一类分数阶奇异q-差分方程边值问题解的存在性和唯一性

主要讨论了一类奇异分数阶q-差分方程边值问题,其中控制函数含有分数阶q-导数.首先利用分数阶q-差分理论将该问题转化为等价的分数阶q-积分方程,得到了相关的格林函数;其次详细地证明了积分算子的全连续性,通过运用Schauder不动点定理和Banach不动点定理,证明了该边值问题解的存在性和唯一性,证明过程中,巧妙地应用了贝塔函数,使奇异问题得以解决;最后为了说明定理的有效性,给出了一个例子.

Existence and Uniqueness of Solution for a Class of Singular Fractional q-Difference Boundary Value Problem

In this paper, we have discussed a class of singular fractional q-difference boundary value problems, in which the control function contains fractional q-derivatives. Firstly, it is transformed into an equivalent fractional q-integral equation by using fractional q-difference theory and the related Green''s function is obtained. Secondly, the complete continuity of the integral operator is proved in detail. Then the existence and uniqueness of the solution for fractional order q-difference boundary value problems is discussed by means of Banach fixed point theorem and Schauder fixed point theorem. In the process of proof, beta function is skillfully applied to solve the singular problem. At last, an example is given to illustrate the validity of theorems.