| Summary: | In this paper, we first introduce a new family of functions like an implicit function called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula>-functions. Furthermore, we introduce a new concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mi>F</mi></mrow></semantics></math></inline-formula>-fuzzy contractive mappings, which is weaker than the class of fuzzy <i>F</i>-contractive mappings. Then, the existence and uniqueness of the fixed point are established for a new type of fuzzy contractive mappings in the setting of fuzzy metric spaces. Moreover, some examples and an application to nonlinear fractional differential equation are given, and these show the importance of the introduced theorems in fuzzy settings.
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