Abstract
In this paper, we introduce a new fuzzy contraction via a new concept of the fuzzy sets called fw-distances initiated in the paper, which is a generalization of a fuzzy contractive mapping initiated in the article (Fuzzy Sets Syst. 159:739-744, 2008). A fixed point theorem is established by using this type of contraction of set-valued mappings in fuzzy metric spaces which are complete in the sense of George and Veeramani. As an application of our results, we give characterizations of fuzzy metric completeness. The results are supported by examples.
Highlights
1 Introduction In Fuzzy metric spaces we refer to as KM-spaces were initiated by Kramosil and Michálek [ ]
In, Gregori and Sapena [ ] have introduced a kind of contractive mappings and proved fuzzy fixed point theorems in GV-spaces and KM-spaces by using a strong condition for completeness, called the completeness in the sense of Grabiec or G-completeness, which can be considered a fuzzy version of the Banach contraction theorem
Let (X, M, ∗)’ be a fuzzy metric space, and let P be an fw-distance on X
Summary
In Fuzzy metric spaces we refer to as KM-spaces were initiated by Kramosil and Michálek [ ]. In , Gregori and Sapena [ ] have introduced a kind of contractive mappings and proved fuzzy fixed point theorems in GV-spaces and KM-spaces by using a strong condition for completeness, called the completeness in the sense of Grabiec or G-completeness, which can be considered a fuzzy version of the Banach contraction theorem. These results have become recently of interest for many authors. Motivated by the works mentioned above, in this paper, we will establish fixed point theorems for weakly fuzzy contractive set-valued mappings on M-complete GV-spaces To this end, we first introduce a new concept called fw-distance here. M is a fuzzy metric, and (X, M, ∗) is a GV-space (see [ ]), P is an fw-distance but not a fuzzy metric on X
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