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Volume 8, Number 2, 2018, Pages 524-531                                                                DOI:10.11948/2018.524
Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences
Yaqiong Cui,Shugui Kang and Huiqin Chen
Keywords:Fractional $q$-difference equation, integral boundary value condition, the first eigenvalue, uniqueness of solutions.
Abstract:
      This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.
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