This paper consists of several fixed point theorems in the fuzzy b-metric spaces. As an important result, we give a sufficient condition for a sequence to be Cauchy in the fuzzy b-metric space. Thus we simplify the proofs of many fixed point theorems in the fuzzy b-metric spaces with the well-known contraction conditions.

We initiate the notion of generalized $\mathcal{M}\mathcal{T}$- cyclic contraction mappings with respect to an auxiliary function $\phi$ and investigate the existence of a best proximity point of such mappings in the setting of metric spaces. We also prove the uniqueness of a fixed point, when we assume an additional condition, named as: property UC. Moreover, we discuss the existence and uniqueness of a fixed point of $\mathcal{M}\mathcal{T}$-cyclic orbital contractions in the context of (b-metric) metric spaces.