10.57647/cna.2025.7hhv-3c31.11

Fixed Point Theorems for Generalized F-Contractions of Rational Type via Iterative Set Methods in Complete Metric Spaces

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  • V. c. Borkar1,
  • Saeed A. A. Al-Salehi*2,
  1. Department of Mathematics, Yeshwant Mahavidyalaya, Swami Ramanand Teerth Marathwada University, India
  2. Department of Mathematics, Science College, Swami Ramanand Teerth Marathwada University, India

Received: 2025-03-24

Revised: 2025-04-15

Accepted: 2025-05-02

Published in Issue 2025-06-30

Copyright (c) 2025 V. c. Borkar, Saeed A. A. Al-Salehi (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Fixed Point Theorems for Generalized F-Contractions of Rational Type via Iterative Set Methods in Complete Metric Spaces. (2025). Communications in Nonlinear Analysis, 13(1). https://doi.org/10.57647/cna.2025.7hhv-3c31.11

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Abstract

In this paper, we investigate a new class of F -contraction mappings defined on complete metric spaces by applying iterative set techniques. Our study focuses on introducing rational-type contractive conditions that extend and generalize several existing approaches in the literature. To ensure clarity and accessibility, we provide precise definitions, detailed step-by-step arguments, and illustrative examples that highlight the main concepts and results. The presentation is designed to make the proofs easier to follow, even for readers who are new to this area, while also offering deeper mathematical insights. Furthermore, the findings reveal potential directions for extending fixed point theory, particularly by formulating new contraction principles and exploring broader applications in mathematical analysis. This work not only contributes to the theoretical development of fixed point results but also emphasizes their relevance for future research in related fields.

Keywords

  • Fixed point theorems,
  • Iterative set methods,
  • Generalized F -contraction of rational type,
  • Metric space,
  • Complete metric space,
  • MSC 2020: 47H10, 54H25
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References

  1. S. Banach. Sur les ope´rations dans les ensembles abstraits et leur application aux e´quations inte´grales. Fundamenta Mathematicae, 3: 133–181, 1922. DOI: https://doi.org/10.4064/fm-3-1-133-181.
  2. M. Geraghty. On contractive mappings. Proceedings of the American Mathematical Society, 40:604–608, 1973. DOI: https://doi.org/10.1090/S0002-9939-1973-0334176-5.
  3. L. B. C´ iric´. A generalization of banach’s contraction principle. Proceedings of the American Mathematical Society, 45:267–273, 1974. DOI: https://doi.org/10.1090/S0002-9939-1974-0356011-2.
  4. W. A. Kirk. Some recent results in metric fixed point theory. Journal of Fixed Point Theory and Applications, 2:195–207, 2007.
  5. C. Semple and M. Steel. Phylogenetics, volume 24 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2003.
  6. M. Abbas and B. E. Rhoades. Fixed and periodic point results in cone metric spaces. Applied Mathematics Letters, 22(4):511–515, 2009. DOI: https://doi.org/10.1016/j.aml.2008.07.001.
  7. B. C. Dhage. Generalized metric spaces with fixed point. Bulletin of the Calcutta Mathematical Society, 84:329–336, 1992.
  8. L. G. Huang and X. Zhang. Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2):1468–1476, 2007. DOI: https://doi.org/10.1016/j.jmaa.2005.03.087.
  9. S. Radenovic´ and B. E. Rhoades. Fixed point theorem for two non-self mappings in cone metric spaces. Computers & Mathematics with Applications, 57(10):1701–1707, 2009. DOI: https://doi.org/10.1016/j.camwa.2009.03.058.
  10. S. Rezapour and R. Hamlbarani. Some notes on the paper ‘cone metric spaces and fixed point theorems of contractive mappings’. Journal of Mathematical Analysis and Applications, 345(2): 719–724, 2008. DOI: https://doi.org/10.1016/j.jmaa.2008.04.049.
  11. A. Branciari. A fixed point theorem of banach–caccioppoli type on a class of generalized metric spaces. Publicationes Mathematicae Debrecen, 57(1-2):31–37, 2000. DOI: https://doi.org/10.5486/PMD.2000.2133.
  12. D. Wardowski. Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, 2012:94, 2012. DOI: https://doi.org/10.1186/1687-1812-2012-94.
  13. N. A. Secelean. Iterated function systems consisting of f -contractions. Fixed Point Theory and Applications, 2013:277, 2013. DOI: https://doi.org/10.1186/1687-1812-2013-277.
  14. H. Piri and P. Kumam. Some fixed point theorems concerning f -contraction in complete metric spaces. Fixed Point Theory and Applications, 2014:210, 2014. DOI: https://doi.org/10.1186/1687-1812-2014-210.
  15. M. Arshad, S. U. Khan, and J. Ahmad. Fixed point results for f -contractions involving some new rational expressions. JP Journal of Fixed Point Theory and Applications, 11(1):79–97, 2016. DOI: https://doi.org/10.17654/FP011010079.