Results on approximate controllability of neutral Sobolev-type delayed inclusion integro-differential system with impulse

The main purpose of this research paper is to explore sufficient conditions for the existence of the mild solution and approximate controllability results for a new class of the Sobolev-type delayed neutral integro-differential inclusion system with nonlocal condition and instantaneous impulse in Banach space. We incorporate infinite delay into the proposed system using the theory of abstract phase space, allowing a realistic representation of systems influenced by their entire past. The Sobolev-type term further enhances the model's flexibility, ensures smoother and more stable solutions. Firstly, the proposed problem is converted into an equivalent fixed point problem. Then, we establish the existence of the mild solution for the proposed system by leveraging semigroup theory, the Bohnenblust–Karlin fixed point theorem, and theory of multivalued map. The main results are derived under the consideration that the corresponding linear system is approximately controllable. A concrete example is provided to illustrate the abstract findings.