Existence and uniqueness of mild solution for fractional-order controlled fuzzy evolution equation (Q2071851)

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