FIXED POINT THEOREMS IN FUZZY 2-METRIC SPACES: THEORY AND APPLICATIONS

Abstract

The study of fixed points in fuzzy metric spaces has gained significant attention due to its ability to model systems characterized by uncertainty and vagueness. In this research, we investigate the existence and uniqueness of common fixed points in fuzzy 2-metric spaces, which generalize classical metric spaces by incorporating fuzzy distances. We present two theorems addressing common fixed points for self-mappings in fuzzy 2-metric spaces. The first theorem establishes the existence and uniqueness of a common fixed point for four self-mappings under the conditions of weak compatibility and specific inequalities involving the fuzzy 2-metric. The second theorem extends this result to six self-mappings, again ensuring uniqueness under weak compatibility and continuity assumptions. The third theorem investigates the common fixed point for two self-mappings, introducing a function ζ that guarantees the existence and uniqueness of the fixed point. The proof of each theorem relies on ordered weak compatibility (OWC) of the mapping pairs, the continuity of the fuzzy-t-norm, and conditions involving fuzzy distances, ensuring convergence and uniqueness of the common fixed point. These results provide valuable insights into the application of fixed-point theory in fuzzy systems and highlight the potential of fuzzy 2-metric spaces in solving problems where classical methods may not apply.