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

Abstract

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 ${}_0^c{D}_{\mathfrak{I}}^{\gamma }\mathfrak{x}\left(\mathfrak{I}\right)=\alpha \mathfrak{x}\left(\mathfrak{I}\right)+\mathfrak{P}\left(\mathfrak{I},\mathfrak{x}\left(\mathfrak{I}\right)\right)+\mathfrak{A}\left(\mathfrak{I}\right)\mathfrak{W}\left(\mathfrak{I}\right),\mathfrak{I}\in \left[0,T\right],\mathfrak{x}\left({\mathfrak{I}}_0\right)={\mathfrak{x}}_0$ , in which $\gamma \in \left(0,1\right)$ , $E^1$ is the fuzzy metric space and $I=\left[0,T\right]$ is a real line interval. With the help of few conditions on functions $\mathfrak{P}:I\times E^1\times E^1⟶E^1$ , $\mathfrak{W}\left(\mathfrak{I}\right)$ is control and it belongs to $E^1$ , $\mathfrak{A}\in F\left(I,L\left(E^1\right)\right)$ , and $\alpha$ stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.