This article is mainly focusing on the existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces. By using the theoretical concepts related to the fractional calculus, cosine, and sine functions of operators and fixed point approach, we prove our main results. By using Kranoselskii's fixed point theorem, we discuss the existence of the mild solution and by applying the Banach contraction principle, we prove the existence and uniqueness of the mild solution of nonlocal fractional delay differential system. Finally, we provide an example to illustrate the obtained theoretical results.

中文翻译：

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Banach空间中1 <r <2阶非局部分数阶时滞微分系统的存在和唯一性的新研究

本文主要关注Banach空间中1 <  r  <2阶的非局部分数阶时滞微分系统的存在和唯一性。通过使用与算子的分数微积分，余弦和正弦函数有关的理论概念和不动点方法，我们证明了我们的主要结果。通过使用Kranoselskii不动点定理，讨论了温和解的存在性，并通过应用Banach压缩原理，证明了非局部分数阶时滞微分系统的温和解的存在性和唯一性。最后，我们提供一个例子来说明所获得的理论结果。