5. Fixed Point Theorems in Fuzzy Metric Spaces

In this chapter, we study the fixed point theory in fuzzy metric spaces. This subject is very important in fuzzy nonlinear operator theory. In Section 5.1, we define weak compatible mappings in fuzzy metric spaces and prove some common fixed point theorems for four mappings satisfying some contractions. In Section 5.2, we define R-weakly commuting mappings in intuitionistic fuzzy metric spaces and prove some common fixed point theorems in these spaces. In Section 5.3, we prove some common fixed point theorems for six mappings in three complete fuzzy metric spaces. In Section 5.4, we consider \(\mathcal {L}\)-fuzzy metric spaces and prove a famous theorem, i.e., Jungck’s Theorem in these spaces. In Section 5.5, we study hyper \(\mathcal {L}\)-fuzzy metric spaces and prove some important fixed point theorems in these spaces. Finally, in Section 5.6, we consider the concept of intuitionistic fuzzy quasi-metric spaces and prove a fixed point theorem to obtain the existence of a solution for a recurrence equation associated with the analysis of Quicksort algorithms.