A new effective technique of nonlocal controllability criteria for state delay with impulsive fractional integro-differential equation

This proposed work concerns the nonlocal controllability criteria for state delay with an impulsive fractional integro-differential equation in n-dimensional Euclidean space in the sense of the Caputo fractional derivative. The mild solution is attained through the standard Laplace transform and iterative process. In particular, we obtained sufficient conditions by using degree theory. In addition, we exhibit the unique solution and nonlocal controllability criteria of our given problem through Gronwall's inequality and appropriate assumptions. At last, we examine the precision of our findings using numerical computations and applications of the adaptive framework we have provided.