Fixed Point Theorems in Metric Spaces Induced by Coprime Relations

In this recent study, we investigated a novel fixed point in a special metric space where the distance between two natural numbers is determined by their coprime relations. We define a metric related to the Greatest Common Divisor (GCD) and explore fixed-point results for standard coprime controlled contraction mappings and their generalizations as well as fixed point results using additional mathematical structures. With the support of artificial intelligence and we also introduced Banach, Cric, and Kannan applications with examples to illustrate this theory.

Fixed point, metric space, contraction mapping

1. Banach, S. (1922). Sur les opérations dans les ensembles abstraits. Fundamenta Mathematicae, V.3, PP.133–181.Kirk, W. A. (1965). [Google Scholar] [Crossref]

2. A fixed point theorem for mappings which do not increase distances. American Mathematical Monthly,V.72, PP.1004–1006. [Google Scholar] [Crossref]

3. Jungck, G.(1976). Commuting mappings and fixed points. American Mathematical Monthly,V.83,PP.261–263. [Google Scholar] [Crossref]

4. Agarwal, R. P. O’Regan, D., and Sahu, D. R. (2009). Fixed point theory for Lipschitzian-type mappings. Springer. pp.1–320. [Google Scholar] [Crossref]

5. Hardy, G. H. and Wright, E. M. (2008). An introduction to the theory of numbers (6th ed.). Oxford University Press. pp. 19–22, pp.PP. 83–90 (coprime properties and Euclid’s algorithm). [Google Scholar] [Crossref]

6. Khamsi, M. A. and Kirk, W. A. (2001). An Introduction to metric spaces and fixed point theory. John Wiley and Sons. pp. 52–60, pp. PP.87–101. [Google Scholar] [Crossref]

7. Gopal, D. and Rhoades, B. E. (2011). Fixed point theorems via altering distance functions. Journal of Mathematical Analysis and Applications, V.375(1),PP. 264–272. [Google Scholar] [Crossref]

8. Smith, M.J.(2014). Coprime structures in metric dynamics. Journal of number theory, V.143, PP 92–106. [Google Scholar] [Crossref]

9. Altun, I. and Sadarangani, K.(2012). Generalized contractions and fixed point theorems. Nonlinear Analysis, V.75(4), PP.2154–2165. [Google Scholar] [Crossref]

10. Berinde, V. (2007). Iterative approximation of fixed points. Springer. pp. 101–115, pp. 148–160. [Google Scholar] [Crossref]

11. Caristi, J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society,V.215, PP.241–251. [Google Scholar] [Crossref]

12. Bruck, R. E. (1973). Nonexpansive projections on subsets of Banach spaces. Pacific Journal of Mathematics,V. 47(2), PP.341–355. [Google Scholar] [Crossref]

13. Kosmol, P.(1987). Optimization and fixed point theory.Springer.PP.65–80. [Google Scholar] [Crossref]

14. Dutta, P. N., and Choudhury, B. S. (2008). A generalization of contraction principle in metric spaces. Fixed point theory and applications,2008, Article ID 406368, PP. 1–8. [Google Scholar] [Crossref]

15. Gregus, M. (1980). A fixed point theorem in complete metric spaces. Mathematica Slovaca,V. 30(2),PP. 107–111. [Google Scholar] [Crossref]

16. Abkar, A. and Eslamian, E. (2010). Common fixed points for weakly compatible maps in fuzzy metric spaces. Fuzzy Sets and Systems, V.161(4), pp.412–420. [Google Scholar] [Crossref]

17. Kirk, W. A. (2003). Fixed point theory in metric spaces. Lecture Notes, University of Iowa. pp. 10–29. [Google Scholar] [Crossref]

18. Singh, S. P. and Mishra, M. K. (1990). Fixed point theory and applications. Macmillan India. pp. 98–115, pp. 135–142. [Google Scholar] [Crossref]

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