Controllability of nonlinear neutral implicit ABC non-integer integro-differential equations with state and control delay

The controllability of ABC nonlinear neutral implicit integro-differential equation of fractional order 0 < 1 with delay in state and control has been examined in this work. To obtain the controllability results of the defined problem, we specifically apply the k-set contraction mapping and Darbo fixed point theorem. The provided example is used to verify the findings. The graphical illustrations are exhibited to view the orthogonal eigen-vectors corresponding to the solution.