Common Fixed-Point Results for Geraghty-Ciric Contraction in Fuzzy Metric Space

Abstract

 The theory of fixed points has undergone significant development over the years through the generalization of contraction conditions and the modification of the axioms of metric spaces. These advancements have broadened the applicability of fixed-point results across various branches of mathematics and applied sciences. In particular, fuzzy metric spaces have provided a flexible and effective framework for dealing with uncertainty, leading to a growing interest in the study of contractive mappings within this setting. In this paper, we investigate common fixed-point results for a pair of weakly compatible mappings satisfying a Geraghty–Ćirić-type contraction condition in the context of complete fuzzy metric spaces. By using the properties of Cauchy sequences and convergence in fuzzy metric spaces, we establish sufficient conditions for the existence and uniqueness of a common fixed point for the mappings under consideration. The results obtained in this work extend and generalize several known theorems in the existing literature, especially those related to Geraghty- and Ćirić-type contractions. To demonstrate the applicability of the main theorem, an illustrative example is provided. Furthermore, a corollary is derived as a special case of the main result, highlighting its versatility and potential for further generalizations.