Semilinear fractional differential equations based on a new integral operator approach

Semilinear fractional differential equations (SFDEs) often arise in some dynamical systems. In most of the existing literature, the fixed point theorems are used to prove the existence and uniqueness of the solutions of SFDEs. In this paper, we give a new way to prove the existence and uniqueness of the solutions by introducing a new integral operator associated with the Mittag–Leffler function.

► We introduce a new integral operator containing the Mittag–Leffler function.
► We establish an integral inequality associated with the integral operator.
► Using the integral operator, we give a new way to study well-posedness of semilinear fractional differential equations.