A comparison of the weakest contractive conditions for Banach and Kannan mappings

TL;DR

This paper compares the weakest convergence conditions for Banach and Kannan fixed point mappings, revealing their equivalence for Kannan mappings and the role of completeness in these conditions.

Contribution

It establishes the equivalence of weakest convergence conditions for Kannan mappings without assuming completeness, contrasting with Banach mappings where this fails.

Findings

Weakest convergence conditions are equivalent for Kannan mappings on complete metric spaces.

Counterexample shows the equivalence fails for Banach contractions.

Discrepancy between Banach and Kannan mappings disappears on G-complete metric spaces.

Abstract

We study the weakest convergence-type conditions for fixed point results for Banach and Kannan mappings. Building on Suzuki's weakest condition for Banach mappings and our previous result for Kannan mappings, we compare convergence conditions defined along Picard sequences. We give a direct proof that several weakest convergence conditions are equivalent for Kannan-type mappings on complete metric spaces. This proof is achieved without assuming the completeness or the convergence of Picard sequences; it deduces the equivalence only from the existence of fixed points. In contrast, we construct a counterexample showing that the corresponding equivalence fails for Banach contractions. Finally, we prove that this discrepancy disappears on G-complete metric spaces, clarifying the role of completeness in weakest fixed point theory.