Abstract
In the present paper, we adopt a short and sharpened approach to prove fixed point results involving fuzzy mathcal{H}-contractive mappings utilized in (Wardowski, Fuzzy Sets Syst. 125:245–252, 2013) and other related articles. In this process, we are able to relax some conditions utilized by earlier authors which in turn yields affirmative answers to some open questions raised by earlier authors.
Highlights
The existing literature on fuzzy sets and systems contains several definitions of fuzzy metric spaces [2,3,4]
The most popular definition of fuzzy metric space is essentially due to Kramosil and Michalek [5]
With a view to having a Hausdorff topology, George and Veeramani [6] modified the concept of fuzzy metric spaces initiated by Kramosil and Michalek [5]
Summary
The existing literature on fuzzy sets and systems contains several definitions of fuzzy metric spaces [2,3,4]. Motivated by Samet et al [7], Salimi et al [8] introduced some classes of fuzzy contractive mappings and gave fixed point results which generalize and extend some comparable results in the existing literature. In 2013, Wardowski [9] generalized the concept of fuzzy contractive mapping by introducing the concept of fuzzy H-contractive mapping and proved a fixed point result in M-complete fuzzy metric space. Thereafter, Shukla [10] defined fuzzy H-weak contractive mapping and utilized the same to extend the fixed point results due to Wardowski [9]. Beg et al [11] defined the notion of α-fuzzy-H-contractive mapping and established some existence and uniqueness of fixed point results in fuzzy M-complete metric spaces. For more results in this direction, we refer the reader to [12,13,14,15,16,17,18,19,20,21,22,23,24]
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