




Volume 37 • Number 4 • 2014 

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Existence of Positive Solutions for a ThreePoint Boundary Value Problem with Fractional $q$Differences
Yueqiang Song
Abstract.
In this paper, we consider the
following nonlinear $q$fractional threepoint 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 $0<\beta\eta^{\alpha2}<1$. 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
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