Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique

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Our purpose in this paper is to study the approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory, the approximating technique, semigroup theory and the theory of fixed point. We also derive a variation of constants formula for representing a solution of the given system by resolvent operators and then study the existence of a mild solution of the system. We prove the main results by taking the impulsive functions to be continuous only and taking the nonlocal initial condition function to be Lipschitz continuous in the first case and continuous only in the second case. Finally, the main results are illustrated using examples.