Existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order 1 < r < 2

In our article, we are primarily concentrating on existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order 1 < r < 2. By applying the results and facts belongs to the cosine function of operators, fractional calculus, the measure of noncompactness and fixed point approach, the main results are established. Initially, we focus the existence of mild solution, and later we establish the controllability of the considered fractional system. Finally, we present an example to demonstrate the theory.