Abstract In this paper, we will introduce some types of Frechet fractional derivative defined on the class of linear correlated fuzzy-valued functions. Firstly, we study Frechet derivative and R-derivative of integer order and investigate their relationship with the well-known generalized Hukuhara derivatives in fuzzy metric space. Secondly, the Riemann-Liouville fractional integral of linear correlated fuzzy-valued functions is well-defined via an isomorphism between R 2 and subspace of fuzzy numbers space R F . That allows us to introduce three types of Frechet fractional derivatives, which are Frechet Caputo derivative, Frechet Riemann-Liouville derivative and Frechet Caputo-Fabrizio derivative. Moreover, some common properties of fuzzy Laplace transform for linear correlated fuzzy-valued function are investigated. Finally, some applications to fuzzy fractional differential equations are presented to demonstrate the usefulness of theoretical results.

中文翻译：

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与 Fréchet 可微性相关的线性相关模糊值函数的分数阶微积分

摘要 在本文中，我们将介绍定义在线性相关模糊值函数类上的一些类型的 Frechet 分数阶导数。首先，我们研究了整数阶的 Frechet 导数和 R 导数，并研究了它们与模糊度量空间中众所周知的广义 Hukuhara 导数的关系。其次，线性相关模糊值函数的黎曼-刘维尔分数积分通过 R 2 和模糊数空间 RF 的子空间之间的同构来明确定义。这允许我们引入三种类型的 Frechet 分数阶导数，它们是 Frechet Caputo 导数、Frechet Riemann-Liouville 导数和 Frechet Caputo-Fabrizio 导数。此外，研究了线性相关模糊值函数的模糊拉普拉斯变换的一些共同性质。最后，