Fractional calculus of linear correlated fuzzy-valued functions related to Fréchet differentiability

In this paper, we will introduce some types of Fréchet fractional derivative defined on the class of linear correlated fuzzy-valued functions. Firstly, we study Fréchet 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 R2 and subspace of fuzzy numbers space RF. That allows us to introduce three types of Fréchet fractional derivatives, which are Fréchet Caputo derivative, Fréchet Riemann-Liouville derivative and Fréchet 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.