### On fractional evolution equations with an extended ψ-fractional derivative

#### Abstract

The main crux of this work is to investigate the existence and uniqueness of solutions for the following fuzzy fractional evolution problems\begin{equation}\label{P1} \left\{ \begin{array}{lrr}^c D_{0^+}^q u(t)=\mathcal{A}(t)u(t),&t\in J=[0,T],\\\\ u(0)=u_0,& \end{array}\right.\end{equation}and \begin{equation}\label{P2} \left\{ \begin{array}{lrr}^c D_{0^+}^q u(t)=\mathcal{A}(t)u(t)+h(t,u(t)),& t\in J=[0,T],\\\\ u(0)=u_0,& \end{array}\right.\end{equation} where $^CD_{0^+,gH}^{q,\Psi}$ is the extended $\Psi-$Caputo-Fuzzy fractional derivative of $u(t)$ at order $q\in(0,1)$, $T>0$, $h$ is a fuzzy continuous function and $\mathcal{A}(t)$ is a bounded linear operator. Our approach is based on the application of an extended $\Psi-$Caputo fractional derivative of order $q\in(0,1)$ valid on fuzzy functions paired with Banach contraction principle. As an example of application, we provide one at the end of this paper to show how the results can be used.

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