The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability

The foundation of the concepts of fuzzy fractional integral and Caputo gH-partial for fuzzy-valued multivariable functions is defined. As a result, fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved. We present two new results on the existence of two kinds of gH-weak solutions of these problems. The first result is based on the Banach fixed point theorem with the Lipschitz condition of functions on the right-hand side of equations. The second result is based on the nonlinear alternative of the Schauder type for fuzzy-valued continuous functions without the Lipschitz condition. Moreover, we indicate the boundedness and continuous dependence of solutions on the initial data of the problems. Our models are embedded in the sense of Caputo gH-differentiability. Some examples are presented to illustrate the results.