Weakest convergence conditions are equivalent for Kannan mappings but not for Banach contractions, with equivalence restored on G-complete metric spaces.
Grabiec,Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems,27, 385–389 (1988)
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A comparison of the weakest contractive conditions for Banach and Kannan mappings
Weakest convergence conditions are equivalent for Kannan mappings but not for Banach contractions, with equivalence restored on G-complete metric spaces.