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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This research investigates the existence of nonlocal impulsive fractional integrodifferential equations of order 1 < r < 2 with infinite delay. To begin with, we discuss the existence of a mild solution for the fractional derivatives by using the sectorial operators, the nonlinear alternative of the Leray-Schauder fixed point theorem, mixed Volterra-Fredholm integrodifferential types, and impulsive systems. Furthermore, we develop the optimal control results for the given system. The application of our findings is demonstrated with the help of an example. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and Sadovskii‐Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

This manuscript mainly focuses on the exact controllability of Sobolev-type Hilfer fractional neutral delay Volterra-Fredholm integro-differential systems. The principal findings of this discussion are established by using the theories on fractional calculus, the measure of noncompactness and Mönch fixed point technique. Initially, the exact controllability of the system is presented and then we improve the discussion to the system with nonlocal conditions. Finally, abstract and filter systems are provided for the illustration. [ABSTRACT FROM AUTHOR]

In this theory, the existence of a mild solution for a neutral partial integrodifferential nonlocal system with finite delay is presented and proved using the techniques of the Monch–Krasnosel'skii type of fixed point theorem, a measure of noncompactness and resolvent operator theory. For this work, we have introduced some sufficient conditions to confirm the existence of the neutral partial integrodifferential system. An illustration of the derived results is offered at the end with a filter system corresponding to our existence result. [ABSTRACT FROM AUTHOR]