New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order 1 < r < 2

In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory.