Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form   0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.