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The existence and uniqueness of solutions to Hilfer fractional neutral delay integro‐differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro‐differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results. [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]

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1 < r < 2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems. [ABSTRACT FROM AUTHOR]

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable. [ABSTRACT FROM AUTHOR]

The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch's fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included. [ABSTRACT FROM AUTHOR]