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.

中文翻译：

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1 < r < 2阶非局部混合Volterra-Fredholm型分数阶时滞积分微分方程的存在性及可控性

在我们的文章中，我们主要关注 1 < r <  2 阶非局部混合 Volterra-Fredholm 型分数阶时滞积分微分方程的存在性和可控性。通过应用属于算子余弦函数、分数阶微积分的结果和事实，非紧性测度和不动点方法，建立了主要结果。最初，我们关注温和解的存在性，后来我们建立了所考虑的分数系统的可控性。最后，我们提出一个例子来证明该理论。