一类分数阶混合型差分与和分方程任意初值问题解的存在唯一性

Existence and Uniqueness of the Solution of the Initial Value Problems for a Class of Riemann-Loiuville Fractional Mixed Difference and Summation Equations

本文研究一类Riemann-Liouville型混合分数阶差分与和分方程的初值问题,通过建立与该类初值问题解等价的Volterra和分方程,运用Banach压缩映射原理,在一定条件下,证明该初值问题解的存在唯一性.另外,还通过构造逐次迭代序列,运用离散Mittag-Leffler函数的性质和离散分数阶Gronwall不等式,在较弱的条件下得到该初值问题解的存在唯一性。

This paper is concerned with the initial value problems for a class of Riemann-Loiuville fractional equations mixed with difference and summation.By establishing the Volterra Summation equation equivalent to the solution of this kind of initial value problem,the existence and uniqueness of the solutions of the initial value problem are proved under certain conditions by using the principle of Banach Compression Mapping.In addition,the existence and uniqueness of solutions of the initial value problem is also obtained under weak conditions by using the successive approximation method combined with the discrete Mittag-Leffler function and the discrete fractional Gronwall inequality.