Answer to a question on Cauchy sequence in fuzzy b-metric spaces and its applications

We give a sufficient condition of Cauchy sequence with contraction constant $\lambda \in (0, 1)$ in fuzzy b-metric spaces, which is a positive answer to the open question proposed by Rakić et al. (J. Inequal. Appl. 2020:99, 2020). Thus we simplify the proofs of many fixed point theorems in fuzzy b-metric spaces. Applying this result, we obtain two single-valued fixed point results and a set-valued fixed point result for Banach type in fuzzy b-metric spaces, which improve and extend the relevant results in the literature.