**1. Introduction**

In this paper, we will study Fuzzy solutions to

$$T\_{\gamma}u(t) = F(t, u(t)), \quad u(t\_0) = u\_0, \ \gamma \in (0, 1], \tag{1}$$

where subject to initial condition *u*<sup>0</sup> for fuzzy numbers, by the use of the concept of conformable fractional *H*-differentiability, we study the Cauchy problem of fuzzy fractional differential equations for the fuzzy valued mappings of a real variable. Several import-extant results are obtained by applying the embedding theorem in [1] which is a generalization of the classical embedding results [2, 3].

In Section 2 we recall some basic results on fuzzy number. In Section 3 we introduce some basic results on the conformable fractional differentiability [4, 5] and conformable integrability [5, 6] for the fuzzy set-valued mapping in [7]. In Section 4 we show the relation between a solution and its approximate solution to the Cauchy problem of the fuzzy fractional differential equation, and furthermore, and we prove the existence and uniqueness theorem for a solution to the Cauchy problem of the fuzzy fractional differential equation.
