Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

Abstract

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag-Leffler type matrix function with a logarithm and delayed Mittag-Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag-Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam-Hyers stability of Hadamard-type fractional time-delay nonlinear equations.