A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case

Abstract

In this paper, the existence results of the solution and the finite-time stability (FTS) are focused for fractional fuzzy differential equations (FFDEs) involving non-instantaneous impulsive effects and perturbance parameters. In view of the definitions of short-memory fractional derivatives and a short-memory fractional modeling approach, the existence of a unique solution is proved through utilizing the fixed point theorem of a weakly contractive mapping on the partially ordered fuzzy space, and the sufficient condition of FTS is obtained by using the generalized fractional Gronwall inequality. The fuzzy fractional derivatives used in this study include the concept of derivatives based on the generalized Hukuhara difference (gH-difference) and the granular difference (gr-difference). It is demonstrated that the obtained results based on the gr-difference can overcome restrictions associated with the previous approaches defined via the gH-difference concept. Finally, some numerical examples are provided to verify our main results.