A comparison of the weakest contractive conditions for Banach and Kannan mappings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914557919756288 |
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| author | Hashimoto, Shunya Kikkawa, Misako Machihara, Shuji Saghir, Aqib |
| author_facet | Hashimoto, Shunya Kikkawa, Misako Machihara, Shuji Saghir, Aqib |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A comparison of the weakest contractive conditions for Banach and Kannan mappings Hashimoto, Shunya Kikkawa, Misako Machihara, Shuji Saghir, Aqib Functional Analysis 54H25, 47H10 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. |
| title | A comparison of the weakest contractive conditions for Banach and Kannan mappings |
| topic | Functional Analysis 54H25, 47H10 |
| url | https://arxiv.org/abs/2605.11883 |