Unique Nontrivial Solutions for a Class of Fractional Boundary Value Problem with Q-Derivatives

In this paper, we investigate unique nontrivial solution for a class nonlinear boundary value problem with fractional q-derivative where
$D_{q}^{\alpha}$
are fractional q-derivative of Riemann-Liouville type,
$b \geq 0, \beta \geq 0$
. Based on the fixed point theorems for increasing
$\varphi-(h, e)$
-concave operators, we establish some sufficient conditions for the existence and uniqueness of nontrivial solution of the problem; on the other hand, an iterative scheme is constructed for approximating the nontrivial solution.