Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations

In this paper, we introduce a concept of delayed two parameters Mittag-Leffler type matrix function, which is an extension of the classical Mittag-Leffler matrix function. With the help of the delayed two parameters Mittag-Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method. In addition, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations. Thereafter, we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term. Finally, an example is presented to illustrate the validity of the main theorems.