Revisiting Suzuki contractions under weak completeness structures

Abstract

This paper investigates Suzuki-type contractive mappings from the viewpoint of convergence structures and completeness requirements. While classical Suzuki fixed point theorems are formulated in complete metric spaces, we show that metric completeness is not intrinsic to the Suzuki contractive mechanism. We introduce a weak completeness framework that emphasizes operator-dependent convergence and prove that orbital completeness suffices for fixed point existence and uniqueness. The results establish fixed point principles that strictly refine the classical Suzuki theorem while preserving the Cauchy and convergence behavior of Picard iteration sequences. In addition, we analyze the structural dependence of Suzuki contractions on completeness properties and derive stability consequences, including well-posedness and data dependence estimates. These findings demonstrate that the essential convergence mechanisms are governed by iterative orbits rather than by global geometric completeness. Nontrivial examples illustrating spaces where metric completeness fails but orbital convergence remains valid are provided. The study yields a sharper interpretation of Suzuki-type contractions and broadens their analytical scope within modern fixed point theory and nonlinear analysis.

1. Introduction

Fixed point theory occupies a central position in modern nonlinear analysis due to its deep theoretical importance and broad applicability across mathematics and applied sciences. The subject is traditionally rooted in the Banach Contraction Principle [1], which established that a contraction mapping on a complete metric space admits a unique fixed point. This principle has inspired extensive research aimed at weakening contractive hypotheses and refining the structural requirements underlying convergence mechanisms.

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References

1. Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 3 (1922), 133–181.Google Scholar
2. Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 136 (2008), 1861–1869.Google Scholar
3. Berinde, V., Iterative Approximation of Fixed Points, Springer, 2007.Google Scholar
4. Kirk, W. A., Fixed Point Theory for Nonexpansive Mappings, Lecture Notes in Mathematics, Springer, 2003.Google Scholar
5. Chirwa, H., Pamba, J., Tembo, I. D., Generalized Rational Suzuki-Type Contractions in Double Controlled Metric Spaces with Applications, Earthline Journal of Mathematical Sciences, 16(3) (2026), 427–449. DOI: 10.34198/ejms.16326.31.427449 .Google Scholar
6. Pamba, J., Tembo, I. D., On fixed point results of Suzuki-type contractions on controlled metric spaces with applications, International Journal of Science, Technology and Society, 13(5) (2025). DOI: 10.11648/j.ijsts.20251305.14 .Google Scholar
7. Reich, S., Some remarks concerning contraction mappings, Canadian Mathematical Bulletin, 14 (1971), 121–124.Google Scholar
8. Rhoades, B. E., Some theorems on weakly contractive maps, Nonlinear Analysis, 47 (2001), 2683–2693.Google Scholar
9. Romaguera, S., Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results, Mathematics, 10(21) (2022), 3931. Available at: https://doi.org/10.3390/math10213931 .Google Scholar
10. Sarwar, M., Mlaiki, N., Iqbal, F., Fixed point theorem for interpolative contraction of Suzuki-type mappings in CAT(0) spaces, Results in Control and Optimization, 19 (2025), 100566. https://doi.org/10.1016/j.rico.2025.100566 .Google Scholar
11. Sezen, M. S., Türkoğlu, A. D., Özcan, S., Etemad, S., Hussain, A., Alubady, R., On the Fixed Point Results for the Suzuki-Type Multivalued δ-Contractions, Journal of Function Spaces, (2025), Article ID 5321847. https://doi.org/10.1155/jofs/5321847 .Google Scholar

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Cite this article

Pamba, J. (2026). Revisiting Suzuki contractions under weak completeness structures. Asia Mathematika, 10(1), 35–48. https://doi.org/10.5281/zenodo.20996418

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