On Langevin Type of Fractional Time - Delay Differential Equations

The existence uniqueness of solutions of the impulsive delayed fractional differential equations(IDFDEs) was proved. The Ulam-Hyers stability of IDFDEs was demonstrated. The controllability of IDFDEs was shown via iterative learning control technique. In the sequel, the neutral fractional multi-delayed differential equations(NFMDDEs) was introduced. The existence and uniqueness of NFMDDEs was investigated in addition to its stability, and relative controllability of NFMDDEs was proved by means of fixed point technique. Lastly, new fractional integral and derivatives, i.e. φ-generalized Riemann Liouville k-fractional integral, φ-generalized Riemann Liouville k-fractional derivative, φ-generalized Caputo k-fractional derivative were defined and some fundamental features were discussed to build theory’s basement.