Local and Global Existence and Uniqueness of Solution for Class of Fuzzy Fractional Functional Evolution Equation

Abstract

For fuzzy fractional functional evolution equations, the concept of global and local existence and uniqueness will be presented in this work. We employ the contraction principle and successive approximations for global and local existence and uniqueness, respectively, as given $\left\{\begin{array}{c}{}_0^c{D}_q^{H}x\left(\mathfrak{I}\right)=f\left(\mathfrak{I},{\text{x}}_{\mathfrak{I}}\right)+{\int }_0^{\mathfrak{I}}g\left(\mathfrak{I},\text{s},{\text{x}}_s\right)\text{ds},\mathfrak{I}\ge {\mathfrak{I}}_0,\mathfrak{I}\in \left[0,T\right],\\ x\left(\mathfrak{I}\right)=\psi \left(\mathfrak{I}-{\mathfrak{I}}_0\right)={\psi }_0\in C_{\sigma },{\mathfrak{I}}_0\ge \mathfrak{I}\ge {\mathfrak{I}}_0-\sigma ,\\ x^{\prime }\left(\mathfrak{I}\right)={\psi }^{\prime }\left(\mathfrak{I}\right)={\psi }_1,\end{array}\right.$ where $C_{\sigma }$ denotes the set of fuzzy continuous mapping defined on $\left[{\mathfrak{I}}_0-\sigma ,T\right]$ and $\sigma >1$ . We also use this method to solve fuzzy fractional functional evolution equations with fuzzy population models and distributed delays using fuzzy fractional functional evolution equations. To explain these results, some theorems are given. Finally, certain fuzzy fractional functional evolution equations are illustrated.