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Abstract
There are some of the classical contractive mappings introduced by Banach, Kannan, Chatterjea, Reich, Hardy--Rogers, and \'{C}iri\'{c}. They remain fundamental in fixed point theory. While their theoretical properties are well documented, a detailed and systematic comparison of the convergence performance of Picard iterations among these classes is still lacking in the literature. This study presents a comparative analysis from a numerical perspective on a complete metric space. We check the sharp constants of their contractions, inclusion relationships, establish a sufficient condition for a Chatterjea contraction to be a Reich contraction, and evaluate the practical performance of Picard iteration through simple numerical experiments. We give two concrete examples, a linear map and a quadratic polynomial for this numerical experiment. They are provided to show that the intersection of all six classical classes is not limited to linear mappings. A hierarchical diagram and structural comparison table are also given to support our study. The integration of theoretical results and numerical validation offers a clearer and more practical reference for students and researchers in studying the concepts of fixed point theory.
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References
- S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. Available: http://kielich.amu.edu.pl/Stefan Banach/pdf/oeuvres2/305.pdf.
- R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968. https://www.scirp.org/reference/referencespapers?referenceid=1119410.
- S. Chatterjea, “Fixed-point theorems,” C. R. Acad. Bulgare Sci., vol. 25, pp. 727–730, 1972. Available: https://zbmath.org/?q=an%3A0274.54033.
- S. Reich, “Some remarks concerning contraction mappings,” Canadian Mathematical Bulletin, vol. 14, pp. 121–124, 1971. Available: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/some-remarks-concerning-contraction-mappings/62ED0CC002E8224C486ABE631A46D721.
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- K. Janngam and R. Wattanataweekul, “A new accelerated fixed-point algorithm for classification and convex minimization problems in hilbert spaces with directed graphs,” Symmetry, vol. 14, p. 1059, May 2022. Available: https://doi.org/10.3390/sym14051059.
- R. Hosseini and S. Sra, “Riemannian stochastic fixed point optimization algorithm,” Numerical Algorithms, vol. 90, pp. 1493–1517, Feb. 2022. Available: https://doi.org/10.1007/s11075-021-01238-y.
- I. Beg, T. Ermis, Ozcan Gelisgen, and M. Abbas, “Applications of asymptotic fixed point theorems in a-metric spaces to integral equations,” Journal of Function Spaces, vol. 2025, p. Article 1727696, 2025. Available: https://onlinelibrary.wiley.com/doi/full/10.1155/jofs/1727696.
- N. Trefethen, “Picard iteration for ode existence proof.” Available: http://www.chebfun.org/examples/ode-nonlin/Picard.html.
- M. Ahmed, “A characterization of the convergence of picard iteration to a fixed point for a continuous mapping and an application,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1298–1304, 2005. Available: https://www.sciencedirect.com/science/article/abs/pii/S0096300304008525.
- E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, 1986. Available: https://link.springer.com/book/10.1007/978-1-4612-4838-5.
- A. Granas and J. Dugundji, Fixed Point Theory. Springer, 2003. Available: https://link.springer.com/book/10.1007/978-0-387-21593-8.
- B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977. Available: https://doi.org/10.1090/S0002-9947-1977-0440503-7.
References
S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. Available: http://kielich.amu.edu.pl/Stefan Banach/pdf/oeuvres2/305.pdf.
R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968. https://www.scirp.org/reference/referencespapers?referenceid=1119410.
S. Chatterjea, “Fixed-point theorems,” C. R. Acad. Bulgare Sci., vol. 25, pp. 727–730, 1972. Available: https://zbmath.org/?q=an%3A0274.54033.
S. Reich, “Some remarks concerning contraction mappings,” Canadian Mathematical Bulletin, vol. 14, pp. 121–124, 1971. Available: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/some-remarks-concerning-contraction-mappings/62ED0CC002E8224C486ABE631A46D721.
G. Hardy and T. Rogers, “A generalization of a fixed point theorem of reich,” Canadian Mathematical Bulletin, vol. 16, pp. 201–206, 1973. Available: https://doi.org/10.4153/CMB-1973-036-0.
L. Ciric, “A generalization of banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974. Available: https://www.jstor.org/stable/2040075.
Z. Bekri, N. Fabiano, M. A. Alomair, and A. K. Alsharidi, “On p-hardy–rogers and p-zamfirescu contractions in complete metric spaces: Existence and uniqueness results,” Mathematics, vol. 13, p. 4011, Dec. 2025. Available: https://doi.org/10.3390/math13244011.
O. T. Wahab, G. R. Ibrahim, and S. A. Musa, “Condensed kannan-type maps and their efficiency measures in complete metric spaces,” Kyungpook Mathematical Journal, vol. 65, pp. 439–451, Sept. 2025. Available: https://doi.org/10.5666/KMJ.2025.65.3.439.
S. Hashimoto, M. Kikkawa, S. Machihara, and A. Saghir, “On the weakest conditions for the existence of fixed points of Kannan and Chatterjea type contractions,” arXiv preprint arXiv:2505.01672, 2025. Version 4, revised March 2026. Available: https://arxiv.org/abs/2505.01672.
B. Rani, J. Kaur, and S. S. Bhatia, “Approximating fixed points via hybrid enriched contractions in convex metric space with an application,” Axioms, vol. 13, p. 815, Dec. 2024. Available: https://doi.org/10.3390/axioms13120815
K. Kankam, P. Cholamjiak, P. Srinet, and N. Pholasa, “On some accelerated optimization algorithms based on fixed point and linesearch techniques for convex minimization problems with applications,” Advances in Continuous and Discrete Models, vol. 2022, p. Article 25, Mar. 2022. Available: https://doi.org/10.1186/s13662-022-03698-5.
K. Janngam and R. Wattanataweekul, “A new accelerated fixed-point algorithm for classification and convex minimization problems in hilbert spaces with directed graphs,” Symmetry, vol. 14, p. 1059, May 2022. Available: https://doi.org/10.3390/sym14051059.
R. Hosseini and S. Sra, “Riemannian stochastic fixed point optimization algorithm,” Numerical Algorithms, vol. 90, pp. 1493–1517, Feb. 2022. Available: https://doi.org/10.1007/s11075-021-01238-y.
I. Beg, T. Ermis, Ozcan Gelisgen, and M. Abbas, “Applications of asymptotic fixed point theorems in a-metric spaces to integral equations,” Journal of Function Spaces, vol. 2025, p. Article 1727696, 2025. Available: https://onlinelibrary.wiley.com/doi/full/10.1155/jofs/1727696.
N. Trefethen, “Picard iteration for ode existence proof.” Available: http://www.chebfun.org/examples/ode-nonlin/Picard.html.
M. Ahmed, “A characterization of the convergence of picard iteration to a fixed point for a continuous mapping and an application,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1298–1304, 2005. Available: https://www.sciencedirect.com/science/article/abs/pii/S0096300304008525.
E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, 1986. Available: https://link.springer.com/book/10.1007/978-1-4612-4838-5.
A. Granas and J. Dugundji, Fixed Point Theory. Springer, 2003. Available: https://link.springer.com/book/10.1007/978-0-387-21593-8.
B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977. Available: https://doi.org/10.1090/S0002-9947-1977-0440503-7.