ON GRAPHICAL FUZZY METRIC SPACES AND RELATED FIXED POINT THEOREMS

The notion of triangular inequality plays an important role in determining the structure of distance spaces. In particular, the structure of fuzzy metric spaces depends on the triangular inequality and the concerned t-norm. In most of the fixed point theorems in fuzzy metric spaces both the triangular inequality and the concerned t-norm have a major impact on the proof of fixed point theorems. Inspired by the concept of graphical metric space, it was recently introduced in N. Saleem et al., On Graphical Fuzzy Metric Spaces with Application to Fractional Differential Equations, , Fractal and Fract., 6:5 (2022), 238:1-12, the notion of graphical fuzzy metric space and proved some fixed point results. The triangular inequality in such spaces is replaced by a weaker one which is directly associated with the graphical structure affine with the space. In this paper some observations on the recent results of Saleem et al. are made and so the results are revisited. Some related topological properties with some new fixed point results in graphical fuzzy metric spaces are also proved. The results of this paper generalize and extend Banach contraction principle and some other known results in this new setting. Several examples are given which support the claims and illustrate the significance of the new concepts and results.