Abstract
We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.
Highlights
In this paper, we study the fractional nonlocal integrodifferential system of the form dαu t dtαA t, B1u tutft, B2u t t g t, s, B3u s ds, B4 u 0 − u0 h u t dt, 1.2 where 0 < α ≤ 1, t ∈ 0, a
We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space
We study the fractional nonlocal integrodifferential system of the form dαu t dtα
Summary
We study the fractional nonlocal integrodifferential system of the form dαu t dtα. The existence results to evolution equations with nonlocal conditions in Banach space were studied first by Byszewski 18, 19 ; subsequently, many authors have been studied the same question, see for instance 20–23. We introduce a new concept in the theory of Semigroup named “implicit evolution system” to show the reader “what is the main difference between the solutions of fractional 0 < α < 1 and classical first order homogeneous evolution equation?” which is based on the work 17 and Pazy 25.
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