Value Problem with Fractional $q$-Differences Yueqiang Song Abstract. In this paper, we consider the following nonlinear $q$-fractional three-point boundary value problem \begin{equation} (D_{q}^{\alpha}u)(t) + f(t,u(t))=0, \quad 0 < t < 1, \quad 2 < \alpha < 3,\\ u(0) = (D_qu)(0) = 0, \quad (D_qu)(1) = \beta (D_qu)(\eta), \end{equation} where $0<\beta\eta^{\alpha-2}<1$. By the properties of the Green function and the lower and upper solution method, some new existence to the above boundary value problem are established. As applications, examples are presented to illustrate the main results. 2010 Mathematics Subject Classification: 26A33, 34B18, 34B27 Full text: PDF