Systems of fuzzy-valued implicit fractional differential equations with nonlocal condition

Abstract

In this paper, two types of fixed point theorems are employed to study the solvability of nonlocal problem for implicit fuzzy fractional differential systems under Caputo gH-fractional differentiability in the framework of generalized metric space. First of all, we extend Krasnoselskii's fixed point theorem to the vector version in the generalized metric space of fuzzy number. Under the Lipschitz conditions, we use Perov's fixed point theorem to prove the global existence of the unique mild fuzzy solution in both types (i) and (ii). When the nonlinearity terms are not Lipschitz, we combine Perov's fixed point theorem with vector version of Krasnoselskii's fixed point theorem to prove the existence of mild fuzzy solutions. Based on the advantage of vector-valued metrics and the role of matrices that converge to zero, we attain some  properties of mild fuzzy solutions such as the boundedness, the attractivity and the Ulam - Hyers  stability. Finally, a computational example is presented to demonstrate the effectivity our main results.