Thesis

The thesis establishes the consistency of the fuzzy topology induced by fuzzy metric spaces with prescribed topologies and introduces the notion of a compatible pair of mutually continuous mappings. This leads to a fixed point theorem in fuzzy metric spaces, with a distinctive feature that it does not necessitate mapping continuity. The exploration further extends to compatible mappings in generalized fuzzy metric spaces, introducing common fixed point theorems. Semi-compatibility and weak compatibility concepts are introduced, offering alternative conditions for mapping continuity.