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In this paper,we are interested in the existence and uniqueness of positive solutions for integral boundary value problem with fractional q-derivative: 			Dqαu(t)+f(t,u(t),u(t))+g(t,u(t))=0,0<t<1,u(0)=Dqu(0)=0,u(1)=μ∫01u(s)dqs, $$\begin{aligned} &D_{q}^{\alpha}u(t)+f\bigl(t,u(t),u(t)\bigr)+g\bigl(t,u(t) \bigr)=0, \quad 0< t< 1, \\ & u(0)=D_{q}u(0)=0, \qquad u(1)=\mu \int_{0}^{1}u(s)\,d_{q}s, \end{aligned}$$  where Dqα$D_{q}^{\alpha}$ is the fractional q-derivative of Riemann–Liouville type, 0<q<1$0< q<1$, 2<α≤3$2<\alpha\leq3 $, and μ is a parameter with 0<μ<[α]q$0<\mu<[\alpha]_{q}$. By virtue of fixed point theorems for mixed monotone operators, we obtain some results on the existence and uniqueness of positive solutions.

Existence and uniqueness results to positive solutions of integral boundary value problem for fractional q-derivatives