A survey on the initial value problems of fuzzy implicit fractional differential equations

Abstract The purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the Caputo-fuzzy-type, Riemann-Liouville-fuzzy-type and Hadamard-fuzzy-type concepts of fractional derivative in the case of the order α ∈ ( 0 , 1 ) . In particular, the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated. In addition, some results concerning continuous dependence of solutions on perturbations of the given parameters, namely, of the initial conditions, of the function and parameters on the right-hand side, of the order of fractional derivative are analyzed. Finally, some examples are given to illustrate our main results.

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