Weakest convergence conditions are equivalent for Kannan mappings but not for Banach contractions, with equivalence restored on G-complete metric spaces.
Matkowski,Fixed point theorems for contractive mappings in metric spaces, ˇCasopis pro pˇ estov´ an´ ı matematiky, 105(4), 341-344 (1980)
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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.