Weakest convergence conditions are equivalent for Kannan mappings but not for Banach contractions, with equivalence restored on G-complete metric spaces.
Jachymski,Equivalent conditions and the Meir-Keeler type theorems, Journal of Mathematical Analysis and Applications,194(1), 293-303 (1995)
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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.