Abstract
In this manuscript, we deal with the nonlocal controllability results for the fractional evolution system of 1< r<2 in a Banach space. The main results of this article are tested by using fractional calculations, the measure of noncompactness, cosine families, Mainardi’s Wright-type function, and fixed point techniques. First, we investigate the controllability results of a mild solution for the fractional evolution system with nonlocal conditions using the Mönch fixed point theorem. Furthermore, we develop the nonlocal controllability results for fractional integrodifferential evolution system by applying the Banach fixed point theorem. Finally, an application is presented for drawing the theory of the main results.
Highlights
1 Introduction Fractional differential equations have arisen as a new branch of applied mathematics that has been utilized to build a variety of mathematical models in science, signal, image processing, biological, control theory, engineering problems, etc
Many authors have addressed the theory of the existence of solutions for fractional differential equations
6 Conclusion The nonlocal controllability results for the fractional differential system of 1 < r < 2 in a Banach space are discussed in this work
Summary
Fractional differential equations have arisen as a new branch of applied mathematics that has been utilized to build a variety of mathematical models in science, signal, image processing, biological, control theory, engineering problems, etc. The researchers established the nonlocal fractional differential systems with or without delay by referring to the nondense domain, semigroup, cosine families, several fixed point techniques, and a measure of noncompactness. Authors have signified controllability results of Caputo fractional evolution systems with order α ∈ (1, 2) referring to the cosine families, Laplace transforms, and different fixed point techniques [56]. For fractional evolution equations of order r ∈ (1, 2) with delay or without delay, numerous researchers have proved their existence, exact and approximate controllability by applying the nonlocal conditions, mixed Volterra–Fredholm type, cosine families, measure of noncompactness, and different fixed point techniques [41, 48, 50, 51, 54]. An application is presented for drawing the law of the main results
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